$A$ motor drives a body along a straight line with a constant force. The power $P$ developed by the motor must vary with time $t$ according to

$A$ ball is falling freely from a height. When it reaches $10 ~m$ height from the ground,its velocity is $v_0$. It collides with the ground,loses $50 \%$ of its energy,and rises back to a height of $10 ~m$. Then the velocity $v_0$ is (in $~m / s$)

The velocity of the bullet becomes one-third after it penetrates $4 \ cm$ in a wooden block. Assuming that the bullet faces a constant resistance during its motion in the block,the bullet stops completely after traveling a total distance of $(4+x) \ cm$ inside the block. The value of $x$ is $.....$ (in $cm$)

$A$ ball of mass $1 \ kg$ moving with a velocity of $0.4 \ m/s$ collides with another ball of the same mass at rest. After the collision,the first ball moves with a velocity of $0.3 \ m/s$ in a direction perpendicular to the initial direction. Find the magnitude of the momentum $\vec{P}$ of the second ball in $kg \cdot m/s$.

Consider the following statements $A$ and $B$. Identify the correct choice in the given answers.
$A$. In an inelastic collision,there is no loss in kinetic energy during collision.
$B$. During a collision,the linear momentum of the entire system of particles is conserved if there is no external force acting on the system.

In a perfectly inelastic collision,two spheres made of the same material with masses $15 \ kg$ and $25 \ kg$,moving in opposite directions with speeds of $10 \ m/s$ and $30 \ m/s$,respectively,strike each other and stick together. The rise in temperature (in $^{\circ}C$),if all the heat produced during the collision is retained by these spheres,is: (specific heat of sphere material $31 \ cal/kg \cdot ^{\circ}C$ and $1 \ cal = 4.2 \ J$)