Question

A rollercoaster starts on the top a tall hill with a height of 50 meters. The mass of the car is 1000kg. The initial velocity of the car at the top of the hill is 5m/s. after the car descends the first hill it goes through a loop with a height of 30 meters and over a second hill with a height of 20 meters. The car comes to rest on a rise 5 meters above the ground and is slowed to a stop with a friction brake the has a kinetic coefficient of friction of Mu k=1. What is the length of the friction brake?

Answer 1

No matter what features the track had, what really important is the difference between the initial and final heights, and the initial speed. According to energy conservation:

$$KE_\text{initial}+PE_\text{initial}=E_\text{friction}+PE_\text{final},\\\space\\ \frac12 mv^2+mgH=\mu mgx+mgh,\\\space\\ x=\frac{2g(H-h)+v^2}{2\mu g}=46.3\text{ m}.$$