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Question

Two friends A and B (each weighing 40kg) are sitting on a frictionless platform some distance d apart. A rolls a ball of mass 4kg 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 5m/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 speeds of A and B after the ball has made 5 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?

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Solution

Two friends A & B (each 40kg) are sitting on a frictionless platform some distance d apart A rolls a ball of mass 4kg 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 & forth between A and B. The ball has a fixed velocity 5m/s.

a) Case - I Total momentum of the man A & the ball will remain constant0=4×540×v v=0.5m/s towards left sluts

b) Case - II When B catches the ball, the momentum between the B & the ball will remain constant.
4×5=44vv=(20/44)m/s
Case - III:- When B throws the ball, then applying L.C.L.M
44×(20/44)=4×5+40×v v=1m/s(towardsright)

c) Case - IV :- When a Catches the ball, the applying L.C.L.M.
4×5+(0.5)×40=44v v=1011m/s towards left. c) Case - V:- When A throws the ball, then applying L.C.L.M.
44×(10/11)=4×540×V V=60/40=3/2m/s towards left.
Case - VI:- When B receives the ball, then applying L.C.L.M
40×1+4×5=44 timesv v=60/44m/s towards right
Case -VII:- When B throws the ball, then applying L.C.L.M.
44×(66/44)=4×5+40×V
V=80/40=2m/s towards right.
Case - VIII :- When A catches the ball, then applying L.C.L.M
4×540×(3/2)=44v v=(80/44)=(20/11)m/s towards left.
Similarly after 5 round trips.
The velocity of A will be (50/11) & velocity of B will be 5 m/s.

d) Since after 6 round trip, the velocity of A is 60/11 i.e. >5m/s. So, it can't catch the ball. So it can only roll the ball six.

e) Let the ball & the body A at the initial position be at origin.
Xc=40×0+40×0+40×d40+40+4=1011d

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