$A$ body of mass $m_1$ moving with an unknown velocity $v_1 \hat{i}$ undergoes a one-dimensional collision with a body of mass $m_2$ moving with velocity $v_2 \hat{i}$. After the collision,the bodies $m_1$ and $m_2$ move with velocities $v_3 \hat{i}$ and $v_4 \hat{i}$ respectively. If $m_2 = 0.5\, m_1$ and $v_3 = 0.5\, v_1$,find $v_1$.

Calculate the work done to pull a block of mass $M$ by a constant force $F$ as shown in the figure. The coefficient of friction between the block and the ground is $\mu$.

$A$ circular well of diameter $2 \ m$ has water up to the ground level. If the bottom of the well is at a depth of $14 \ m$,the time taken in seconds to empty the well using a $1.4 \ kW$ motor is (Acceleration due to gravity $= 10 \ m \ s^{-2}$)

Answer carefully,with reasons:
$(a)$ In an elastic collision of two billiard balls,is the total kinetic energy conserved during the short time of collision of the balls (i.e.,when they are in contact)?
$(b)$ Is the total linear momentum conserved during the short time of an elastic collision of two balls?
$(c)$ What are the answers to $(a)$ and $(b)$ for an inelastic collision?
$(d)$ If the potential energy of two billiard balls depends only on the separation distance between their centres,is the collision elastic or inelastic?
(Note: We are talking here of potential energy corresponding to the force during collision,not gravitational potential energy.)

$A$ block of mass $m$ starts at rest at height $h$ on a frictionless inclined plane. The block slides down the plane,travels across a rough horizontal surface with coefficient of kinetic friction $\mu$,and compresses a spring with force constant $k$ a distance $x$ before momentarily coming to rest. Then the spring extends and the block travels back across the rough surface,sliding up the plane. The block travels a total distance $d$ on the rough horizontal surface. The correct expression for the maximum height $h'$ that the block reaches on its return is

This question has Statement-$1$ and Statement-$2$. Of the four choices given after the statements, choose the one that best describes the two statements.
Statement-$1$: $A$ point particle of mass $m$ moving with speed $u$ collides with a stationary point particle of mass $M$. If the maximum energy loss possible is given as $f \left( \frac{1}{2} m u^2 \right)$, then $f = \left( \frac{m}{M + m} \right)$.
Statement-$2$: Maximum energy loss occurs when the particles get stuck together as a result of the collision.