An object falls from a height of $20 \ cm$ on the surface of the Earth. If $75\%$ of its mechanical energy is lost after striking the Earth,the object will rebound to a height of ......... $cm$.

Three objects $A, B$ and $C$ are kept in a straight line on a frictionless horizontal surface. The masses of $A, B$ and $C$ are $m, 2m$ and $2m$ respectively. $A$ moves towards $B$ with a speed of $9 \ m/s$ and makes an elastic collision with it. Thereafter,$B$ makes a completely inelastic collision with $C$. All motions occur along the same straight line. The final speed of $C$ is $.... \ m/s$.

Two blocks $A(5 \ kg)$ and $B(2 \ kg)$ attached to the ends of a spring of spring constant $k = 1120 \ N/m$ are placed on a smooth horizontal plane with the spring undeformed. Simultaneously,velocities of $3 \ m/s$ and $10 \ m/s$ along the line of the spring in the same direction are imparted to $A$ and $B$. Then:

The diagram shows the velocity-time graph for two masses $R$ and $S$ that collided elastically. Which of the following statements is true?
$(I)$ $R$ and $S$ moved in the same direction after the collision.
$(II)$ Kinetic energy of the system $(R \text{ & } S)$ is minimum at $t = 2 \text{ ms}$.
$(III)$ The mass of $R$ was greater than the mass of $S$.

Which physical quantity is conserved in both elastic and inelastic collisions?

$A$ 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: