$A$ bullet of mass $10 \,g$ is fired horizontally with a velocity $1000 \,ms^{-1}$ from a rifle situated at a height $50 \,m$ above the ground. If the bullet reaches the ground with a velocity $500 \,ms^{-1}$, the work done against air resistance in the trajectory of the bullet is : $(g=10 \,ms^{-2})$ (in $\,J$)

$A$ man who is running has half the kinetic energy of a boy of half his mass. The man speeds up by $1 \, m/s$ and then has the same kinetic energy as the boy. The original speed of the man was:

In a one-dimensional collision between two particles,their relative velocity is $\vec{v}_1$ before the collision and $\vec{v}_2$ after the collision.

$A$ rifle bullet loses $\left(\frac{1}{20}\right)^{th}$ of its velocity in passing through a plank. Assuming that the plank exerts a constant retarding force, the least number of such planks required just to stop the bullet is .............

$A$ body of mass $0.5 \ kg$ is supplied with a power $P$ (in watt) which varies with time $t$ (in second) as $P = 3t^2 + 3$. If the velocity of the body at time $t = 0$ is zero,then the velocity of the body at time $t = 3 \ s$ is (in $ms^{-1}$)

Three blocks $A, B$ and $C$ are kept as shown in the figure. The coefficient of friction between $A$ and $B$ is $0.2$,$B$ and $C$ is $0.1$,and $C$ and the ground is $0.0$. The masses of $A, B$ and $C$ are $3\, kg, 2\, kg$ and $1\, kg$ respectively. $A$ is given a horizontal velocity of $10\, m/s$. Blocks $A, B$ and $C$ always remain in contact and move together as a single system. The total work done by friction will be ........ $J$.