Question

Determine the required height h of the roller coaster so that when it is essentially at rest at the crest of the hill A it will reach a speed of 111 km/h when it comes to the bottom B. Express your answer to three significant figures and include the appropriate units

Answer 1

`h=48.5m`

Step-by-step explanation:

The loss of gravitational potential energy must be equal to the gain of kinetic energy since there are no other interactions at play (no friction, etc).

We know that the change of potential energy will be written as
`\Delta U=mgh`, where h is the difference between the different heights.

We know that the change of kinetic energy will be written as
`\Delta K=(mv^2)/(2)`, where v is the final velocity and we have assumed it departs from rest.

We then have:

`mgh=(mv^2)/(2)`

`gh=(v^2)/(2)`

`h=(v^2)/(2g)`

We transform km/h to m/s by multiplying by the conversion factors (which are equal to 1, so don't alter the result):

`((1000m)/(1km))((1h)/(3600s))`

So for our values we have:

`h=(((111km/h)((1000m)/(1km))((1h)/(3600s)))^2)/(2(9.81m/s^2))=48.45537m=48.5m`

to three significant figures.