[{"data":1,"prerenderedAt":4076},["ShallowReactive",2],{"post-\u002Fblog\u002Fkut-kydka-maksymalna-dalnist":3,"post-rel-\u002Fblog\u002Fkut-kydka-maksymalna-dalnist":4075},{"id":4,"title":5,"body":6,"date":4056,"description":4057,"draft":4058,"extension":4059,"grade":3410,"image":4060,"meta":4061,"navigation":4062,"path":4063,"rawbody":4064,"seo":4065,"stem":4066,"subject":4067,"tags":4068,"topic":4073,"updated":4060,"__hash__":4074},"blog\u002Fblog\u002Fkut-kydka-maksymalna-dalnist.md","Рух тіла, кинутого під кутом, 10 клас: чому найдалі — під 45°",{"type":7,"value":8,"toc":4043},"minimark",[9,13,18,194,198,201,657,661,732,1157,1161,1341,1773,1777,2057,2365,2745,3136,3372,3376,3379,3382,3386,3445,4036,4040],[10,11,12],"p",{},"Ця задача — ідеальний місток між фізикою і математикою: фізика дає рівняння руху, а математика знаходить, де функція досягає максимуму. Нижче в статті є інтерактивна мішень — перевірте формулу на практиці.",[14,15,17],"h2",{"id":16},"умова","Умова",[10,19,20,21,129,130,162,163,193],{},"М’яч кидають з поверхні землі зі швидкістю ",[22,23,26,56],"span",{"className":24},[25],"katex",[22,27,30],{"className":28},[29],"katex-mathml",[31,32,34],"math",{"xmlns":33},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[35,36,37,51],"semantics",{},[38,39,40],"mrow",{},[41,42,43,47],"msub",{},[44,45,46],"mi",{},"v",[48,49,50],"mn",{},"0",[52,53,55],"annotation",{"encoding":54},"application\u002Fx-tex","v_0",[22,57,61],{"className":58,"ariaHidden":60},[59],"katex-html","true",[22,62,65,70],{"className":63},[64],"base",[22,66],{"className":67,"style":69},[68],"strut","height:0.5806em;vertical-align:-0.15em;",[22,71,74,79],{"className":72},[73],"mord",[22,75,46],{"className":76,"style":78},[73,77],"mathnormal","margin-right:0.0359em;",[22,80,83],{"className":81},[82],"msupsub",[22,84,88,120],{"className":85},[86,87],"vlist-t","vlist-t2",[22,89,92,115],{"className":90},[91],"vlist-r",[22,93,97],{"className":94,"style":96},[95],"vlist","height:0.3011em;",[22,98,100,105],{"style":99},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[22,101],{"className":102,"style":104},[103],"pstrut","height:2.7em;",[22,106,112],{"className":107},[108,109,110,111],"sizing","reset-size6","size3","mtight",[22,113,50],{"className":114},[73,111],[22,116,119],{"className":117},[118],"vlist-s","​",[22,121,123],{"className":122},[91],[22,124,127],{"className":125,"style":126},[95],"height:0.15em;",[22,128],{}," під кутом ",[22,131,133,148],{"className":132},[25],[22,134,136],{"className":135},[29],[31,137,138],{"xmlns":33},[35,139,140,145],{},[38,141,142],{},[44,143,144],{},"α",[52,146,147],{"encoding":54},"\\alpha",[22,149,151],{"className":150,"ariaHidden":60},[59],[22,152,154,158],{"className":153},[64],[22,155],{"className":156,"style":157},[68],"height:0.4306em;",[22,159,144],{"className":160,"style":161},[73,77],"margin-right:0.0037em;"," до горизонту. Опором повітря нехтуємо. Знайти дальність польоту ",[22,164,166,180],{"className":165},[25],[22,167,169],{"className":168},[29],[31,170,171],{"xmlns":33},[35,172,173,178],{},[38,174,175],{},[44,176,177],{},"L",[52,179,177],{"encoding":54},[22,181,183],{"className":182,"ariaHidden":60},[59],[22,184,186,190],{"className":185},[64],[22,187],{"className":188,"style":189},[68],"height:0.6833em;",[22,191,177],{"className":192},[73,77]," і кут, при якому вона максимальна.",[14,195,197],{"id":196},"розкладаємо-рух-на-дві-осі","Розкладаємо рух на дві осі",[10,199,200],{},"По горизонталі сили не діють — рух рівномірний. По вертикалі діє сила тяжіння — рух рівноприскорений:",[22,202,205],{"className":203},[204],"katex-display",[22,206,208,313],{"className":207},[25],[22,209,211],{"className":210},[29],[31,212,214],{"xmlns":33,"display":213},"block",[35,215,216,310],{},[38,217,218,221,226,229,232,235,241,244,247,249,252,254,257,261,264,266,268,270,272,278,281,283,285,287,289,292],{},[44,219,220],{},"x",[222,223,225],"mo",{"stretchy":224},"false","(",[44,227,228],{},"t",[222,230,231],{"stretchy":224},")",[222,233,234],{},"=",[41,236,237,239],{},[44,238,46],{},[48,240,50],{},[44,242,243],{},"cos",[222,245,246],{},"⁡",[44,248,144],{},[222,250,251],{},"⋅",[44,253,228],{},[222,255,256],{"separator":60},",",[258,259],"mspace",{"width":260},"2em",[44,262,263],{},"y",[222,265,225],{"stretchy":224},[44,267,228],{},[222,269,231],{"stretchy":224},[222,271,234],{},[41,273,274,276],{},[44,275,46],{},[48,277,50],{},[44,279,280],{},"sin",[222,282,246],{},[44,284,144],{},[222,286,251],{},[44,288,228],{},[222,290,291],{},"−",[293,294,295,308],"mfrac",{},[38,296,297,300],{},[44,298,299],{},"g",[301,302,303,305],"msup",{},[44,304,228],{},[48,306,307],{},"2",[48,309,307],{},[52,311,312],{"encoding":54},"x(t) = v_0 \\cos\\alpha \\cdot t, \\qquad y(t) = v_0 \\sin\\alpha \\cdot t - \\frac{g t^2}{2}",[22,314,316,348,420,461,529,548],{"className":315,"ariaHidden":60},[59],[22,317,319,323,326,330,333,337,341,345],{"className":318},[64],[22,320],{"className":321,"style":322},[68],"height:1em;vertical-align:-0.25em;",[22,324,220],{"className":325},[73,77],[22,327,225],{"className":328},[329],"mopen",[22,331,228],{"className":332},[73,77],[22,334,231],{"className":335},[336],"mclose",[22,338],{"className":339,"style":340},[258],"margin-right:0.2778em;",[22,342,234],{"className":343},[344],"mrel",[22,346],{"className":347,"style":340},[258],[22,349,351,355,395,399,403,406,409,413,417],{"className":350},[64],[22,352],{"className":353,"style":354},[68],"height:0.5945em;vertical-align:-0.15em;",[22,356,358,361],{"className":357},[73],[22,359,46],{"className":360,"style":78},[73,77],[22,362,364],{"className":363},[82],[22,365,367,387],{"className":366},[86,87],[22,368,370,384],{"className":369},[91],[22,371,373],{"className":372,"style":96},[95],[22,374,375,378],{"style":99},[22,376],{"className":377,"style":104},[103],[22,379,381],{"className":380},[108,109,110,111],[22,382,50],{"className":383},[73,111],[22,385,119],{"className":386},[118],[22,388,390],{"className":389},[91],[22,391,393],{"className":392,"style":126},[95],[22,394],{},[22,396],{"className":397,"style":398},[258],"margin-right:0.1667em;",[22,400,243],{"className":401},[402],"mop",[22,404],{"className":405,"style":398},[258],[22,407,144],{"className":408,"style":161},[73,77],[22,410],{"className":411,"style":412},[258],"margin-right:0.2222em;",[22,414,251],{"className":415},[416],"mbin",[22,418],{"className":419,"style":412},[258],[22,421,423,426,429,433,437,440,443,446,449,452,455,458],{"className":422},[64],[22,424],{"className":425,"style":322},[68],[22,427,228],{"className":428},[73,77],[22,430,256],{"className":431},[432],"mpunct",[22,434],{"className":435,"style":436},[258],"margin-right:2em;",[22,438],{"className":439,"style":398},[258],[22,441,263],{"className":442,"style":78},[73,77],[22,444,225],{"className":445},[329],[22,447,228],{"className":448},[73,77],[22,450,231],{"className":451},[336],[22,453],{"className":454,"style":340},[258],[22,456,234],{"className":457},[344],[22,459],{"className":460,"style":340},[258],[22,462,464,468,508,511,514,517,520,523,526],{"className":463},[64],[22,465],{"className":466,"style":467},[68],"height:0.8179em;vertical-align:-0.15em;",[22,469,471,474],{"className":470},[73],[22,472,46],{"className":473,"style":78},[73,77],[22,475,477],{"className":476},[82],[22,478,480,500],{"className":479},[86,87],[22,481,483,497],{"className":482},[91],[22,484,486],{"className":485,"style":96},[95],[22,487,488,491],{"style":99},[22,489],{"className":490,"style":104},[103],[22,492,494],{"className":493},[108,109,110,111],[22,495,50],{"className":496},[73,111],[22,498,119],{"className":499},[118],[22,501,503],{"className":502},[91],[22,504,506],{"className":505,"style":126},[95],[22,507],{},[22,509],{"className":510,"style":398},[258],[22,512,280],{"className":513},[402],[22,515],{"className":516,"style":398},[258],[22,518,144],{"className":519,"style":161},[73,77],[22,521],{"className":522,"style":412