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:
[tex]E=K+U[/tex]
where the kinetic energy depends on the speed v of the ball:
[tex]K= \frac{1}{2} mv^2[/tex]
while the potential energy depends on the height h at which the ball is:
[tex]U=mgh[/tex]
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.
