Rollercoasters generate speed by converting gravitational potential energy into kinetic energy by taking the cart to the top of a large hill and letting it go. The conversion of a perfectly efficient system would be like so:
PE = KE
and using the formulas for potential and kinetic energy:
mgh = 1/2mv^2.
However, the efficiency of this system is 50%, meaning that the kinetic energy obtained from this conversion would appear as so:
PE=0.5 KE
mgh=0.5(1/2mv^2)
mgh=1/4mv^2.
The masses cancel out, leaving:
gh=1/4v^2
The goal is to achieve 65 m/s, and with Earth’s innate gravity of 9.806 m/s^2, we have:
gh=1/4v^2
(9.806)h=1/4(65)^2
h=107.71 meters
The height of the first hill must be 107.7 meters to generate a speed of 65 m/s with a conversion efficiency of 50%.
Hope this helps!
