Chapter 11.2, Problem 16PP

To determine

The initial kinetic energy of the system and the height at which the bike will come to rest if the rider coasts up the hill.

Answer to Problem 16PP

  $3070.625\text{J}$ , and $3.7\text{m}$ , respectively

Explanation of Solution

Given:

The combined mass of the bike and rider, $m=85.0\text{kg}$

Initial velocity of the bike, $v_i=8.5\text{}\text{m/s}$

Formula used:

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

  $KE=\frac{1}{2}m{v}^2$

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.

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

  $PE=mgh$

Calculation:

Using the given values, the initial kinetic energy of the rider and bike will be,

  $\begin{array}{c}K{E}_i=\frac{1}{2}m{v}^2\\ =\frac{1}{2}\left(85\right){\left(8.5\right)}^2\\ =3070.625\text{J}\end{array}$

Let the height at which the bike stops be $h$ , then using conservation of mechanical energy principle,

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

Using the calculated and given values in above,

  $\begin{array}{c}3070.625=\left(85\right)\left(9.8\right)h\\ h\approx 3.7\text{m}\end{array}$

Conclusion:

Thus, initial kinetic energy of the bike and rider is $3070.625\text{J}$ , and the height at which the bike stops is $3.7\text{m}$ .