Question

Chuck and Jackie stand on separate carts, both of which can slide without friction. The combined mass of Chuck and his cart, mcart, is identical to the combined mass of Jackie and her cart. Initially, Chuck and Jackie and their carts are at rest. Chuck then picks up a ball of mass mball and throws it to Jackie, who catches it. Assume that the ball travels in a straight line parallel to the ground (ignore the effect of gravity). After Chuck throws the ball, his speed relative to the ground is vc. The speed of the thrown ball relative to the ground is vb. Jackie catches the ball when it reaches her, and she and her cart begin to move. Jackie's speed relative to the ground after she catches the ball is vj. When answering the questions in this problem, keep the following in mind: The original mass mcart of Chuck and his cart does not include the mass of the ball. The speed of an object is the magnitude of its velocity. An object's speed will always be a nonnegative quantity. Find the relative speed u between Chuck and the ball after Chuck has thrown theball.
Express the speed in terms ofv_c and v_b.

Answer 1

As per momentum conservation we can say that Chuck and ball as a system has no external force

So here momentum of them must be conserved

now here initially if they are at rest

so we can say

$P_i = P_f$

$0 = m_{cart}v_c + m_{ball} v_b$

now we have

$v_b = \frac{m_{cart}v_c}{m_{ball}$

now the relative velocity of the cart and ball is given as

$v_r = v_c + \frac{m_{cart}v_c}{m_{ball}$

also we can say since they are moving in opposite direction so relative speed will be

$v_r = v_c + v_b$