$A$ boy of mass $40 \, kg$ runs up a flight of $50$ steps,each $10 \, cm$ high,in $5 \, s$. Find: $(i)$ the work done by the boy,and $(ii)$ the power developed. $(g = 9.8 \, m s^{-2})$

$(a)$ Two bodies have equal masses and are moving with uniform velocities of $2\,v$ and $3\,v$. What is the ratio of their kinetic energies?
$(b)$ $A$ mass is dropped from height $h$. At halfway to the ground,what is the total energy?

The velocity of a body moving in a straight line is increased by applying a constant force $F$, for some distance in the direction of the motion. Prove that the increase in the kinetic energy of the body is equal to the work done by the force on the body.

Two children $A$ and $B$ each weighing $20 \, kg$ climb a rope up to a height of $10 \, m$. Child $A$ takes $10 \, s$ and child $B$ takes $20 \, s$ to climb. State whether the work done by both is equal or unequal. Who has more power?

$A$ spring which has been kept compressed by tying its ends together is allowed to be dissolved in an acid. What happens to the potential energy of the spring?

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.