},[258],[22,524,251],{"className":525},[416],[22,527],{"className":528,"style":412},[258],[22,530,532,536,539,542,545],{"className":531},[64],[22,533],{"className":534,"style":535},[68],"height:0.6984em;vertical-align:-0.0833em;",[22,537,228],{"className":538},[73,77],[22,540],{"className":541,"style":412},[258],[22,543,291],{"className":544},[416],[22,546],{"className":547,"style":412},[258],[22,549,551,555],{"className":550},[64],[22,552],{"className":553,"style":554},[68],"height:2.1771em;vertical-align:-0.686em;",[22,556,558,562,654],{"className":557},[73],[22,559],{"className":560},[329,561],"nulldelimiter",[22,563,565],{"className":564},[293],[22,566,568,645],{"className":567},[86,87],[22,569,571,642],{"className":570},[91],[22,572,575,588,599],{"className":573,"style":574},[95],"height:1.4911em;",[22,576,578,582],{"style":577},"top:-2.314em;",[22,579],{"className":580,"style":581},[103],"height:3em;",[22,583,585],{"className":584},[73],[22,586,307],{"className":587},[73],[22,589,591,594],{"style":590},"top:-3.23em;",[22,592],{"className":593,"style":581},[103],[22,595],{"className":596,"style":598},[597],"frac-line","border-bottom-width:0.04em;",[22,600,602,605],{"style":601},"top:-3.677em;",[22,603],{"className":604,"style":581},[103],[22,606,608,611],{"className":607},[73],[22,609,299],{"className":610,"style":78},[73,77],[22,612,614,617],{"className":613},[73],[22,615,228],{"className":616},[73,77],[22,618,620],{"className":619},[82],[22,621,623],{"className":622},[86],[22,624,626],{"className":625},[91],[22,627,630],{"className":628,"style":629},[95],"height:0.8141em;",[22,631,633,636],{"style":632},"top:-3.063em;margin-right:0.05em;",[22,634],{"className":635,"style":104},[103],[22,637,639],{"className":638},[108,109,110,111],[22,640,307],{"className":641},[73,111],[22,643,119],{"className":644},[118],[22,646,648],{"className":647},[91],[22,649,652],{"className":650,"style":651},[95],"height:0.686em;",[22,653],{},[22,655],{"className":656},[336,561],[14,658,660],{"id":659},"час-польоту","Час польоту",[10,662,663,664,731],{},"М’яч падає на землю, коли ",[22,665,667,691],{"className":666},[25],[22,668,670],{"className":669},[29],[31,671,672],{"xmlns":33},[35,673,674,688],{},[38,675,676,678,680,682,684,686],{},[44,677,263],{},[222,679,225],{"stretchy":224},[44,681,228],{},[222,683,231],{"stretchy":224},[222,685,234],{},[48,687,50],{},[52,689,690],{"encoding":54},"y(t) = 0",[22,692,694,721],{"className":693,"ariaHidden":60},[59],[22,695,697,700,703,706,709,712,715,718],{"className":696},[64],[22,698],{"className":699,"style":322},[68],[22,701,263],{"className":702,"style":78},[73,77],[22,704,225],{"className":705},[329],[22,707,228],{"className":708},[73,77],[22,710,231],{"className":711},[336],[22,713],{"className":714,"style":340},[258],[22,716,234],{"className":717},[344],[22,719],{"className":720,"style":340},[258],[22,722,724,728],{"className":723},[64],[22,725],{"className":726,"style":727},[68],"height:0.6444em;",[22,729,50],{"className":730},[73],":",[22,733,735],{"className":734},[204],[22,736,738,819],{"className":737},[25],[22,739,741],{"className":740},[29],[31,742,743],{"xmlns":33,"display":213},[35,744,745,816],{},[38,746,747,749,779,781,783,786,789,791,794,796],{},[44,748,228],{},[38,750,751,753,759,761,763,765,767,777],{},[222,752,225],{"fence":60},[41,754,755,757],{},[44,756,46],{},[48,758,50],{},[44,760,280],{},[222,762,246],{},[44,764,144],{},[222,766,291],{},[293,768,769,775],{},[38,770,771,773],{},[44,772,299],{},[44,774,228],{},[48,776,307],{},[222,778,231],{"fence":60},[222,780,234],{},[48,782,50],{},[258,784],{"width":785},"1em",[222,787,788],{},"⇒",[258,790],{"width":785},[44,792,793],{},"T",[222,795,234],{},[293,797,798,814],{},[38,799,800,802,808,810,812],{},[48,801,307],{},[41,803,804,806],{},[44,805,46],{},[48,807,50],{},[44,809,280],{},[222,811,246],{},[44,813,144],{},[44,815,299],{},[52,817,818],{"encoding":54},"t\\left(v_0 \\sin\\alpha - \\frac{g t}{2}\\right) = 0 \\quad\\Rightarrow\\quad T = \\frac{2 v_0 \\sin\\alpha}{g}",[22,820,822,990,1015,1034],{"className":821,"ariaHidden":60},[59],[22,823,825,829,832,835,981,984,987],{"className":824},[64],[22,826],{"className":827,"style":828},[68],"height:2.4em;vertical-align:-0.95em;",[22,830,228],{"className":831},[73,77],[22,833],{"className":834,"style":398},[258],[22,836,839,848,888,891,894,897,900,903,906,909,975],{"className":837},[838],"minner",[22,840,844],{"className":841,"style":843},[329,842],"delimcenter","top:0em;",[22,845,225],{"className":846},[847,110],"delimsizing",[22,849,851,854],{"className":850},[73],[22,852,46],{"className":853,"style":78},[73,77],[22,855,857],{"className":856},[82],[22,858,860,880],{"className":859},[86,87],[22,861,863,877],{"className":862},[91],[22,864,866],{"className":865,"style":96},[95],[22,867,868,871],{"style":99},[22,869],{"className":870,"style":104},[103],[22,872,874],{"className":873},[108,109,110,111],[22,875,50],{"className":876},[73,111],[22,878,119],{"className":879},[118],[22,881,883],{"className":882},[91],[22,884,886],{"className":885,"style":126},[95],[22,887],{},[22,889],{"className":890,"style":398},[258],[22,892,280],{"className":893},[402],[22,895],{"className":896,"style":398},[258],[22,898,144],{"className":899,"style":161},[73,77],[22,901],{"className":902,"style":412},[258],[22,904,291],{"className":905},[416],[22,907],{"className":908,"style":412},[258],[22,910,912,915,972],{"className":911},[73],[22,913],{"className":914},[329,561],[22,916,918],{"className":917},[293],[22,919,921,964],{"className":920},[86,87],[22,922,924,961],{"className":923},[91],[22,925,928,939,947],{"className":926,"style":927},[95],"height:1.2921em;",[22,929,930,933],{"style":577},[22,931],{"className":932,"style":581},[103],[22,934,936],{"className":935},[73],[22,937,307],{"className":938},[73],[22,940,941,944],{"style":590},[22,942],{"className":943,"style":581},[103],[22,945],{"className":946,"style":598},[597],[22,948,949,952],{"style":601},[22,950],{"className":951,"style":581},[103],[22,953,955,958],{"className":954},[73],[22,956,299],{"className":957,"style":78},[73,77],[22,959,228],{"className":960},[73,77],[22,962,119],{"className":963},[118],[22,965,967],{"className":966},[91],[22,968,970],{"className":969,"style":651},[95],[22,971],{},[22,973],{"className":974},[336,561],[22,976,978],{"className":977,"style":843},[336,842],[22,979,231],{"className":980},[847,110],[22,982],{"className":983,"style":340},[258],[22,985,234],{"className":986},[344],[22,988],{"className":989,"style":340},[258],[22,991,993,996,999,1003,1006,1009,1012],{"className":992},[64],[22,994],{"className":995,"style":727},[68],[22,997,50],{"className":998},[73],[22,1000],{"className":1001,"style":1002},[258],"margin-right:1em;",[22,1004],{"className":1005,"style":340},[258],[22,1007,788],{"className":1008},[344],[22,1010],{"className":1011,"style":1002},[258],[22,1013],{"className":1014,"style":340},[258],[22,1016,1018,1021,1025,1028,1031],{"className":1017},[64],[22,1019],{"className":1020,"style":189},[68],[22,1022,793],{"className":1023,"style":1024},[73,77],"margin-right:0.1389em;",[22,1026],{"className":1027,"style":340},[258],[22,1029,234],{"className":1030},[344],[22,1032],{"className":1033,"style":340},[258],[22,1035,1037,1041],{"className":1036},[64],[22,1038],{"className":1039,"style":1040},[68],"height:2.2253em;vertical-align:-0.8804em;",[22,1042,1044,1047,1154],{"className":1043},[73],[22,1045],{"className":1046},[329,561],[22,1048,1050],{"className":1049},[293],[22,1051,1053,1145],{"className":1052},[86,87],[22,1054,1056,1142],{"className":1055},[91],[22,1057,1060,1071,1079],{"className":1058,"style":1059},[95],"height:1.3449em;",[22,1061,1062,1065],{"style":577},[22,1063],{"className":1064,"style":581},[103],[22,1066,1068],{"className":1067},[73],[22,1069,299],{"className":1070,"style":78},[73,77],[22,1072,1073,1076],{"style":590},[22,1074],{"className":1075,"style":581},[103],