A horizontal frictionless rod is threaded through a bead of mass \( m \). The length of the cart...

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A horizontal frictionless rod is threaded through a bead of mass \( m \). The length of the cart is \( L \) and the radius of the bead, \( r \), is very small in comparison with \( L(rL) \). Initially at \( (t=0) \) the bead is at the right edge of the cart. The cart is struck and as a result, it moves with velocity \( v_{0} \) towards right. When the bead collides with the carts walls, the collisions are always completely elastic.
Velocity of bead just after the first collision is
(1) \( \frac{-m v_{0}}{m+M} \)
(2) \( \frac{M v_{0}}{m+M} \)
(3) \( \frac{M-m}{M+m} v_{0} \)
(4) \( \frac{2 M}{m+M} v_{0} \)
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