$A$ wooden block of mass $M$ is suspended by a cord and is at rest. $A$ bullet of mass $m$,moving with a velocity $v$,passes through the block and emerges with a velocity $v/2$ in the same direction. If the block rises to a height $H$,calculate the value of $H$.

State whether the following statements are true or false:
$(a)$ If $\overrightarrow P \cdot \overrightarrow Q = 0$,then the angle between $\overrightarrow P$ and $\overrightarrow Q$ is $0^o$.
$(b)$ If two bodies stick together after a collision,the collision is called an elastic collision.
$(c)$ When an equal force is applied to a heavy and a light object on the same surface,more work is done by the force on the heavy object.

The potential energy in joules of a particle of mass $1\, kg$ moving in a plane is given by $U = 3x + 4y$,where the position coordinates $x$ and $y$ are measured in metres. If the particle is initially at rest at $(6, 4)$,then:

Two balls at the same temperature collide. What is conserved?

Two bodies of masses $m$ and $2m$ are attached to the two ends of an ideal spring. The spring is compressed. The total energy stored in the spring is $60 \ J$. If the spring is released,then:

$A$ ball strikes against the floor and returns with double the velocity; in which type of collision is it possible?