I am having difficulty in conserving momentum for objects moving with relative velocities. For example:
A block of mass $m$ with a semicircular track of radius $r$ rests on a frictionless horizontal surface. A ball of radius $R$ and mass $M$ is released from the top point. What is the speed of the block when the ball has reached the bottom of the track?
The ball falls from a height of $r$ to $(r-R)$. Let the velocities of block and ball be $v_2$ and $v_1$ resp (from the ground frame).
By energy conservation: $$Mg(r-R)=\left(\frac12mv_2^2+\frac12Mv_1^2\right)$$
By momentum conservation: $$0=mv_2+Mv_1$$
Am I wrong here?
Let final velocity of ball wrt to block be v. Let final velocity of block be v2(wrt ground). by energy conservation(wrt block):
By momentum conservation: $$0=mv_2+M(v+v_2)$$
Will the answer be same, or am I wrong somewhere?