Conservation of Energy 4

Question

A roller coaster begins at rest at the top of a 40 m hill. Find the velocity at a point 5 m above the bottom of the hill.

Answer 1

Assume no loss of energy due to heating of tracks, air, etc.

Conservation of Energy book keeping states the book columns are labeled as potential and mechanical energy. The roller coaster at rest means that the initial mechanical energy for the coaster is zero. The initial potential energy is that energy that would be developed If the hill dropped out from under the coaster. The potential for gravity to do work. That is the mass of the coaster times gravity acceleration x distance to the top of our hill( mgh) on reaching the five meter point above ground the new potential energy is mg(5m) and the mechanical energy is the kinetic energy given by 1/2 m v².

v² = 2 x g x vertical displacement, the vertical displacement being 45 meters

This is the same equation that is developed for constant acceleration in kinematics where we remove the time dependence between velocity and displacement equations. We don't require energy book keeping to arrive at this equation. We only require constant acceleration of the object.

Answer 2

m = mass

g = acceleration of gravity

Potential energy is m * g * height. So the change in PE from and height of 40m to 5m is:

m * 9.8 * 35 = 343m

That change shows up as kinetic energy. KE = (m*v²) / 2

So the KE = the change in PE = 343M

343m = (m*v²) / 2

v² / 2 = 343

v² = 686

v = √686 = 26.19m/s