Calculate the acceleration (in $m/s^{2}$) of the block and trolley system shown in the figure. The coefficient of kinetic friction between the trolley and the surface is $0.05$. (Given: $g = 10\; m/s^{2}$,mass of the string is negligible and no other friction exists).

In the figures $(i), (ii)$ and $(iii)$ shown, the objects $A, B$ and $C$ have the same mass $m$. The string, spring, and pulley are massless. Object $C$ strikes object $B$ with velocity $u$ in each case and sticks to it. Find the ratio of the velocity of $B$ in case $(i)$ to $(ii)$ to $(iii)$.

$A$ man can move on a horizontal plank supported symmetrically as shown. The variation of normal reaction on support $A$ with distance $x$ of the man from the end of the plank is best represented by:

An object of mass $10 \,kg$ is projected from the ground with a speed of $40 \,m/s$ at an angle of $60^{\circ}$ with the horizontal. The rate of change of momentum of the object one second after projection in $SI$ units is ........ [Take $g = 9.8 \,m/s^2$]

$A$ system consists of three masses $m_1, m_2$ and $m_3$ connected by a string passing over a pulley $P$. The mass $m_1$ hangs freely and $m_2$ and $m_3$ are on a rough horizontal table (the coefficient of friction $= \mu$). The pulley is frictionless and of negligible mass. The downward acceleration of mass $m_1$ is (Assume $m_1 = m_2 = m_3 = m$)

$A$ bullet of mass $20 \, g$ travelling horizontally with a speed of $500 \, m/s$ passes through a wooden block of mass $10.0 \, kg$ initially at rest on a surface. The bullet emerges with a speed of $100 \, m/s$ and the block slides $20 \, cm$ on the surface before coming to rest. Find the coefficient of friction between the block and the surface. $(g = 10 \, m/s^2)$