[22,1077],{"className":1078,"style":598},[597],[22,1080,1081,1084],{"style":601},[22,1082],{"className":1083,"style":581},[103],[22,1085,1087,1090,1130,1133,1136,1139],{"className":1086},[73],[22,1088,307],{"className":1089},[73],[22,1091,1093,1096],{"className":1092},[73],[22,1094,46],{"className":1095,"style":78},[73,77],[22,1097,1099],{"className":1098},[82],[22,1100,1102,1122],{"className":1101},[86,87],[22,1103,1105,1119],{"className":1104},[91],[22,1106,1108],{"className":1107,"style":96},[95],[22,1109,1110,1113],{"style":99},[22,1111],{"className":1112,"style":104},[103],[22,1114,1116],{"className":1115},[108,109,110,111],[22,1117,50],{"className":1118},[73,111],[22,1120,119],{"className":1121},[118],[22,1123,1125],{"className":1124},[91],[22,1126,1128],{"className":1127,"style":126},[95],[22,1129],{},[22,1131],{"className":1132,"style":398},[258],[22,1134,280],{"className":1135},[402],[22,1137],{"className":1138,"style":398},[258],[22,1140,144],{"className":1141,"style":161},[73,77],[22,1143,119],{"className":1144},[118],[22,1146,1148],{"className":1147},[91],[22,1149,1152],{"className":1150,"style":1151},[95],"height:0.8804em;",[22,1153],{},[22,1155],{"className":1156},[336,561],[14,1158,1160],{"id":1159},"дальність-польоту","Дальність польоту",[10,1162,1163,1164,1192,1193,1237,1238,731],{},"Підставляємо ",[22,1165,1167,1180],{"className":1166},[25],[22,1168,1170],{"className":1169},[29],[31,1171,1172],{"xmlns":33},[35,1173,1174,1178],{},[38,1175,1176],{},[44,1177,793],{},[52,1179,793],{"encoding":54},[22,1181,1183],{"className":1182,"ariaHidden":60},[59],[22,1184,1186,1189],{"className":1185},[64],[22,1187],{"className":1188,"style":189},[68],[22,1190,793],{"className":1191,"style":1024},[73,77]," у ",[22,1194,1196,1216],{"className":1195},[25],[22,1197,1199],{"className":1198},[29],[31,1200,1201],{"xmlns":33},[35,1202,1203,1213],{},[38,1204,1205,1207,1209,1211],{},[44,1206,220],{},[222,1208,225],{"stretchy":224},[44,1210,228],{},[222,1212,231],{"stretchy":224},[52,1214,1215],{"encoding":54},"x(t)",[22,1217,1219],{"className":1218,"ariaHidden":60},[59],[22,1220,1222,1225,1228,1231,1234],{"className":1221},[64],[22,1223],{"className":1224,"style":322},[68],[22,1226,220],{"className":1227},[73,77],[22,1229,225],{"className":1230},[329],[22,1232,228],{"className":1233},[73,77],[22,1235,231],{"className":1236},[336]," і згадуємо формулу синуса подвійного кута ",[22,1239,1241,1277],{"className":1240},[25],[22,1242,1244],{"className":1243},[29],[31,1245,1246],{"xmlns":33},[35,1247,1248,1274],{},[38,1249,1250,1252,1254,1256,1258,1260,1262,1264,1266,1268,1270,1272],{},[48,1251,307],{},[44,1253,280],{},[222,1255,246],{},[44,1257,144],{},[44,1259,243],{},[222,1261,246],{},[44,1263,144],{},[222,1265,234],{},[44,1267,280],{},[222,1269,246],{},[48,1271,307],{},[44,1273,144],{},[52,1275,1276],{"encoding":54},"2\\sin\\alpha\\cos\\alpha = \\sin 2\\alpha",[22,1278,1280,1323],{"className":1279,"ariaHidden":60},[59],[22,1281,1283,1287,1290,1293,1296,1299,1302,1305,1308,1311,1314,1317,1320],{"className":1282},[64],[22,1284],{"className":1285,"style":1286},[68],"height:0.6679em;",[22,1288,307],{"className":1289},[73],[22,1291],{"className":1292,"style":398},[258],[22,1294,280],{"className":1295},[402],[22,1297],{"className":1298,"style":398},[258],[22,1300,144],{"className":1301,"style":161},[73,77],[22,1303],{"className":1304,"style":398},[258],[22,1306,243],{"className":1307},[402],[22,1309],{"className":1310,"style":398},[258],[22,1312,144],{"className":1313,"style":161},[73,77],[22,1315],{"className":1316,"style":340},[258],[22,1318,234],{"className":1319},[344],[22,1321],{"className":1322,"style":340},[258],[22,1324,1326,1329,1332,1335,1338],{"className":1325},[64],[22,1327],{"className":1328,"style":1286},[68],[22,1330,280],{"className":1331},[402],[22,1333],{"className":1334,"style":398},[258],[22,1336,307],{"className":1337},[73],[22,1339,144],{"className":1340,"style":161},[73,77],[22,1342,1344],{"className":1343},[204],[22,1345,1347,1422],{"className":1346},[25],[22,1348,1350],{"className":1349},[29],[31,1351,1352],{"xmlns":33,"display":213},[35,1353,1354,1419],{},[38,1355,1356,1358,1360,1366,1368,1370,1372,1374,1394,1396],{},[44,1357,177],{},[222,1359,234],{},[41,1361,1362,1364],{},[44,1363,46],{},[48,1365,50],{},[44,1367,243],{},[222,1369,246],{},[44,1371,144],{},[222,1373,251],{},[293,1375,1376,1392],{},[38,1377,1378,1380,1386,1388,1390],{},[48,1379,307],{},[41,1381,1382,1384],{},[44,1383,46],{},[48,1385,50],{},[44,1387,280],{},[222,1389,246],{},[44,1391,144],{},[44,1393,299],{},[222,1395,234],{},[293,1397,1398,1417],{},[38,1399,1400,1409,1411,1413,1415],{},[1401,1402,1403,1405,1407],"msubsup",{},[44,1404,46],{},[48,1406,50],{},[48,1408,307],{},[44,1410,280],{},[222,1412,246],{},[48,1414,307],{},[44,1416,144],{},[44,1418,299],{},[52,1420,1421],{"encoding":54},"L = v_0 \\cos\\alpha \\cdot \\frac{2 v_0 \\sin\\alpha}{g} = \\frac{v_0^2 \\sin 2\\alpha}{g}",[22,1423,1425,1443,1510,1639],{"className":1424,"ariaHidden":60},[59],[22,1426,1428,1431,1434,1437,1440],{"className":1427},[64],[22,1429],{"className":1430,"style":189},[68],[22,1432,177],{"className":1433},[73,77],[22,1435],{"className":1436,"style":340},[258],[22,1438,234],{"className":1439},[344],[22,1441],{"className":1442,"style":340},[258],[22,1444,1446,1449,1489,1492,1495,1498,1501,1504,1507],{"className":1445},[64],[22,1447],{"className":1448,"style":354},[68],[22,1450,1452,1455],{"className":1451},[73],[22,1453,46],{"className":1454,"style":78},[73,77],[22,1456,1458],{"className":1457},[82],[22,1459,1461,1481],{"className":1460},[86,87],[22,1462,1464,1478],{"className":1463},[91],[22,1465,1467],{"className":1466,"style":96},[95],[22,1468,1469,1472],{"style":99},[22,1470],{"className":1471,"style":104},[103],[22,1473,1475],{"className":1474},[108,109,110,111],[22,1476,50],{"className":1477},[73,111],[22,1479,119],{"className":1480},[118],[22,1482,1484],{"className":1483},[91],[22,1485,1487],{"className":1486,"style":126},[95],[22,1488],{},[22,1490],{"className":1491,"style":398},[258],[22,1493,243],{"className":1494},[402],[22,1496],{"className":1497,"style":398},[258],[22,1499,144],{"className":1500,"style":161},[73,77],[22,1502],{"className":1503,"style":412},[258],[22,1505,251],{"className":1506},[416],[22,1508],{"className":1509,"style":412},[258],[22,1511,1513,1516,1630,1633,1636],{"className":1512},[64],[22,1514],{"className":1515,"style":1040},[68],[22,1517,1519,1522,1627],{"className":1518},[73],[22,1520],{"className":1521},[329,561],[22,1523,1525],{"className":1524},[293],[22,1526,1528,1619],{"className":1527},[86,87],[22,1529,1531,1616],{"className":1530},[91],[22,1532,1534,1545,1553],{"className":1533,"style":1059},[95],[22,1535,1536,1539],{"style":577},[22,1537],{"className":1538,"style":581},[103],[22,1540,1542],{"className":1541},[73],[22,1543,299],{"className":1544,"style":78},[73,77],[22,1546,1547,1550],{"style":590},[22,1548],{"className":1549,"style":581},[103],[22,1551],{"className":1552,"style":598},[597],[22,1554,1555,1558],{"style":601},[22,1556],{"className":1557,"style":581},[103],[22,1559,1561,1564,1604,1607,1610,1613],{"className":1560},[73],[22,1562,307],{"className":1563},[73],[22,1565,1567,1570],{"className":1566},[73],[22,1568,46],{"className":1569,"style":78},[73,77],[22,1571,1573],{"className":1572},[82],[22,1574,1576,1596],{"className":1575},[86,87],[22,1577,1579,1593],{"className":1578},[91],[22,1580,1582],{"className":1581,"style":96},[95],[22,1583,1584,1587],{"style":99},[22,1585],{"className":1586,"style":104},[103],[22,1588,1590],{"className":1589},[108,109,110,111],[22,1591,50],{"className":1592},[73,111],[22,1594,119],{"className":1595},[118],[22,1597,1599],{"className":1598},[91],[22,1600,1602],{"className":1601,"style":126},[95],[22,1603],{},[22,1605],{"className":1606,"style":398},[258],[22,1608,280],{"className":1609},[402],[22,1611],{"className":1612,"style":398},[258],[22,1614,144],{"className