A cart of mass `3m` is on a smooth horizontal ground. The cart surface consists of an incline plane `AB` and horizontal surface `BD` both of which are frictionless. The cart is initially at rest and a small block of mass `m` is released from point `A`. Assume that the corner `B` is rounded so that mass in passes it smoothly. The length of part `BD` of car be `L` and the vartical height of incline plane `AB` be `h` as shown in digram

Select correct option(s)

Options

A. Time taken by the block to slide from `B` to `D` on the cart is `sqrt((3L^(2))/(8gh))`
B. Time taken by the block to slide from `B` to `D` on the cart is `sqrt((L^(2))/(8gh))`
C. component of momentum (in horizontal direction) of system is conserved from `A` to `B`
D. component of momentum (in vertical direction) of system is conserved from `A` to `B`

Answer

A, C

Explanation

Conversation of momentum.
`0 = -mv_(1) + 3mv_(2)`
`v_(1) = 3v_(2)`
`mgh = (1)/(2)mv_(1)^(2) + (1)/(2)3mv_(2)^(2)`
`2gh = v_(1)^(2) + 3v_(2)^(2)`
`2gh = 9v_(2)^(2) + 3v_(2)^(2)`
`v_(2) = sqrt((gh)/(6))` and `v_(1) = sqrt((3gh)/(2))`

`t = (L)/(V_(1) + V_(2)) = (L)/(sqrt((3gh)/(2)) + sqrt((gh)/(6))) rArr t = sqrt((3L^(2))/(8gh))`