To simulate car accidents,auto manufacturers study the collisions of moving cars with mounted springs of different spring constants. Consider a typical simulation with a car of mass $1000 \; kg$ moving with a speed $18.0 \; km/h$ on a rough road having $\mu = 0.5$ and colliding with a horizontally mounted spring of spring constant $6.25 \times 10^{3} \; N m^{-1}$. What is the maximum compression of the spring in $m$?

In the figure shown,two identical balls of mass $M$ and radius $R$ each are placed in contact with each other on a frictionless horizontal surface. $A$ third ball of mass $M$ and radius $R$ is coming down vertically with a velocity $v_0$. It simultaneously hits the two balls. The smaller ball (top ball) does not stop after the collision but continues to move downwards with a speed $v_0/2$. Find the speed of each bigger ball after the collision.

Which of the following statements is $NOT$ true?

$A$ 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 $\left[\frac{2 J}{P}+1\right]$.

Two bodies of masses $m_{1}$ and $m_{2}$ are acted upon by a constant force $F$ for a time $t$. They start from rest and acquire kinetic energies,$E_{1}$ and $E_{2}$ respectively. Then $\frac{E_{1}}{E_{2}}$ is

$A$ person trying to lose weight by burning fat lifts a mass of $10 \ kg$ up to a height of $1 \ m$, $1000$ times. Assume that the potential energy lost each time he lowers the mass is dissipated. How much fat will he use up considering the work done only when the weight is lifted up? Fat supplies $3.8 \times 10^7 \ J$ of energy per $kg$, which is converted to mechanical energy with a $20\%$ efficiency rate. Take $g = 9.8 \ m/s^2$.