The figure shows a man of mass $m$ standing at the end $A$ of a trolley of mass $M$ placed at rest on a smooth horizontal surface. The man starts moving towards the end $B$ with a velocity $u_{rel}$ with respect to the trolley. The length of the trolley is $L$. The distance moved by the man with respect to ground is

$A$ piece of wood of mass $0.03\, kg$ is dropped from the top of a $100\, m$ height building. At the same time,a bullet of mass $0.02\, kg$ is fired vertically upward,with a velocity $100\, ms^{-1}$,from the ground. The bullet gets embedded in the wood. Then the maximum height to which the combined system reaches above the top of the building before falling below is ........ $m$. $(g = 10\, ms^{-2})$

$A$ body of mass $2 \,kg$ is moving towards north with a velocity of $20 \,m \,s^{-1}$ and another body of mass $3 \,kg$ is moving towards east with a velocity of $10 \,m \,s^{-1}$. The magnitude of the velocity of the centre of mass of the system of the two bodies is

Two particles of masses $2 \ kg$ and $3 \ kg$ are moving along the $X$-axis with velocities $3 \ m/s$ and $2 \ m/s$ respectively. The velocity of the center of mass of this system is ........ $m/s$.

$A$ man of mass $m$ is suspended in air by holding the rope of a balloon of mass $M$. As the man climbs up the rope,the balloon

Two blocks of masses $2 \text{ kg}$ and $1 \text{ kg}$ respectively, are tied to the ends of a string which passes over a light frictionless pulley as shown in the figure below. The masses are held at rest at the same horizontal level and then released. The distance traversed by the centre of mass in $2 \text{ s}$ is . . . . . . $\text{m}$. (Take $g = 10 \text{ m/s}^2$)