Question

A car travelling at 80km/h gets a stone caught in one of its tires. Determine an equation that models the height of the stone over time if the radius is 25cm.

Answer 1

A typical equation for the y-coordinate over time of a point during its circular motion looks like this:

$$Y=A\text{sin}(\omega t),$$

where $A$ - amplitude, $\omega$ - angular frequency.

In our case amplitude is the radius of the wheel, the angular frequency can be determined through the velocity and wheel radius as well:

$$v=\omega R.$$

Hence:

$$Y(t)=R\cdot\text{sin}\Big(\frac{v}{R}\cdot t\Big),$$

make sure you use SI, i.e. you converted km/h to m/s and centimeters to meters:

$$Y(t)=0.25\cdot\text{sin}\Big(\frac{80\cdot1000/3600}{0.25}\cdot t\Big),$$

so the height over time is

$$Y(t)=0.25\cdot\text{sin}(88.89 t)\text{ m}.$$