$A$ $2 \ kg$ block slides on a horizontal floor with a speed of $4 \ m/s$. It strikes an uncompressed spring and compresses it until the block is motionless. The kinetic friction force is $15 \ N$ and the spring constant is $10,000 \ N/m$. The spring compresses by ............. $cm$.

$A$ body $A$ moving with momentum $P$ collides one-dimensionally with another stationary body $B$ of same mass. During impact,$A$ gives impulse $J$ to $B$. Then which of the following is/are correct?
$(a)$ The total momentum of $A$ and $B$ is $P$ before and after impact and $(P-J)$ during the impact.
$(b)$ During the impact,$B$ gives impulse of magnitude $J$ to $A$.
$(c)$ The coefficient of restitution is $\left[\frac{2 J}{P}-1\right]$.
$(d)$ The coefficient of restitution is $\left[\frac{2 J}{P}+1\right]$.

If a skater of mass $3 \,kg$ has an initial speed of $32 \,m/s$ and a second skater of mass $4 \,kg$ has a speed of $5 \,m/s$. After a perfectly inelastic collision,they move together with a speed of $5 \,m/s$. Calculate the loss in kinetic energy.

$A$ small block of mass $M$ moves on a frictionless surface of an inclined plane,as shown in the figure. The angle of the incline suddenly changes from $60^{\circ}$ to $30^{\circ}$ at point $B$. The block is initially at rest at $A$. Assume that collisions between the block and the incline are totally inelastic $\left(g=10 \ m/s^2\right)$.
$1.$ The speed of the block at point $B$ immediately after it strikes the second incline is
$(A) \sqrt{60} \ m/s$ $(B) \sqrt{45} \ m/s$ $(C) \sqrt{30} \ m/s$ $(D) \sqrt{15} \ m/s$
$2.$ The speed of the block at point $C$,immediately before it leaves the second incline is
$(A) \sqrt{120} \ m/s$ $(B) \sqrt{105} \ m/s$ $(C) \sqrt{90} \ m/s$ $(D) \sqrt{75} \ m/s$
$3.$ If the collision between the block and the incline is completely elastic,then the vertical (upward) component of the velocity of the block at point $B$,immediately after it strikes the second incline is
$(A) \sqrt{30} \ m/s$ $(B) \sqrt{15} \ m/s$ $(C) 0$ $(D) -\sqrt{15} \ m/s$
Give the answers for questions $1, 2,$ and $3.$

$A$ free body of mass $8 \, kg$ is travelling at $2 \, m/s$ in a straight line. At a certain instant,the body splits into two equal parts due to an internal explosion which releases $16 \, J$ of energy. Neither part leaves the original line of motion. Finally:

$A$ block of mass $5 \ kg$ is lifted to a height of $5 \ m$ by a force of $60 \ N$. Find:
$(1)$ Work done by the force.
$(2)$ Potential energy of the block at $5 \ m$.
$(3)$ Kinetic energy of the block at $5 \ m$.
$(4)$ Velocity of the block at $5 \ m$. (Take $g = 9.8 \ m/s^2$)