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?

Answer

A, B

Explanation

Two frieds A and B (each 40 kg ) ar sitting on a friction less platform some distance d apart. A rolls a ball of mass 4 kg on the platform towards B, which B cathces. Then B rolls the ball towards A and A cathces it. The ball keeps on moving back and forth between A and B. The ball has a fixed velocity 5 m/s.
a. Case I: Total momentum of the man Aand ball will remain constant
`:. 0=4xx5-40xxv`
`rarr v=0.5`m/s towards Left
b. Case Ii: When B catches the ball the momentum betweenteh B and teh ball will remin constant.
`rarr 4xx5=44v`
`rarr v=(20/44)m/s`
Case III: When B throws the ball then applying L.C.L.M.
`rarr 44xx(20/44)=-4xx5+40v`
`rarr v=1m/s(towards right)`
Case IV: When A catches the ball then applying L.C.L.M :
`rarr -4xx5+(-0.5)xx10=44v`
`_gt v=40/44=10/11 m/s` towards left
c. Case V: When A throws the balls then applying L.C.L.M.
`rarr 44xx(10/11)=4xx5+40xxv`
`rarr v=60/40=3/2m/s` (towards left)
Case VI: When B receives the ball then applying L.C.L.M
`rarr 40xx1+4xx5=44xxv`
`rarr v=60/44`m/s towards right
Case VII : When B throws th ball, then applying L.C.L.M.
`rarr v=44xx(60/44)`m/s towards right
Case VIII: When a catchs the ball tehn applying L.C.L.M
`-4xx5+40(3/2)=-44upsilon`
`rarrV=-40/44`
=(10/11)m/s` towards left Similarly after 5 round trips
`The velocity of A will be (50/11)` and velocity of B will be 5 m/s.

d. since after 6 round trips, the velocity of A is `60/11` i.e.gt5m/s. so it can't catch tehball. So it can only roll theball six times.
e. Let the ball and tehbody A at the initial position be at origin.
`:. X_C=(40xx0+4xx040xxd)/(4040+4)`
`=10/21d`