A small block of mass `2m` initially rests at the bottom of a fixed circular, vertical track, which has a radius of `R`. The contact surface between the mass and the loop is frictionless. `A` bullet of mass `m` strikes the block horizontally with initial speed `v_(0)` and remain embedded in the block as the block and the bullect circle the loop. Determine each of the following in terms of `m,R` and `g`.
(a) The speed of the masses immediately after the impact.
(b) The minimum initial speed of the bullet if the block and the bullet are to successfully execute a complete ride on the loop

Answer

(a)`v_(0)//3,(b)3sqrt(5gR)`

Explanation

Using conservaion of momentum
`mv_(0)+2m(0)=3mv`
`rArr" "v=(v_(0))/(3)`
(b) `v=sqrt(5gR)" "rArr(v_(0))/(3)=sqrt(5gR)" "rArr" "v_(0)=3sqrt(5gR)`