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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=4×540×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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