Describe the relationship between the potential and kinetic energies of the tennis ball as it travels the length of the roller coaster.

If we neglect frictional force, the total mechanical energy of the ball is conserved.
The total mechanical energy of the ball is the sum of its kinetic energy K and its potential energy U:

where the kinetic energy depends on the speed v of the ball: 
 
while the potential energy depends on the height h at which the ball is:


As the ball travels along the roller coaster, there is a continuous conversion between kinetic and potential energy, because the total mechanical energy E has always the same value. Therefore, when the ball goes on top of a hill, its height h increases and its potential energy U increases as well, while the speed v decreases and K decreases. Vice-versa, when the ball reaches the bottom of a hill, its height h decreases and therefore the potential energy U decreases, while the speed v increases and therefore the kinetic energy K of the ball increases as well.