Seema tried to push a heavy rock of $100 \, kg$ for $200 \, s$ but could not move it. Find the work done by Seema at the end of $200 \, s$. (in $, J$)

$(a)$ Define potential energy. Write an expression for potential energy of an object of mass $m$ raised through a height $h$.
$(b)$ Find the energy possessed by an object of mass $10 \, kg$ when it is raised to a height of $6 \, m$ above the ground. (Given $g = 9.8 \, m s^{-2}$)

An automobile engine propels a $1000 \,kg$ car $(A)$ along a levelled road at a speed of $36 \,km \,h^{-1}$. Find the power if the opposing frictional force is $100 \,N$. Now,suppose after travelling a distance of $200 \,m$,this car collides with another stationary car $(B)$ of the same mass and comes to rest. Let its engine also stop at the same time. Now car $(B)$ starts moving on the same level road without its engine started. Find the speed of the car $(B)$ just after the collision.

$A$ rocket is moving up with a velocity $v$. If the velocity of this rocket is suddenly tripled,what will be the ratio of the initial kinetic energy to the final kinetic energy?

Justify,giving proper reasoning,whether the work done in the following case is positive or negative: Work done by friction on a body sliding down an inclined plane.

$(a)$ $A$ person is holding a bucket of water by applying a vertical force of $100 \, N$. He first moves a horizontal distance of $5 \, m$ and then climbs up through stairs a vertical distance of $10 \, m$. Calculate the total work done by the person on the bucket.
$(b)$ Establish the relationship between kinetic energy and momentum of a body.