":1615,"style":161},[73,77],[22,1617,119],{"className":1618},[118],[22,1620,1622],{"className":1621},[91],[22,1623,1625],{"className":1624,"style":1151},[95],[22,1626],{},[22,1628],{"className":1629},[336,561],[22,1631],{"className":1632,"style":340},[258],[22,1634,234],{"className":1635},[344],[22,1637],{"className":1638,"style":340},[258],[22,1640,1642,1646],{"className":1641},[64],[22,1643],{"className":1644,"style":1645},[68],"height:2.3715em;vertical-align:-0.8804em;",[22,1647,1649,1652,1770],{"className":1648},[73],[22,1650],{"className":1651},[329,561],[22,1653,1655],{"className":1654},[293],[22,1656,1658,1762],{"className":1657},[86,87],[22,1659,1661,1759],{"className":1660},[91],[22,1662,1664,1675,1683],{"className":1663,"style":574},[95],[22,1665,1666,1669],{"style":577},[22,1667],{"className":1668,"style":581},[103],[22,1670,1672],{"className":1671},[73],[22,1673,299],{"className":1674,"style":78},[73,77],[22,1676,1677,1680],{"style":590},[22,1678],{"className":1679,"style":581},[103],[22,1681],{"className":1682,"style":598},[597],[22,1684,1685,1688],{"style":601},[22,1686],{"className":1687,"style":581},[103],[22,1689,1691,1744,1747,1750,1753,1756],{"className":1690},[73],[22,1692,1694,1697],{"className":1693},[73],[22,1695,46],{"className":1696,"style":78},[73,77],[22,1698,1700],{"className":1699},[82],[22,1701,1703,1735],{"className":1702},[86,87],[22,1704,1706,1732],{"className":1705},[91],[22,1707,1709,1721],{"className":1708,"style":629},[95],[22,1710,1712,1715],{"style":1711},"top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;",[22,1713],{"className":1714,"style":104},[103],[22,1716,1718],{"className":1717},[108,109,110,111],[22,1719,50],{"className":1720},[73,111],[22,1722,1723,1726],{"style":632},[22,1724],{"className":1725,"style":104},[103],[22,1727,1729],{"className":1728},[108,109,110,111],[22,1730,307],{"className":1731},[73,111],[22,1733,119],{"className":1734},[118],[22,1736,1738],{"className":1737},[91],[22,1739,1742],{"className":1740,"style":1741},[95],"height:0.2481em;",[22,1743],{},[22,1745],{"className":1746,"style":398},[258],[22,1748,280],{"className":1749},[402],[22,1751],{"className":1752,"style":398},[258],[22,1754,307],{"className":1755},[73],[22,1757,144],{"className":1758,"style":161},[73,77],[22,1760,119],{"className":1761},[118],[22,1763,1765],{"className":1764},[91],[22,1766,1768],{"className":1767,"style":1151},[95],[22,1769],{},[22,1771],{"className":1772},[336,561],[14,1774,1776],{"id":1775},"чому-саме-45","Чому саме 45°",[10,1778,1779,1780,1849,1850,1894,1895,1962,1963,2056],{},"При заданій швидкості ",[22,1781,1783,1800],{"className":1782},[25],[22,1784,1786],{"className":1785},[29],[31,1787,1788],{"xmlns":33},[35,1789,1790,1798],{},[38,1791,1792],{},[41,1793,1794,1796],{},[44,1795,46],{},[48,1797,50],{},[52,1799,55],{"encoding":54},[22,1801,1803],{"className":1802,"ariaHidden":60},[59],[22,1804,1806,1809],{"className":1805},[64],[22,1807],{"className":1808,"style":69},[68],[22,1810,1812,1815],{"className":1811},[73],[22,1813,46],{"className":1814,"style":78},[73,77],[22,1816,1818],{"className":1817},[82],[22,1819,1821,1841],{"className":1820},[86,87],[22,1822,1824,1838],{"className":1823},[91],[22,1825,1827],{"className":1826,"style":96},[95],[22,1828,1829,1832],{"style":99},[22,1830],{"className":1831,"style":104},[103],[22,1833,1835],{"className":1834},[108,109,110,111],[22,1836,50],{"className":1837},[73,111],[22,1839,119],{"className":1840},[118],[22,1842,1844],{"className":1843},[91],[22,1845,1847],{"className":1846,"style":126},[95],[22,1848],{}," дальність залежить лише від ",[22,1851,1853,1873],{"className":1852},[25],[22,1854,1856],{"className":1855},[29],[31,1857,1858],{"xmlns":33},[35,1859,1860,1870],{},[38,1861,1862,1864,1866,1868],{},[44,1863,280],{},[222,1865,246],{},[48,1867,307],{},[44,1869,144],{},[52,1871,1872],{"encoding":54},"\\sin 2\\alpha",[22,1874,1876],{"className":1875,"ariaHidden":60},[59],[22,1877,1879,1882,1885,1888,1891],{"className":1878},[64],[22,1880],{"className":1881,"style":1286},[68],[22,1883,280],{"className":1884},[402],[22,1886],{"className":1887,"style":398},[258],[22,1889,307],{"className":1890},[73],[22,1892,144],{"className":1893,"style":161},[73,77],". Синус не перевищує 1, а ",[22,1896,1898,1923],{"className":1897},[25],[22,1899,1901],{"className":1900},[29],[31,1902,1903],{"xmlns":33},[35,1904,1905,1920],{},[38,1906,1907,1909,1911,1913,1915,1917],{},[44,1908,280],{},[222,1910,246],{},[48,1912,307],{},[44,1914,144],{},[222,1916,234],{},[48,1918,1919],{},"1",[52,1921,1922],{"encoding":54},"\\sin 2\\alpha = 1",[22,1924,1926,1953],{"className":1925,"ariaHidden":60},[59],[22,1927,1929,1932,1935,1938,1941,1944,1947,1950],{"className":1928},[64],[22,1930],{"className":1931,"style":1286},[68],[22,1933,280],{"className":1934},[402],[22,1936],{"className":1937,"style":398},[258],[22,1939,307],{"className":1940},[73],[22,1942,144],{"className":1943,"style":161},[73,77],[22,1945],{"className":1946,"style":340},[258],[22,1948,234],{"className":1949},[344],[22,1951],{"className":1952,"style":340},[258],[22,1954,1956,1959],{"className":1955},[64],[22,1957],{"className":1958,"style":727},[68],[22,1960,1919],{"className":1961},[73]," при ",[22,1964,1966,1992],{"className":1965},[25],[22,1967,1969],{"className":1968},[29],[31,1970,1971],{"xmlns":33},[35,1972,1973,1989],{},[38,1974,1975,1977,1979,1981],{},[48,1976,307],{},[44,1978,144],{},[222,1980,234],{},[301,1982,1983,1986],{},[48,1984,1985],{},"90",[222,1987,1988],{},"∘",[52,1990,1991],{"encoding":54},"2\\alpha = 90^\\circ",[22,1993,1995,2016],{"className":1994,"ariaHidden":60},[59],[22,1996,1998,2001,2004,2007,2010,2013],{"className":1997},[64],[22,1999],{"className":2000,"style":727},[68],[22,2002,307],{"className":2003},[73],[22,2005,144],{"className":2006,"style":161},[73,77],[22,2008],{"className":2009,"style":340},[258],[22,2011,234],{"className":2012},[344],[22,2014],{"className":2015,"style":340},[258],[22,2017,2019,2023,2027],{"className":2018},[64],[22,2020],{"className":2021,"style":2022},[68],"height:0.6741em;",[22,2024,2026],{"className":2025},[73],"9",[22,2028,2030,2033],{"className":2029},[73],[22,2031,50],{"className":2032},[73],[22,2034,2036],{"className":2035},[82],[22,2037,2039],{"className":2038},[86],[22,2040,2042],{"className":2041},[91],[22,2043,2045],{"className":2044,"style":2022},[95],[22,2046,2047,2050],{"style":632},[22,2048],{"className":2049,"style":104},[103],[22,2051,2053],{"className":2052},[108,109,110,111],[22,2054,1988],{"className":2055},[416,111],", тобто",[22,2058,2060],{"className":2059},[204],[22,2061,2063,2118],{"className":2062},[25],[22,2064,2066],{"className":2065},[29],[31,2067,2068],{"xmlns":33,"display":213},[35,2069,2070,2115],{},[38,2071,2072,2074,2076,2083,2085,2087,2101,2103],{},[44,2073,144],{},[222,2075,234],{},[301,2077,2078,2081],{},[48,2079,2080],{},"45",[222,2082,1988],{},[222,2084,256],{"separator":60},[258,2086],{"width":260},[41,2088,2089,2091],{},[44,2090,177],{},[38,2092,2093,2096,2099],{},[44,2094,2095],{},"m",[44,2097,2098],{},"a",[44,2100,220],{},[222,2102,234],{},[293,2104,2105,2113],{},[1401,2106,2107,2109,2111],{},[44,2108,46],{},[48,2110,50],{},[48,2112,307],{},[44,2114,299],{},[52,2116,2117],{"encoding":54},"\\alpha = 45^\\circ, \\qquad