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_n` are heights of projectile from inclined and `t_1, t_2,....t_n` are their corresponding time of flights, then choose the correct options.

Options

A. `t_1, t_2, ……t_n` form a geometric progression of common ratio e
B. `h_1 gt h_2 gt h_3 gt ….. gt h_n`
C. `t_1, t_2,……t_n` form a geometric progression of common ratio `e^2`
D. `h_1, h_2 ……..h_n` form a geometric progression of common ratio `e^2`

Answer

A, B, D

Explanation

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`.