A hill has a height h. A child on a sled (total mass m) slide down starting from rest at the top. Does the speed at the bottom depends on the angle of hill if (a) it is icy and there is no friction, and (b) there is friction (deep snow)? Explain your answers.

As gravitational force is conservative in nature, net work done by gravitational force only depends on the initial and final positions (and not on the path followed).

As there is no friction, kinetic energy gained at the bottom will equal the change in potential energy. As gravitational potential energy is conservative in nature, it will not depend on the angle of the hill and will only depend upon the initial height from which the child started.

When friction is present, final kinetic energy will equal the summation of work done by friction force and gravitational force (by work-energy theorem). As friction is a non-conservative force, work done by friction will depend on the length of the path followed and thus, will depend on \(\theta \) (angle of the hill).

That implies that the final kinetic energy of the child will depend on the value of \(\theta \) in this case.