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:

In the figure shown,two identical balls of mass $M$ and radius $R$ each are placed in contact with each other on a frictionless horizontal surface. $A$ third ball of mass $M$ and radius $R$ is coming down vertically with a velocity $v_0$. It simultaneously hits the two balls. The smaller ball (top ball) does not stop after the collision but continues to move downwards with a speed $v_0/2$. Find the speed of each bigger ball after the collision.

In an inelastic collision,

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:

$A$ 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 $= 125 \, J \, kg^{-1} K^{-1}$)

Two blocks $A$ and $B$,each of mass $m$,are connected by a massless spring of natural length $L$ and spring constant $K$. The blocks are initially resting on a smooth horizontal floor with the spring at its natural length as shown in the figure. $A$ third identical block $C$,also of mass $m$,moves on the floor with a speed $v$ along the line joining $A$ and $B$ and collides with $A$. Then: