24. (II) A brother and sister live near a lake. In the middle of winter when the

Question

24. (II) A brother and sister live near a lake. In the middle of winter when the lake is solidly and smoothly frozen, they play catch by sliding a rock back and forth between them across the ice. Ignore all friction. The mass of the two children is 40 kg each, and the rock has a mass of 2.0 kg. Each time the rock is given a push by the child who has just caught it, it leaves the hand with a speed of 1.0 m/s (relative to the hand). (a) The children are initially at rest, and the sister first slides the rock toward her brother. What is the initial motion of the sister after she releases the rock? (b) The brother receives the rock. What is his initial velocity after he has received it? (c) The brother then slides the rock to his sister. What is his velocity now? (d) The sister catches the rock. What is her velocity now? (e) Will either child ever havea speed such that the rock will never reach the other player? In other words, can the game go on forever?

Explanation / Answer

We assume Intially sister is on the left and Br. is on the right both at rest. we take velcoity and momentum to the right as +ve and to left as -ve.

a) Si. pushes the rock with a vel. of 1m/s to the right

momentum of the rock = 2*1 = 2 kg-m/s

conserving the momentum , momentum of sis. = -2

velocity of sis = -2/40 = -0.05 m/s

b) Now the boy on the right recives the rock , initially at rest.

momentum of the rock = 2 kg-m/s

total momentum of the boy and the rock after he recives the rock = 2 kg-m/s

velocity = 2/42 m/s

Now when the boy pushes the rock to the left (to sis.) its momentum = -2

Total change in momentum of the rock = -4

c) conseerving the total momnetum, momentum of the boy = 4

velocity of the Br. = 4/40 = 0.1 m/s

d) intial momentum of sis. = -2

    momentum of the rock when she recieved it = -2

   Total momentum = -4

vel. fo sis +rock = -4/42 = -2/21 m/s

e) with each push the vel. of the child is increasing in opposite direction, sis. to the left and the br. to the right, at some stage velocity of one of the children reaches equal to that of the rock and they will never be able to recive the rock and push it back and the game cannot continue any more.

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