$A$ lead bullet penetrates into a solid object and melts. Assuming that $40 \%$ of its kinetic energy is used to heat it,the initial speed of the bullet is ............ $m \, s^{-1}$.
(Given: Initial temperature of the bullet $= 127^{\circ} C$,
Melting point of the bullet $= 327^{\circ} C$,
Latent heat of fusion of lead $= 2.5 \times 10^{4} \, J \, kg^{-1}$,
Specific heat capacity of lead $= 125 \, J \, kg^{-1} K^{-1}$)

$A$ point mass of $1 \, kg$ collides elastically with a stationary point mass of $5 \, kg$. After their collision, the $1 \, kg$ mass reverses its direction and moves with a speed of $2 \, m/s$. Which of the following statement(s) is (are) correct for the system of these two masses?
$(A)$ Total momentum of the system is $3 \, kg \cdot m/s$
$(B)$ Momentum of $5 \, kg$ mass after collision is $4 \, kg \cdot m/s$
$(C)$ Kinetic energy of the centre of mass is $0.75 \, J$
$(D)$ Total kinetic energy of the system is $4 \, J$

$A$ block of mass $1 \ kg$ is moved by a distance of $1 \ m$ on a horizontal surface by applying a force of $8 \ N$. Then it is lifted to a height of $2 \ m$. The total work done is ... $J$. (Take $g = 10 \ m/s^2$)

$A$ block of mass $5 \,kg$ is moved by a distance of $10 \,m$ on a surface with a coefficient of friction $\mu = 0.2$ by applying a force of $25 \,N$. The kinetic energy gained by the block is ...... $J$. (Take $g = 10 \,m/s^2$)

$A$ particle $P$ is sliding down a frictionless hemispherical bowl. It passes the point $A$ at $t = 0$. At this instant of time, the horizontal component of its velocity is $v$. $A$ bead $Q$ of the same mass as $P$ is ejected from $A$ at $t = 0$ along the horizontal string $AB$ (see figure) with the speed $v$. Friction between the bead and the string may be neglected. Let ${t_P}$ and ${t_Q}$ be the respective time taken by $P$ and $Q$ to reach the point $B$. Then

$A$ sphere of mass $m$ travelling at constant speed $v$ strikes another sphere of the same mass. If the coefficient of restitution is $e$,then the ratio of the velocities of both spheres just after the collision is: