Chapter 11, Problem 80A

To determine

The height of the first hill.

Answer to Problem 80A

The height of the first hill must be increased by four times.

Explanation of Solution

Introduction:

The kinetic energy of an object of mass $m$ travelling at a speed of $v$ is given by,

  $KE=\frac{1}{2}m{v}^2$

And, conservation of Mechanical Energy is given as,

  $K{E}_{before}+P{E}_{before}=K{E}_{after}+P{E}_{after}$

Where $KE$ is the kinetic energy and $PE$ is the potential energy.

Also, the PE of an object of mass, m , due to its height, h , and gravity is given by,

  $PE=mgh$

The initial velocity at the top of the roller coaster will be zero, while the height at the bottom of the roller coaster will be 0, thus

  $\begin{array}{c}K{E}_{before}+P{E}_{before}=K{E}_{after}+P{E}_{after}\\ \frac{1}{2}m{\left(0\right)}^2+mgh=\frac{1}{2}m{v}_f^2+mg\left(0\right)\\ mgh=\frac{1}{2}m{v}_f^2\end{array}$

Then to have the velocity at the bottom of the roller coaster doubled, the height of the first hill must be increased by four times.

Conclusion:

The height of the first hill must be increased by four times to have the velocity at the bottom of the hill doubled.