A (trolley$+$child) of total mass $200 \mathrm{kg}$ is moving with a uniform speed of $36 \mathrm{km} {\mathrm{h}}^{-1}$ on a frictionless track. The child of mass $20 \mathrm{kg}$ starts running on the trolley from one end to the other $(10 \mathrm{m} \text{away})$ with a speed of $10 \mathrm{m} {\mathrm{s}}^{-1}$ relative to the trolley in the direction of the trolley’s motion and jumps out of the trolley with the same relative velocity. What is the final speed of the trolley? How much has the trolley moved from the time the child begins to run and just before jump?

A small cube of mass $m$ slides down a circular path of radius $R$ formed from a large block of mass $M$ as shown in the figure $M$ rests on a table and both blocks move without friction. The blocks are initially at rest and $m$ starts from the top of the path. Find the velocity $v$ of the cube as it leaves the block. Initially the line joining m and the centre is horizontal.

A small block of mass $m$ moving with speed $v$ on a smooth horizontal part of a bigger block of mass $M$ which is further kept on smooth floor. The curved part of the surface shown is one fourth of a circle. Find the speed of the bigger block when the smaller block reaches the point $A$ of the surface.

Two block of masses $m_1 \text{and} m_2$ are connected with the help of a spring of spring constant $k$ initially the spring in its natural length as shown. A sharp impulse is given to mass $m_2$ so that it acquires a velocity $v_0$ towards right. If the system is kept on smooth floor then find

(a) the velocity of the centre of mass,
(b) the maximum elongation that the spring will suffer?

A ball of mass $100 \mathrm{g}$ moving with a speed of $4 \mathrm{m} {\mathrm{s}}^{-1}$ strikes a horizontal surface at an angle of $30°$ from the surface. The ball is reflected back with same speed and same angle of reflection find

(a) The impulse imparted to the ball,

(b) change in magnitude of momentum of the ball.