# A roller coaster has a hill that is 20m tall. If the roller coaster cart is moving at 20m/s as it enters the bottom of the hill, how do you answer the following questions?

## a) Will it make it all the way over the hill? (assume no friction) b) If the roller coaster cart had a mass of $5000 k g$ and friction does $80 k J$ of work on the cart, how high up the hill does the cart go?

May 13, 2016

the answer to (a) is yes and (b) is 18.76m (both assuming $g = 9.81 m {s}^{-} 2$ )

#### Explanation:

For part (a) we are to assume no friction. Hence all kinetic energy will be converted to potential energy.

Kinetic energy = KE=$\frac{1}{2} m {v}^{2} = \frac{1}{2} m {20}^{2} = m \cdot 200$
Potential energy is given by $P E = m g h$ and we need to get to $h = 20 m$ and since $g = 9.81 m {s}^{-} 2$, then at the top of the hill we will have

$P E = m \cdot 9.81 \cdot 20 = m \cdot 196.2$

So we can see that there is sufficient KE to get tot the top of the hill. (if you take $g = 10 m {s}^{-} 2$ then it will only just get to the top and then stop since KE=PE)

For part (b), some of the energy is used in working against friction, so energy available for conversion into PE is given by:
$\text{KE-work done} = 200 \cdot m - \left(80 \times {10}^{3}\right) = 200 \cdot 5000 - 80000 = 920000$

This will be converted to PE, so
$P E = 5000 \cdot 9.81 \cdot h = 920000$
Solving gives $h = 18.76 m$