Two friends `A` and `B` (each weighing `40 kg`) are sitting on a frictionless platform some distance d apart. A rolls a ball of mass `4 kg` on the platform towards `B` which `B` catches. Then `B` rolls the ball towards `A` and `A` catches it. The ball keeps on moving back and forth between `A` and `B`. The ball has a fixed speed of `5 m//s` on the platform.
a. Find the speed of `A` after he rolls the ball for the first time.
b. Find the speed of `A` after he catches the ball for the first time.
c. Find the speed of `A` and `B` after the ball has made five round trips and is held by `A`.
d. How many times can `A` roll the ball?
e. Where is the centre of mass of the system `A + B +` ball at. the end of the nth trip?

Explanation

After Every throw `A` or `B` gains a momentum of `20 kg m//s` in the direction opposite to the motion of the ball.
First round

Second Round

At the end of each round, both men receive a momentum `40 kg m//s`
a. `pA=20=40 v_(A)`
`v_(A)=1/2m//s`
`b. p_(A,1)=40=(40+4)v_(A,1)`
`v_(A,1)=40/44=10/11m//s`
c. At the end of five round trips, the linear momentum of (ball `+ A` ) system,
`p_(A,S)=40xx5=200kgm/s`
`200=(40+4)v_(A,5)`
`V_(A,5)=50/11m/s`
Momenum transfered to `B` in five trips
`=40xx5=200impliesv_(B,5)=5m//s`
d. A can catch the ball until the velocity of `A` is less than `5 m/s`. after fie rounds when `A` throuws the balls, i.e., `6 th` time, the velocity of `A` becomes greater than `5m//s` hence `A` can throw the ball only six times.
e. `A` no external force is action on As, `B` and ball system hence the postion of centre of mass of the system should remain constant always. Let the centre of mass be at a distance `x` cm from the position of `A`.
`(40+4+10)x_(CM)=40xx0+4xx0+40d`
`x_(CM)=(10d)/21m` from initial position of `A`.