In the situation discussed above, find
(a) velocity of combined mass just after collision at the bottommost point `(or u)`.
(b) loss of mechanical energy during collision.
(c) minimum value of `v_0` so that the combined mass completes the vertical circular motion.

Answer

A, B

Explanation

(a) Appying conservation of linear momentum, just before and just after collision.
`p_i=p_fimpliesmv_0=(M+m)uimpliesu=(mv_0)/(M+m)`

(b) Loss of mechanical energy during collision,
`=E_i-E_f` (E=mechanical energy)
`=1/2mv_0-1/2(M+m)u^2`
`=1/2mv_0^2-1/2(M+m)[(mv_0)/(M+m)]^2`
`=1/2(Mmv_0^2)/((M+m))`
(c) For completing the vertical circular motion, velocity at bottommost point, `ugesqrt(5gl)`
`:. (mv_0)/(M+m)gesqrt(5gl)implies:. v_0ge(M+m)/(m)sqrt(5gl)`
or `(v_0)_(m "in")=(M+m)/(m)sqrt(5gl)`