L_{max} = \\frac{v_0^2}{g}",[22,2119,2121,2139,2249],{"className":2120,"ariaHidden":60},[59],[22,2122,2124,2127,2130,2133,2136],{"className":2123},[64],[22,2125],{"className":2126,"style":157},[68],[22,2128,144],{"className":2129,"style":161},[73,77],[22,2131],{"className":2132,"style":340},[258],[22,2134,234],{"className":2135},[344],[22,2137],{"className":2138,"style":340},[258],[22,2140,2142,2146,2150,2182,2185,2188,2191,2240,2243,2246],{"className":2141},[64],[22,2143],{"className":2144,"style":2145},[68],"height:0.9186em;vertical-align:-0.1944em;",[22,2147,2149],{"className":2148},[73],"4",[22,2151,2153,2157],{"className":2152},[73],[22,2154,2156],{"className":2155},[73],"5",[22,2158,2160],{"className":2159},[82],[22,2161,2163],{"className":2162},[86],[22,2164,2166],{"className":2165},[91],[22,2167,2170],{"className":2168,"style":2169},[95],"height:0.7241em;",[22,2171,2173,2176],{"style":2172},"top:-3.113em;margin-right:0.05em;",[22,2174],{"className":2175,"style":104},[103],[22,2177,2179],{"className":2178},[108,109,110,111],[22,2180,1988],{"className":2181},[416,111],[22,2183,256],{"className":2184},[432],[22,2186],{"className":2187,"style":436},[258],[22,2189],{"className":2190,"style":398},[258],[22,2192,2194,2197],{"className":2193},[73],[22,2195,177],{"className":2196},[73,77],[22,2198,2200],{"className":2199},[82],[22,2201,2203,2232],{"className":2202},[86,87],[22,2204,2206,2229],{"className":2205},[91],[22,2207,2210],{"className":2208,"style":2209},[95],"height:0.1514em;",[22,2211,2213,2216],{"style":2212},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[22,2214],{"className":2215,"style":104},[103],[22,2217,2219],{"className":2218},[108,109,110,111],[22,2220,2222,2226],{"className":2221},[73,111],[22,2223,2225],{"className":2224},[73,77,111],"ma",[22,2227,220],{"className":2228},[73,77,111],[22,2230,119],{"className":2231},[118],[22,2233,2235],{"className":2234},[91],[22,2236,2238],{"className":2237,"style":126},[95],[22,2239],{},[22,2241],{"className":2242,"style":340},[258],[22,2244,234],{"className":2245},[344],[22,2247],{"className":2248,"style":340},[258],[22,2250,2252,2255],{"className":2251},[64],[22,2253],{"className":2254,"style":1645},[68],[22,2256,2258,2261,2362],{"className":2257},[73],[22,2259],{"className":2260},[329,561],[22,2262,2264],{"className":2263},[293],[22,2265,2267,2354],{"className":2266},[86,87],[22,2268,2270,2351],{"className":2269},[91],[22,2271,2273,2284,2292],{"className":2272,"style":574},[95],[22,2274,2275,2278],{"style":577},[22,2276],{"className":2277,"style":581},[103],[22,2279,2281],{"className":2280},[73],[22,2282,299],{"className":2283,"style":78},[73,77],[22,2285,2286,2289],{"style":590},[22,2287],{"className":2288,"style":581},[103],[22,2290],{"className":2291,"style":598},[597],[22,2293,2294,2297],{"style":601},[22,2295],{"className":2296,"style":581},[103],[22,2298,2300],{"className":2299},[73],[22,2301,2303,2306],{"className":2302},[73],[22,2304,46],{"className":2305,"style":78},[73,77],[22,2307,2309],{"className":2308},[82],[22,2310,2312,2343],{"className":2311},[86,87],[22,2313,2315,2340],{"className":2314},[91],[22,2316,2318,2329],{"className":2317,"style":629},[95],[22,2319,2320,2323],{"style":1711},[22,2321],{"className":2322,"style":104},[103],[22,2324,2326],{"className":2325},[108,109,110,111],[22,2327,50],{"className":2328},[73,111],[22,2330,2331,2334],{"style":632},[22,2332],{"className":2333,"style":104},[103],[22,2335,2337],{"className":2336},[108,109,110,111],[22,2338,307],{"className":2339},[73,111],[22,2341,119],{"className":2342},[118],[22,2344,2346],{"className":2345},[91],[22,2347,2349],{"className":2348,"style":1741},[95],[22,2350],{},[22,2352,119],{"className":2353},[118],[22,2355,2357],{"className":2356},[91],[22,2358,2360],{"className":2359,"style":1151},[95],[22,2361],{},[22,2363],{"className":2364},[336,561],[10,2366,2367,2368,2461,2462,2526,2527,2744],{},"Для ",[22,2369,2371,2394],{"className":2370},[25],[22,2372,2374],{"className":2373},[29],[31,2375,2376],{"xmlns":33},[35,2377,2378,2391],{},[38,2379,2380,2386,2388],{},[41,2381,2382,2384],{},[44,2383,46],{},[48,2385,50],{},[222,2387,234],{},[48,2389,2390],{},"20",[52,2392,2393],{"encoding":54},"v_0 = 20",[22,2395,2397,2452],{"className":2396,"ariaHidden":60},[59],[22,2398,2400,2403,2443,2446,2449],{"className":2399},[64],[22,2401],{"className":2402,"style":69},[68],[22,2404,2406,2409],{"className":2405},[73],[22,2407,46],{"className":2408,"style":78},[73,77],[22,2410,2412],{"className":2411},[82],[22,2413,2415,2435],{"className":2414},[86,87],[22,2416,2418,2432],{"className":2417},[91],[22,2419,2421],{"className":2420,"style":96},[95],[22,2422,2423,2426],{"style":99},[22,2424],{"className":2425,"style":104},[103],[22,2427,2429],{"className":2428},[108,109,110,111],[22,2430,50],{"className":2431},[73,111],[22,2433,119],{"className":2434},[118],[22,2436,2438],{"className":2437},[91],[22,2439,2441],{"className":2440,"style":126},[95],[22,2442],{},[22,2444],{"className":2445,"style":340},[258],[22,2447,234],{"className":2448},[344],[22,2450],{"className":2451,"style":340},[258],[22,2453,2455,2458],{"className":2454},[64],[22,2456],{"className":2457,"style":727},[68],[22,2459,2390],{"className":2460},[73]," м\u002Fс і ",[22,2463,2465,2484],{"className":2464},[25],[22,2466,2468],{"className":2467},[29],[31,2469,2470],{"xmlns":33},[35,2471,2472,2481],{},[38,2473,2474,2476,2478],{},[44,2475,299],{},[222,2477,234],{},[48,2479,2480],{},"9,8",[52,2482,2483],{"encoding":54},"g = 9{,}8",[22,2485,2487,2506],{"className":2486,"ariaHidden":60},[59],[22,2488,2490,2494,2497,2500,2503],{"className":2489},[64],[22,2491],{"className":2492,"style":2493},[68],"height:0.625em;vertical-align:-0.1944em;",[22,2495,299],{"className":2496,"style":78},[73,77],[22,2498],{"className":2499,"style":340},[258],[22,2501,234],{"className":2502},[344],[22,2504],{"className":2505,"style":340},[258],[22,2507,2509,2513,2516,2522],{"className":2508},[64],[22,2510],{"className":2511,"style":2512},[68],"height:0.8389em;vertical-align:-0.1944em;",[22,2514,2026],{"className":2515},[73],[22,2517,2519],{"className":2518},[73],[22,2520,256],{"className":2521},[432],[22,2523,2525],{"className":2524},[73],"8"," м\u002Fс²: ",[22,2528,2530,2572],{"className":2529},[25],[22,2531,2533],{"className":2532},[29],[31,2534,2535],{"xmlns":33},[35,2536,2537,2569],{},[38,2538,2539,2551,2553,2563,2566],{},[41,2540,2541,2543],{},[44,2542,177],{},[38,2544,2545,2547,2549],{},[44,2546,2095],{},[44,2548,2098],{},[44,2550,220],{},[222,2552,234],{},[2554,2555,2556],"mstyle",{"scriptlevel":50,"displaystyle":60},[293,2557,2558,2561],{},[48,2559,2560],{},"400",[48,2562,2480],{},[222,2564,2565],{},"≈",[48,2567,2568],{},"40,8",[52,2570,2571],{"encoding":54},"L_{max} = \\dfrac{400}{9{,}8} \\approx 40{,}8",[22,2573,2575,2637,2725],{"className":2574,"ariaHidden":60},[59],[22,2576,2578,2582,2628,2631,2634],{"className":2577},[64],[22,2579],{"className":2580,"style":2581},[68],"height:0.8333em;vertical-align:-0.15em;",[22,2583,2585,2588],{"className":2584},[73],[22,2586,177],{"className":2587},[73,77],[22,2589,2591],{"className":2590},[82],[22,2592,2594,2620],{"className":2593},[86,87],[22,2595,2597,2617],{"className":2596},[91],[22,2598,2600],{"className":2599,"style":2209},[95],[22,2601,2602,2605],{"style":2212},[22,2603],{"className":2604,"style":104},[103],[22,2606,2608],{"className":2607},[108,109,110,111],[22,2609,2611,2614],{"className":2610},[73,111],[22,2612,2225],{"className":2613},[73,77,111],[22,2615,220],{"className":2616},[73,77,111],[22,2618,119],{"className":2619},[118],[22,2621,2623],{"className":2622},[91],[22,2624,2626],{"className":2625,"style":126},[95],[22,2627],{},[22,2629],{"className":2630,"style":340},[258],[22,2632,234],{"className":2633},[344],[22,2635],{"className":2636,"style":340},[258],[22,2638,2640,2644,2716,2719,2722],{"className":2639},[64],[22,2641],{"className":2642,"style":2643},[68],"height:2.2019em;vertical-align:-0.8804em;",[22,2645,2647,2650,2713],{"className":2646},[73],[22,2648],{"className":2649},[329,561],[22,2651,2653],{"className":2652},[293],[22,2654,2656,2705],{"className":2655},[86,87],[22,2657,2659,2702],{"className":2658},[91],[22,2660,2663,2683,2691],{"className":2661,"style":2662},[95],"height:1.3214em;",[22,2664,2665,2668],{"style":577},[22,2666],{"className":2667,"style":581},[103],[22,2669,2671,2674,2680],{"className":2670},[73],[22,2672,2026],{"className":2673},[73],[22,2675,2677],{"className":2676},[73],[22,2678,256],{"className":2679},[432],[22,2681,2525],{"className":2682},[73],[22,2684,2685,2688],{"style":590},[22,2686],{"className":2687,"style":581},[103],[22,2689],{"className":2690,"style":598},[597],[22,2692,2693,2696],{"style":601},[22,2694],{"className":2695,"style":581},[103],[22,2697,2699],{"className":2698},[73],[22,2700,2560],{"className":2701},[73],[22,2703,119],{"className":2704},[118],[22,2706,2708],{"className":2707},[91],[22,2709,2711],{"className":2710,"style":1151},[95],[22,2712],{},[22,2714],{"className":2715},[336,561],[22,2717],{"className":2718,"style":340},[258],[22,2720,2565],{"className":2721},[344],[22,2723],{"className":2724,"style":340},[258],[22,2726,2728,2731,2735,2741],{"className":2727},[64],[22,2729],{"className":2730,"style":2512},[68],[22,2732,2734],{"className":2733},[73],"40",[22,2736,2738],{"className":2737},[73],[22,2739,256],{"className":2740},[432],[22,2742,2525],{"className":2743},[73]," м.",[10,2746,2747,2751,2752,2816,2817,2881,2882,3021,3022,2816,3050,3135],{},[2748,2749,2750],"strong",{},"Цікавий