Lançamento na horizontal de uma bola

Lançamento na Horizontal
Considere um móvel P lançado horizontalmente nas proximidades da superfície terrestre.[br]Vamos desprezar a resistência do ar. O movimento de P pode ser considerado como a composição de dois movimentos, um horizontal (P[sub]x[/sub]) e outro vertical (P[sub]y[/sub]). [br]O móvel vai sair do repouso : vo = 0 m/s[br]i) Na horizontal tem-se M.U. de função horária:    x = vo*tii) Na vertical tem-se M.U.V.   de funções horárias da velocidade v = f(t) e de y = f(t)[br]a) Da posição:   y = - (g/ 2) * t^2 + h    b) Da velocidade: vy =-  g*t    [br][br]iii) Cálculo do tempo de queda (tq)t = tq quando y = 0 è   h = (g/2) t^2 [b][u]tq = sqrt( 2*h/g)[/u][/b][b][u]iv) Alcance máximo (D)[/u][/b]x = D para t = tq è D = vo * t v) Em cada instante m-se que[img]data:image/png;base64,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[/img][br][br]vi) Foram feios os seguintes gráficos no Geogebra[br][br]a) sx = f(t) , b) sy = f(t) [br]c) s = f(x) usando as equações paramétricas sx e sy, com o seguinte comando:[br][br]CURVA = (sx=f(t) , sy = f(t) , t , 0 , 5)[br]O gráfico s = f(x) permite mostrar o alcance máximo obtido no lançamento. enquanto os gráficos de sx = f(t), sy = f(t) e vy = f(t) permite analisar o movimento realizado.[br][br]

Information: Lançamento na horizontal de uma bola