1.

A ball falls on an inclined plane as shown. The ball is dropped from height h. Coefficient of restitution for collision is e and the surface is frictionless. If h_1, h_2,......h_nare heights of projectile from inclined and t_1, t_2,....t_nare their corresponding time of flights, then choose the correct options.

Answer»

`t_1, t_2, ……t_n` FORM a geometric progression of COMMON ratio e
`h_1 gt h_2 gt h_3 gt ….. gt h_n`
`t_1, t_2,……t_n` form a geometric progression of common ratio `e^2`
`h_1, h_2 ……..h_n`form a geometric progression of common ratio `e^2`

Solution :First TIME it COLLIDES with velocity`u = sqrt(2 gh)`, after collision,`a_x = g sin THETA, a_y = - g cos theta`
`u_(x_1) = u sin theta & u_(y_1) = e u sin theta`
As between two successive collision, Sy = 0
`implies `(v_y = u_y) "" implies u_(y_2) = e^2u cos theta`
`u_(y_3) = e^3 u cos theta.... and so on.
`implies t_(n) = (2 e^n u cos theta)/(g cos theta) = e^n (2u)/g`..... common ratio = e
`& ""h_(n) = ((u_(yn))^(2))/(2a_y) = (e^(2n) u^2 cos theta)/(2 g) implies common ratio = e^2`.


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