A light in extensible thread passes over a small frictionless pully. Two blocks of masses `m = kg` and `M = 3kg` respectively are attached with the thread as shown in the fig. The heavier block rests on a horizontal surface. A shell of mass `1 kg` moving upward with a velocity `10m//s` collides and sticks with the block of mass `'m'` as shown in the fig at `t = 0`. If the long inclined plane is smooth.

Find the maximum height ascended by `'M'`

Options

A. `(1)/(4)m`
B. `(1)/(2)m`
C. `1m`
D. `(1)/(6) m`

Answer

B

Explanation

`(m+1)u = 1 xx 10`
`u = 5m//s`
So, the combined body will move up the plane and after a certain time it will come back to its initial position with a velocity `5m//s` and time `t^(1) = 2s`. Now `M` is jerked into motion thread become taut, a sharp impluse is exerted by thread to `M` and combined body.
So `(m+1) v_(o) = (m+1) u-M v_(o)`
`therefore V_(o) = 2m//s`
Now, `M` will move with retardation from free body diagrams
`T - (m+1)g, sin 30 = (m+1)a`
`Mg - T = Ma`
`a = 4m//s^(2)`
Maximum height ascended by `M`
`v^(2) - u^(2) = 2 as`
`0-4 = -2 (4) xx H`
`H = (1)/(2)m`
time taken to reach that height
`v = u+at, 0 = 5-4t , t = 0.5s`
Total time `= t' + t = 2+0.5 = 2.5S`