наслідок:"," кути ",[22,2753,2755,2774],{"className":2754},[25],[22,2756,2758],{"className":2757},[29],[31,2759,2760],{"xmlns":33},[35,2761,2762,2771],{},[38,2763,2764],{},[301,2765,2766,2769],{},[48,2767,2768],{},"30",[222,2770,1988],{},[52,2772,2773],{"encoding":54},"30^\\circ",[22,2775,2777],{"className":2776,"ariaHidden":60},[59],[22,2778,2780,2783,2787],{"className":2779},[64],[22,2781],{"className":2782,"style":2022},[68],[22,2784,2786],{"className":2785},[73],"3",[22,2788,2790,2793],{"className":2789},[73],[22,2791,50],{"className":2792},[73],[22,2794,2796],{"className":2795},[82],[22,2797,2799],{"className":2798},[86],[22,2800,2802],{"className":2801},[91],[22,2803,2805],{"className":2804,"style":2022},[95],[22,2806,2807,2810],{"style":632},[22,2808],{"className":2809,"style":104},[103],[22,2811,2813],{"className":2812},[108,109,110,111],[22,2814,1988],{"className":2815},[416,111]," і ",[22,2818,2820,2839],{"className":2819},[25],[22,2821,2823],{"className":2822},[29],[31,2824,2825],{"xmlns":33},[35,2826,2827,2836],{},[38,2828,2829],{},[301,2830,2831,2834],{},[48,2832,2833],{},"60",[222,2835,1988],{},[52,2837,2838],{"encoding":54},"60^\\circ",[22,2840,2842],{"className":2841,"ariaHidden":60},[59],[22,2843,2845,2848,2852],{"className":2844},[64],[22,2846],{"className":2847,"style":2022},[68],[22,2849,2851],{"className":2850},[73],"6",[22,2853,2855,2858],{"className":2854},[73],[22,2856,50],{"className":2857},[73],[22,2859,2861],{"className":2860},[82],[22,2862,2864],{"className":2863},[86],[22,2865,2867],{"className":2866},[91],[22,2868,2870],{"className":2869,"style":2022},[95],[22,2871,2872,2875],{"style":632},[22,2873],{"className":2874,"style":104},[103],[22,2876,2878],{"className":2877},[108,109,110,111],[22,2879,1988],{"className":2880},[416,111]," дають однакову дальність, бо ",[22,2883,2885,2920],{"className":2884},[25],[22,2886,2888],{"className":2887},[29],[31,2889,2890],{"xmlns":33},[35,2891,2892,2917],{},[38,2893,2894,2896,2898,2904,2906,2908,2910],{},[44,2895,280],{},[222,2897,246],{},[301,2899,2900,2902],{},[48,2901,2833],{},[222,2903,1988],{},[222,2905,234],{},[44,2907,280],{},[222,2909,246],{},[301,2911,2912,2915],{},[48,2913,2914],{},"120",[222,2916,1988],{},[52,2918,2919],{"encoding":54},"\\sin 60^\\circ = \\sin 120^\\circ",[22,2921,2923,2976],{"className":2922,"ariaHidden":60},[59],[22,2924,2926,2929,2932,2935,2938,2967,2970,2973],{"className":2925},[64],[22,2927],{"className":2928,"style":2022},[68],[22,2930,280],{"className":2931},[402],[22,2933],{"className":2934,"style":398},[258],[22,2936,2851],{"className":2937},[73],[22,2939,2941,2944],{"className":2940},[73],[22,2942,50],{"className":2943},[73],[22,2945,2947],{"className":2946},[82],[22,2948,2950],{"className":2949},[86],[22,2951,2953],{"className":2952},[91],[22,2954,2956],{"className":2955,"style":2022},[95],[22,2957,2958,2961],{"style":632},[22,2959],{"className":2960,"style":104},[103],[22,2962,2964],{"className":2963},[108,109,110,111],[22,2965,1988],{"className":2966},[416,111],[22,2968],{"className":2969,"style":340},[258],[22,2971,234],{"className":2972},[344],[22,2974],{"className":2975,"style":340},[258],[22,2977,2979,2982,2985,2988,2992],{"className":2978},[64],[22,2980],{"className":2981,"style":2022},[68],[22,2983,280],{"className":2984},[402],[22,2986],{"className":2987,"style":398},[258],[22,2989,2991],{"className":2990},[73],"12",[22,2993,2995,2998],{"className":2994},[73],[22,2996,50],{"className":2997},[73],[22,2999,3001],{"className":3000},[82],[22,3002,3004],{"className":3003},[86],[22,3005,3007],{"className":3006},[91],[22,3008,3010],{"className":3009,"style":2022},[95],[22,3011,3012,3015],{"style":632},[22,3013],{"className":3014,"style":104},[103],[22,3016,3018],{"className":3017},[108,109,110,111],[22,3019,1988],{"className":3020},[416,111],". Загалом однаково далеко летять тіла, кинуті під кутами ",[22,3023,3025,3038],{"className":3024},[25],[22,3026,3028],{"className":3027},[29],[31,3029,3030],{"xmlns":33},[35,3031,3032,3036],{},[38,3033,3034],{},[44,3035,144],{},[52,3037,147],{"encoding":54},[22,3039,3041],{"className":3040,"ariaHidden":60},[59],[22,3042,3044,3047],{"className":3043},[64],[22,3045],{"className":3046,"style":157},[68],[22,3048,144],{"className":3049,"style":161},[73,77],[22,3051,3053,3075],{"className":3052},[25],[22,3054,3056],{"className":3055},[29],[31,3057,3058],{"xmlns":33},[35,3059,3060,3072],{},[38,3061,3062,3068,3070],{},[301,3063,3064,3066],{},[48,3065,1985],{},[222,3067,1988],{},[222,3069,291],{},[44,3071,144],{},[52,3073,3074],{"encoding":54},"90^\\circ - \\alpha",[22,3076,3078,3126],{"className":3077,"ariaHidden":60},[59],[22,3079,3081,3085,3088,3117,3120,3123],{"className":3080},[64],[22,3082],{"className":3083,"style":3084},[68],"height:0.7574em;vertical-align:-0.0833em;",[22,3086,2026],{"className":3087},[73],[22,3089,3091,3094],{"className":3090},[73],[22,3092,50],{"className":3093},[73],[22,3095,3097],{"className":3096},[82],[22,3098,3100],{"className":3099},[86],[22,3101,3103],{"className":3102},[91],[22,3104,3106],{"className":3105,"style":2022},[95],[22,3107,3108,3111],{"style":632},[22,3109],{"className":3110,"style":104},[103],[22,3112,3114],{"className":3113},[108,109,110,111],[22,3115,1988],{"className":3116},[416,111],[22,3118],{"className":3119,"style":412},[258],[22,3121,291],{"className":3122},[416],[22,3124],{"className":3125,"style":412},[258],[22,3127,3129,3132],{"className":3128},[64],[22,3130],{"className":3131,"style":157},[68],[22,3133,144],{"className":3134,"style":161},[73,77],". А от висота польоту різна:",[22,3137,3139],{"className":3138},[204],[22,3140,3142,3189],{"className":3141},[25],[22,3143,3145],{"className":3144},[29],[31,3146,3147],{"xmlns":33,"display":213},[35,3148,3149,3186],{},[38,3150,3151,3154,3156],{},[44,3152,3153],{},"H",[222,3155,234],{},[293,3157,3158,3180],{},[38,3159,3160,3168,3178],{},[1401,3161,3162,3164,3166],{},[44,3163,46],{},[48,3165,50],{},[48,3167,307],{},[301,3169,3170,3176],{},[38,3171,3172,3174],{},[44,3173,280],{},[222,3175,246],{},[48,3177,307],{},[44,3179,144],{},[38,3181,3182,3184],{},[48,3183,307],{},[44,3185,299],{},[52,3187,3188],{"encoding":54},"H = \\frac{v_0^2 \\sin^2\\alpha}{2g}",[22,3190,3192,3211],{"className":3191,"ariaHidden":60},[59],[22,3193,3195,3198,3202,3205,3208],{"className":3194},[64],[22,3196],{"className":3197,"style":189},[68],[22,3199,3153],{"className":3200,"style":3201},[73,77],"margin-right:0.0813em;",[22,3203],{"className":3204,"style":340},[258],[22,3206,234],{"className":3207},[344],[22,3209],{"className":3210,"style":340},[258],[22,3212,3214,3218],{"className":3213},[64],[22,3215],{"className":3216,"style":3217},[68],"height:2.4293em;vertical-align:-0.8804em;",[22,3219,3221,3224,3369],{"className":3220},[73],[22,3222],{"className":3223},[329,561],[22,3225,3227],{"className":3226},[293],[22,3228,3230,3361],{"className":3229},[86,87],[22,3231,3233,3358],{"className":3232},[91],[22,3234,3237,3251,3259],{"className":3235,"style":3236},[95],"height:1.5489em;",[22,3238,3239,3242],{"style":577},[22,3240],{"className":3241,"style":581},[103],[22,3243,3245,3248],{"className":3244},[73],[22,3246,307],{"className":3247},[73],[22,3249,299],{"className":3250,"style":78},[73,77],[22,3252,3253,3256],{"style":590},[22,3254],{"className":3255,"style":581},[103],[22,3257],{"className":3258,"style":598},[597],[22,3260,3261,3264],{"style":601},[22,3262],{"className":3263,"style":581},[103],[22,3265,3267,3318,3321,3352,3355],{"className":3266},[73],[22,3268,3270,3273],{"className":3269},[73],[22,3271,46],{"className":3272,"style":78},[73,77],[22,3274,3276],{"className":3275},[82],[22,3277,3279,3310],{"className":3278},[86,87],[22,3280,3282,3307],{"className":3281},[91],[22,3283,3285,3296],{"className":3284,"style":629},[95],[22,3286,3287,3290],{"style":1711},[22,3288],{"className":3289,"style":104},[103],[22,3291,3293],{"className":3292},[108,109,110,111],[22,3294,50],{"className":3295},[73,111],[22,3297,3298,3301],{"style":632},[22,3299],{"className":3300,"style":104},[103],[22,3302,3304],{"className":3303},[108,109,110,111],[22,3305,307],{"className":3306},[73,111],[22,3308,119],{"className":3309},[118],[22,3311,3313],{"className":3312},[91],[22,3314,3316],{"className":3315,"style":1741},[95],[22,3317],{},[22,3319],{"className":3320,"style":398},[258],[22,3322,3324,3327],{"className":3323},[402],[22,3325,280],{"className":3326},[402],[22,3328,3330],{"className":3329},[82],[22,3331,3333],{"className":3332},[86],[22,3334,3336],{"className":3335},[91],[22,3337,3340],{"className":3338,"style":3339},[95],"height:0.8719em;",[22,3341,3343,3346],{"style":3342},"top:-3.1208em;margin-right:0.05em;",[22,3344],{"className":3345,"style":104},[103],[22,3347,3349],{"className":3348},[108,109,110,111],[22,3350,307],{"className":3351},[73,111],[22,3353],{"className":3354,"style":398},[258],[22,3356,144],{"className":3357,"style":161},[73,77],[22,3359,119],{"className":3360},[118],[22,3362,3364],{"className":3363},[91],[22,3365,3367],{"className":3366,"style":1151},[95],[22,3368],{},[22,3370],{"className":3371},[336,561],[14,3373,3375],{"id":3374},"перевірте-на-практиці","Перевірте на практиці",[10,3377,3378],{},"Тепер, знаючи формулу, влучте в мішень — спершу розрахуйте швидкість, а потім стріляйте:",[3380,3381],"lab-target",{},[14,3383,3385],{"id":3384},"задача-для-тренування","Задача для тренування",[3387,3388,3389],"blockquote",{},[10,3390,3391,3392,3444],{},"Тіло кинули зі швидкістю 15 м\u002Fс під кутом 30° до горизонту. Знайдіть час польоту, дальність і максимальну висоту. Вважайте ",[22,3393,3395,3414],{"className":3394},[25],[22,3396,3398],{"className":3397},[29],[31,3399,3400],{"xmlns":33},[35,3401,3402,3411],{},[38,3403,3404,3406,3408],{},[44,3405,299],{},[222,3407,234],{},[48,3409,3410],{},"10",[52,3412,3413],{"encoding":54},"g = 10",[22,3415,3417,3435],{"className":3416,"ariaHidden":60},[59],[22,3418,3420,3423,3426,3429,3432],{"className":3419},[64],[22,3421],{"className":3422,"style":2493},[68],[22,3424,299],{"className":3425,"style":78},[73,77],[22,3427],{"className":3428,"style":340},[258],[22,3430,234],{"className":3431},[344],[22,3433],{"className":3434,"style":340},[258],[22,3436,3438,3441],{"className":3437},[64],[22,3439],{"className":3440,"style":727},[68],[22,3442,3410],{"className":3443},[73]," м\u002Fс².",[3446,3447,3448],"answer",{},[10,3449,3450,3644,3645,3858,3859,2744],{},[22,3451,3453,3494],{"className":3452},[25],[22,3454,3456],{"className":3455},[29],[31,3457,3458],{"xmlns":33},[35,3459,3460,3491],{},[38,3461,3462,3464,3466,3486,3488],{},[44,3463,793],{},[222,3465,234],{},[2554,3467,3468],{"scriptlevel":50,"displaystyle":60},[293,3469,3470,3484],{},[38,3471,3472,3474,3476,3479,3481],{},[48,3473,307],{},[222,3475,251],{},[48,3477,3478],{},"15",[222,3480,251],{},[48,3482,3483],{},"0,5",[48,3485,3410],{},[222,3487,234],{},[48,3489,3490],{},"1,5",[52,3492,3493],{"encoding":54},"T = \\dfrac{2 \\cdot 15 \\cdot 0{,}5}{10} = 1{,}5",[22,3495,3497,3515,3626],{"className":3496,"ariaHidden":60},[59],[22,3498,3500,3503,3506,3509,3512],{"className":3499},[64],[22,3501],{"className":3502,"style":189},[68],[22,3504,793],{"className":3505,"style":1024},[73,77],[22,3507],{"className":3508,"style":340},[258],[22,3510,234],{"className":3511},[344],[22,3513],{"className":3514,"style":340},[258],[22,3516,3518,3522,3617,3620,3623],{"className":3517},[64],[22,3519],{"className":3520,"style":3521},[68],"height:2.0074em;vertical-align:-0.686em;",[22,3523,3525,3528,3614],{"className":3524},[73],[22,3526],{"className":3527},[329,561],[22,3529,3531],{"className":3530},[293],[22,3532,3534,3606],{"className":3533},[86,87],[22,3535,3537,3603],{"className":3536},[91],[22,3538,3540,3551,3559],{"className":3539,"style":2662},[95],[22,3541,3542,3545],{"style":577},[22,3543],{"className":3544,"style":581},[103],[22,3546,3548],{"className":3547},[73],[22,3549,3410],{"className":3550},[73],[22,3552,3553,3556],{"style":590},[22,3554],{"className":3555,"style":581},[103],[22,3557],{"className":3558,"style":598},[597],[22,3560,3561,3564],{"style":601},[22,3562],{"className":3563,"style":581},[103],[22,3565,3567,3570,3573,3576,3579,3582,3585,3588,3591,3594,3600],{"className":3566},[73],[22,3568,307],{"className":3569},[73],[22,3571],{"className":3572,"style":412},[258],[22,3574,251],{"className":3575},[416],[22,3577],{"className":3578,"style":412},[258],[22,3580,3478],{"className":3581},[73],[22,3583],{"className":3584,"style":412},[258],[22,3586,251],{"className":3587},[416],[22,3589],{"className":3590,"style":412},[258],[22,3592,50],{"className":3593},[73],[22,3595,3597],{"className":3596},[73],[22,3598,256],{"className":3599},[432],[22,3601,2156],{"className":3602},[73],[22,3604,119],{"className":3605},[118],[22,3607,3609],{"className":3608},[91],[22,3610,3612],{"className":3611,"style":651},[95],[22,3613],{},[22,3615],{"className":3616},[336,561],[22,3618],{"className":3619,"style":340},[258],[22,3621,234],{"className":3622},[344],[22,3624],{"className":3625,"style":340},[258],[22,3627,3629,3632,3635,3641],{"className":3628},[64],[22,3630],{"className":3631,"style":2512},[68],[22,3633,1919],{"className":3634},[73],[22,3636,3638],{"className":3637},[73],[22,3639,256],{"className":3640},[432],[22,3642,2156],{"className":3643},[73]," с; ",[22,3646,3648,3692],{"className":3647},[25],[22,3649,3651],{"className":3650},[29],[31,3652,3653],{"xmlns":33},[35,3654,3655,3689],{},[38,3656,3657,3659,3661,3684,3686],{},[44,3658,177],{},[222,3660,234],{},[2554,3662,3663],{"scriptlevel":50,"displaystyle":60},[293,3664,3665,3682],{},[38,3666,3667,3670,3672,3674,3676],{},[48,3668,3669],{},"225",[222,3671,251],{},[44,3673,280],{},[222,3675,246],{},[301,3677,3678,3680],{},[48,3679,2833],{},[222,3681,1988],{},[48,3683,3410],{},[222,3685,2565],{},[48,3687,3688],{},"19,5",[52,3690,3691],{"encoding":54},"L = \\dfrac{225 \\cdot \\sin 60^\\circ}{10} \\approx 19{,}5",[22,3693,3695,3713,3839],{"className":3694,"ariaHidden":60},[59],[22,3696,3698,3701,3704,3707,3710],{"className":3697},[64],[22,3699],{"className":3700,"style":189},[68],[22,3702,177],{"className":3703},[73,77],[22,3705],{"className":3706,"style":340},[258],[22,3708,234],{"className":3709},[344],[22,3711],{"className":3712,"style":340},[258],[22,3714,3716,3720,3830,3833,3836],{"className":3715},[64],[22,3717],{"className":3718,"style":3719},[68],"height:2.0371em;vertical-align:-0.686em;",[22,3721,3723,3726,3827],{"className":3722},[73],[22,3724],{"className":3725},[329,561],[22,3727,3729],{"className":3728},[293],[22,3730,3732,3819],{"className":3731},[86,87],[22,3733,3735,3816],{"className":3734},[91],[22,3736,3739,3750,3758],{"className":3737,"style":3738},[95],"height:1.3511em;",[22,3740,3741,3744],{"style":577},[22,3742],{"className":3743,"style":581},[103],[22,3745,3747],{"className":3746},[73],[22,3748,3410],{"className":3749},[73],[22,3751,3752,3755],{"style":590},[22,3753],{"className":3754,"style":581},[103],[22,3756],{"className":3757,"style":598},[597],[22,3759,3760,3763],{"style":601},[22,3761],{"className":3762,"style":581},[103],[22,3764,3766,3769,3772,3775,3778,3781,3784,3787],{"className":3765},[73],[22,3767,3669],{"className":3768},[73],[22,3770],{"className":3771,"style":412},[258],[22,3773,251],{"className":3774},[416],[22,3776],{"className":3777,"style":412},[258],[22,3779,280],{"className":3780},[402],[22,3782],{"className":3783,"style":398},[258],[22,3785,2851],{"className":3786},[73],[22,3788,3790,3793],{"className":3789},[73],[22,3791,50],{"className":3792},[73],[22,3794,3796],{"className":3795},[82],[22,3797,3799],{"className":3798},[86],[22,3800,3802],{"className":3801},[91],[22,3803,3805],{"className":3804,"style":2022},[95],[22,3806,3807,3810],{"style":632},[22,3808],{"className":3809,"style":104},[103],[22,3811,3813],{"className":3812},[108,109,110,111],[22,3814,1988],{"className":3815},[416,111],[22,3817,119],{"className":3818},[118],[22,3820,3822],{"className":3821},[91],[22,3823,3825],{"className":3824,"style":651},[95],[22,3826],{},[22,3828],{"className":3829},[336,561],[22,3831],{"className":3832,"style":340},[258],[22,3834,2565],{"className":3835},[344],[22,3837],{"className":3838,"style":340},[258],[22,3840,3842,3845,3849,3855],{"className":3841},[64],[22,3843],{"className":3844,"style":2512},[68],[22,3846,3848],{"className":3847},[73],"19",[22,3850,3852],{"className":3851},[73],[22,3853,256],{"className":3854},[432],[22,3856,2156],{"className":3857},[73]," м; ",[22,3860,3862,3898],{"className":3861},[25],[22,3863,3865],{"className":3864},[29],[31,3866,3867],{"xmlns":33},[35,3868,3869,3895],{},[38,3870,3871,3873,3875,3890,3892],{},[44,3872,3153],{},[222,3874,234],{},[2554,3876,3877],{"scriptlevel":50,"displaystyle":60},[293,3878,3879,3888],{},[38,3880,3881,3883,3885],{},[48,3882,3669],{},[222,3884,251],{},[48,3886,3887],{},"0,25",[48,3889,2390],{},[222,3891,2565],{},[48,3893,3894],{},"2,8",[52,3896,3897],{"encoding":54},"H = \\dfrac{225 \\cdot 0{,}25}{20} \\approx 2{,}8",[22,3899,3901,3919,4018],{"className":3900,"ariaHidden":60},[59],[22,3902,3904,3907,3910,3913,3916],{"className":3903},[64],[22,3905],{"className":3906,"style":189},[68],[22,3908,3153],{"className":3909,"style":3201},[73,77],[22,3911],{"className":3912,"style":340},[258],[22,3914,234],{"className":3915},[344],[22,3917],{"className":3918,"style":340},[258],[22,3920,3922,3925,4009,4012,4015],{"className":3921},[64],[22,3923],{"className":3924,"style":3521},[68],[22,3926,3928,3931,4006],{"className":3927},[73],[22,3929],{"className":3930},[329,561],[22,3932,3934],{"className":3933},[293],[22,3935,3937,3998],{"className":3936},[86,87],[22,3938,3940,3995],{"className":3939},[91],[22,3941,3943,3954,3962],{"className":3942,"style":2662},[95],[22,3944,3945,3948],{"style":577},[22,3946],{"className":3947,"style":581},[103],[22,3949,3951],{"className":3950},[73],[22,3952,2390],{"className":3953},[73],[22,3955,3956,3959],{"style":590},[22,3957],{"className":3958,"style":581},[103],[22,3960],{"className":3961,"style":598},[597],[22,3963,3964,3967],{"style":601},[22,3965],{"className":3966,"style":581},[103],[22,3968,3970,3973,3976,3979,3982,3985,3991],{"className":3969},[73],[22,3971,3669],{"className":3972},[73],[22,3974],{"className":3975,"style":412},[258],[22,3977,251],{"className":3978},[416],[22,3980],{"className":3981,"style":412},[258],[22,3983,50],{"className":3984},[73],[22,3986,3988],{"className":3987},[73],[22,3989,256],{"className":3990},[432],[22,3992,3994],{"className":3993},[73],"25",[22,3996,119],{"className":3997},[118],[22,3999,4001],{"className":4000},[91],[22,4002,4004],{"className":4003,"style":651},[95],[22,4005],{},[22,4007],{"className":4008},[336,561],[22,4010],{"className":4011,"style":340},[258],[22,4013,2565],{"className":4014},[344],[22,4016],{"className":4017,"style":340},[258],[22,4019,4021,4024,4027,4033],{"className":4020},[64],[22,4022],{"className":4023,"style":2512},[68],[22,4025,307],{"className":4026},[73],[22,4028,4030],{"className":4029},[73],[22,4031,256],{"className":4032},[432],[22,4034,2525],{"className":4035},[73],[14,4037,4039],{"id":4038},"а-в-реальному-житті","А в реальному житті?",[10,4041,4042],{},"Через опір повітря оптимальний кут для м’яча чи ядра трохи менший за 45°. Але на НМТ опором повітря нехтують — сміливо пишіть 45°.",{"title":4044,"searchDepth":4045,"depth":4046,"links":4047},"",2,3,[4048,4049,4050,4051,4052,4053,4054,4055],{"id":16,"depth":4045,"text":17},{"id":196,"depth":4045,"text":197},{"id":659,"depth":4045,"text":660},{"id":1159,"depth":4045,"text":1160},{"id":1775,"depth":4045,"text":1776},{"id":3374,"depth":4045,"text":3375},{"id":3384,"depth":4045,"text":3385},{"id":4038,"depth":4045,"text":4039},"2026-10-14","Виводимо формулу дальності польоту тіла, кинутого під кутом до горизонту, доводимо, що максимум досягається при 45°, і розв’язуємо задачу НМТ.",false,"md",null,{},true,"\u002Fblog\u002Fkut-kydka-maksymalna-dalnist","---\ntitle: 'Рух тіла, кинутого під кутом, 10 клас: чому найдалі — під 45°'\ndescription: 'Виводимо формулу дальності польоту тіла, кинутого під кутом до горизонту, доводимо, що максимум досягається при 45°, і розв’язуємо задачу НМТ.'\ndate: '2026-10-14'\nsubject: physics\ngrade: 10\ntopic: 'Рух тіла, кинутого під кутом'\ntags: [кінематика, рух під кутом, НМТ, механіка]\n---\n\nЦя задача — ідеальний місток між фізикою і математикою: фізика дає рівняння руху, а математика знаходить, де функція досягає максимуму. Нижче в статті є інтерактивна мішень — перевірте формулу на практиці.\n\n## Умова\n\nМ’яч кидають з поверхні землі зі швидкістю $v_0$ під кутом $\\alpha$ до горизонту. Опором повітря нехтуємо. Знайти дальність польоту $L$ і кут, при якому вона максимальна.\n\n## Розкладаємо рух на дві осі\n\nПо горизонталі сили не діють — рух рівномірний. По вертикалі діє сила тяжіння — рух рівноприскорений:\n\n$$\nx(t) = v_0 \\cos\\alpha \\cdot t, \\qquad y(t) = v_0 \\sin\\alpha \\cdot t - \\frac{g t^2}{2}\n$$\n\n## Час польоту\n\nМ’яч падає на землю, коли $y(t) = 0$:\n\n$$\nt\\left(v_0 \\sin\\alpha - \\frac{g t}{2}\\right) = 0 \\quad\\Rightarrow\\quad T = \\frac{2 v_0 \\sin\\alpha}{g}\n$$\n\n## Дальність польоту\n\nПідставляємо $T$ у $x(t)$ і згадуємо формулу синуса подвійного кута $2\\sin\\alpha\\cos\\alpha = \\sin 2\\alpha$:\n\n$$\nL = v_0 \\cos\\alpha \\cdot \\frac{2 v_0 \\sin\\alpha}{g} = \\frac{v_0^2 \\sin 2\\alpha}{g}\n$$\n\n## Чому саме 45°\n\nПри заданій швидкості $v_0$ дальність залежить лише від $\\sin 2\\alpha$. Синус не перевищує 1, а $\\sin 2\\alpha = 1$ при $2\\alpha = 90^\\circ$, тобто\n\n$$\n\\alpha = 45^\\circ, \\qquad L_{max} = \\frac{v_0^2}{g}\n$$\n\nДля $v_0 = 20$ м\u002Fс і $g = 9{,}8$ м\u002Fс²: $L_{max} = \\dfrac{400}{9{,}8} \\approx 40{,}8$ м.\n\n**Цікавий наслідок:** кути $30^\\circ$ і $60^\\circ$ дають однакову дальність, бо $\\sin 60^\\circ = \\sin 120^\\circ$. Загалом однаково далеко летять тіла, кинуті під кутами $\\alpha$ і $90^\\circ - \\alpha$. А от висота польоту різна:\n\n$$\nH = \\frac{v_0^2 \\sin^2\\alpha}{2g}\n$$\n\n## Перевірте на практиці\n\nТепер, знаючи формулу, влучте в мішень — спершу розрахуйте швидкість, а потім стріляйте:\n\n::lab-target\n::\n\n## Задача для тренування\n\n> Тіло кинули зі швидкістю 15 м\u002Fс під кутом 30° до горизонту. Знайдіть час польоту, дальність і максимальну висоту. Вважайте $g = 10$ м\u002Fс².\n\n::answer\n$T = \\dfrac{2 \\cdot 15 \\cdot 0{,}5}{10} = 1{,}5$ с; $L = \\dfrac{225 \\cdot \\sin 60^\\circ}{10} \\approx 19{,}5$ м; $H = \\dfrac{225 \\cdot 0{,}25}{20} \\approx 2{,}8$ м.\n::\n\n## А в реальному житті?\n\nЧерез опір повітря оптимальний кут для м’яча чи ядра трохи менший за 45°. Але на НМТ опором повітря нехтують — сміливо пишіть 45°.\n",{"title":5,"description":4057},"blog\u002Fkut-kydka-maksymalna-dalnist","physics",[4069,4070,4071,4072],"кінематика","рух під кутом","НМТ","механіка","Рух тіла, кинутого під кутом","AeIDNBYsUIb-nHOKvODI6G4Tif-KaP2HPTnKfWIUn5s",[],1792103491839]