A bag of sand of mass $M$ is suspended by a string. A bullet of mass $m$ is fired at it with velocity $v$ and gets embedded into it. The loss of kinetic energy in this process is

The diagram to the right 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 $(R$ & $S)$ is minimum at $t = 2$ milli sec.

$(III)$ The mass of $R$ was greater than mass of $S.$

A mass $m$ moving horizontally with velocity $v_0$ strikes a pendulum of mass $m$. If the two masses stick together after the collision, then the maximum height reached by the pendulum is

State if each of the following statements is true or false. Give reasons for your answer.

$(a)$ In an elastic collision of two bodies, the momentum and energy of each body is conserved.

$(b)$ Total energy of a system is always conserved, no matter what internal and external forces on the body are present.

$(c)$ Work done in the motion of a body over a closed loop is zero for every force in nature.

$(d)$ In an inelastic collision, the final kinetic energy is always less than the initial kinetic energy of the system.

A block of mass $0.50\, kg$ is moving with a speed of $2.00\, ms^{-1}$ on a smooth surface. It strikes another mass of $1.00\, kg$ and then they move together as a single body. The energy loss during the collision is ………….. $\mathrm{J}$