$A$ particle is moving in a circular path of radius $a$ under the action of an attractive potential $U = - \frac{k}{2r^2}$. Its total energy is

$A$ $2 \, kg$ block is placed on a frictionless inclined plane with an angle of $30^\circ$ and a length of $2 \, m$. After reaching the bottom of the incline,it travels on a surface with a coefficient of friction of $0.25$. What distance (in $m$) will it travel on the surface before coming to rest?

$A$ person of mass $60 \, kg$ wants to lose $5 \, kg$ by going up and down a $10 \, m$ high stairs. Assume he burns twice as much fat while going up than coming down. If $1 \, kg$ of fat is burnt on expending $7000 \, kcal$,how many times must he go up and down to reduce his weight by $5 \, kg$?

$A$ body of mass $5\; kg$ is moving with a momentum of $10\; kg\; m/s$. $A$ force of $0.2\; N$ acts on it in the direction of motion of the body for $10\; s$. Find the increase in its kinetic energy. (in $; J$)

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.)

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: