how is mechanical energy is related to roller coasters?

Answer 1

Imagine you have a roller coaster which starts moving from point A down to point B, which is at ground level (where height, h, is equal to zero). It then moves up to point C, which is at about half the height of point A, then down to point D, which is slightly above ground level. Then it moves up again to point E, which is at a greater height than point A, and in doing so passes point F, which is at the same height as point A (drawing this out will help or look at the related link below for a diagram).

TE=total energy

PE=potential energy

KE=kinetic energy

Assuming friction and air resistance are negligible and that the roller coaster starts from rest, then the TE of the roller coaster is equal to its PE at point A.

TE=PE at A

As the roller coaster moves from A to B, its PE changes into KE. Since h=0 at B, then all the PE of the roller coaster at A is turned into KE at B.

The change in PE=the change in KE from A to B.

Here it is useful to note that at A, KE is a minimum (0) and PE is a maximum; at B, KE is a maximum and PE is a minimum (0). Thus, the KE at B is also equal to the TE.

TE=KE at B

Also note that TE remains constant, being the sum of the PE and KE possessed by the roller coaster.

PE at A=KE at B

At A, TE=PE+0

At B, TE=KE+0

Hence, TE is constant.

As the roller coaster moves from B to C, its KE changes into PE as its height above the ground increases. However, when it reaches C, it does not possess only PE, but a combination of PE and KE.

TE at C=PE at C + KE at C

The reason why PE is not a maximum at C is because C is lower in height than A. We know that PE at A is the TE of the roller coaster for the entire course. Since PE is dependent on height, in order for the roller coaster to reach maximum PE, it must be at a height equal to the starting height. C is at roughly half the height of A, hence the roller coaster will possess only about half the PE it had compared to when it was at A. The rest of the energy is KE since TE=KE+PE.

D is not at the same level as B, but is slightly higher. Hence, the roller coaster will not move as fast at D than it did at B. This is because it has less KE at D, due to the fact that it still possesses some PE (since h is not equal to 0 at D).

Since TE=KE+PE and PE is not equal to 0, then KE will not be maximum and thus the roller coaster will move less quickly at D than it did at B.

Using the same principle, the roller coaster will not be able to reach E. This is because it reaches maximum PE when it is at F, since F is at the same height as A. We know that at A, PE=TE. Hence, at F, PE=TE. Energy can neither be created nor destroyed, hence the energy of the roller coaster cannot exceed the TE it had at the start. Therefore, it will not reach E, but it will be at rest momentarily at F before moving down again and back to A (remember friction and air resistance are negligible), and continue moving back and forth between A and F.

However, the roller coaster will be able to reach E if it is given KE in addition to the PE at A. In other words, if the roller coaster is already moving at a sufficient speed as it passes A, then it will be able to reach E. This is because the TE at A will now be equal to the sum of KE and PE at A, and KE is not equal to zero as it was in the previous example. The additional KE that would need to be supplied in order for the roller coaster to reach E would be equal to the difference in the PE at E and the PE at A (or F).

PE at E - PE at A = KE at A

which is the same thing as

TE - PE at A = KE at A; or TE=PE at A + KE at A

That's pretty much all of it.