Determine the required height h of the crest of the roller coaster to thebottom so that when it is essentially at rest at the crest of the hill it will reacha speed of 28 m/s when it comes to the bottom.

h = 40 m

Explanation:

Assuming no friction present, total mechanical energy must be conserved, so the following expression stands:ΔK + ΔU = 0 (1)Now, if the car is at rest at the crest of the hill, the change in kinetic energy is just as follows:

         (2)

       where vb = speed at the bottom = 28 m/s

If we define the bottom as our zero reference level for the gravitational potential energy, we can write the following equation:

        (3)

From (1) we get:ΔK = -ΔUReplacing by (2) and (3), we get:

       

Simplifying and rearranging terms, we can solve for h (height required) as follows:

       

Wavelength divides in half and speed stays the same.