Question

# Two friends A and B (each weighing 40 kg) are 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 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 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 ?

Solution

## Two friends A and B (each 40 kg) are 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 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. (a) Case-I:Total momentum of the man A and ball will remain constant ∴          0=4×5−40×v ⇒v=0.5   ms towards left (b) Case-II:When B catches the ball,the  momentum between the B and the ball will  remain constant. ⇒4×5=44 v ⇒v=(2044)ms Case-III:When B throws the ball,then applying L.C.L.M: ⇒44×(2044)=−4×5+40×v ⇒   v=1 ms (towards right)  Case-IV:When A catches the ball,then applying L.C.L.M: ⇒−4×5+(−0.5)×40=44 v ⇒   v=4044=1011 ms towards left (c)Case-V:When A throws  the ball,then applying L.C.L.M: ⇒   44×(1011)=4×5+40×v ⇒ v=6040=32 ms towards left Case-VI:When B receives the ball,then applying L.C.L.M: ⇒40×1+4×5=44×v ⇒v=6044 ms towards right Case-VII:When B throws the ball,then applying L.C.L.M: ⇒v=44×6044 ms towards right Case-VIII:When A catches the ball,then applying L.C.L.M: ⇒−4×5+40(32)=−44v ⇒V=−4044          =(1011) ms towards left Similarly after 5 round trips The velocity of A will be (5011) and velocity of B will be 5 ms. (d) Since after 6 round trips,the velocity of A is 6011i.e.>5 ms.So it can't catch the ball.So it can only roll the ball six times. (e) Let the ball and the body A at the initial position be at origin. ∴          Xc=40×0+4×0+40×d40+40+4                                            =1021d

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