A cart of mass \( M \) is kept on a smooth horizontal surface. As shown in figure a horizontal smooth massless rod \( A B \) of length \( L \) is threaded through a bead of mass \( m \), whose radius \( r \) is very small in comparison to \( L \). Initially the bead is at end \( A \) of the rod. Now cart is given an impulse such that it starts moving with speed \( v_{0} \). If all the collisions between bead and cart's walls are elastic, then choose the correct statement(s). After first collision the speed of cart in frame of centre of mass of system is \( \frac{m v_{0}}{m+M} \) Time spent between first and second collision is \( \frac{L}{v_{0}} \) Time spent between first and second collision is \( \frac{2 L}{v_{0}} \)

Question

A cart of mass \( M \) is kept on a smooth horizontal surface. As shown in figure a horizontal smooth massless rod \( A B \) of length \( L \) is threaded through a bead of mass \( m \), whose radius \( r \) is very small in comparison to \( L \). Initially the bead is at end \( A \) of the rod. Now cart is given an impulse such that it starts moving with speed \( v_{0} \). If all the collisions between bead and cart's walls are elastic, then choose the correct statement(s). After first collision the speed of cart in frame of centre of mass of system is \( \frac{m v_{0}}{m+M} \) Time spent between first and second collision is \( \frac{L}{v_{0}} \) Time spent between first and second collision is \( \frac{2 L}{v_{0}} \)