The following exercise explores an application of the multivariable chain rule in the context of a well-known fact from physics.

Take to be the height associated to the ground, and suppose that a ball is dropped from an initial height of . Initially, the ball has only potential energy, which depends on the height of the ball only and is given by

where is the mass of the ball, is the acceleration due to gravity, and is the height of the ball.

As the ball falls, its velocity increases, and it gains kinetic energy, which depends only on the velocity of the ball and is given by

The total energy of the ball during its fall is given by

Now, as the ball falls, and will depend on time, so we may think of them as functions of time and write and to make this dependence explicit. Note that as the ball continues to fall, increasesdecreases and increasesdecreases .

We can thus think of the total energy as a function of time and write

We can use the chain rule to compute .

Select the correct expression for .

. .
Note that and , so the chain rule tells us that

In the absence of air resistance or any other external forces, the work-energy theorem guarantees that the total energy of the ball is conserved; thus is constant although the amount of kinetic and potential energy changes in time.

Since is constant in time, we must have that since the derivative of a constant is .

Substituting this gives

Writing and gives

from which we conclude for any time during the fall, .

This should not be surprising; the only force acting on the ball is gravity, and this force is directed downwards. Hence, Newton’s second law produces the result

which s exactly what we found before!