This week’s Riddler Classic is a simple-looking question about the turning radius of a truck.

Suppose I’m driving a very long truck (with length L) with two front wheels and two rear wheels. (The truck is so long compared to its width that I can consider the two front wheels as being a single wheel, and the two rear wheels as being a single wheel.)

Suppose I can also rotate the front wheels by $\alpha$ and the back wheels — independently from the front wheels — by $\beta$. What is the truck’s turning radius?

Here is my solution:

[Show Solution]

When operating normally (no slippage), wheels can never slide sideways. They can only lead to motion in the direction they are pointing. In a steady turn (front and rear wheel angles constant), the wheels each trace out a circle and each wheel is tangent to its circle. Thinking about this a bit more carefully, we come to the conclusion that the front and rear wheels cannot trace out the same circle, since that would cause either the front or rear to slip. Therefore, there are actually *two* turning radii (inner and outer). Here is a diagram illustrating the situation:

By the law of sines, we have:

\[

\frac{r}{\sin(\tfrac{\pi}{2}-\alpha)} = \frac{R}{\sin(\tfrac{\pi}{2}-\beta)}

\]and by the law of cosines, we have:

\[

L^2 = R^2 + r^2-2rR \cos(\alpha+\beta).

\]Putting these two together and solving for $r$ and $R$, we obtain:

$\displaystyle

\frac{r}{L} = \frac{\cos(\alpha)}{\sin(\alpha+\beta)}

\quad\text{and}\quad

\frac{R}{L} = \frac{\cos(\beta)}{\sin(\alpha+\beta)}

$

We have a few cases depending on the relative size of $\alpha$ and $\beta$:

- If $\alpha \gt \beta$ (front wheel can turn more than the back wheel), then $R \gt r$, so the turning radius of the front wheel is larger than that of the back wheel.
- If $\alpha \lt \beta$ (back wheel can turn more than the front wheel), then $R \lt r$, so the turning radius of the back wheel is larger than that of the front wheel.
- If $\alpha=\beta$ then $R=r$ and both wheels turn on the same circle.

Here is a live GeoGebra script I made so you can play around with different values of the front and rear angles; click and drag to pan, use the mouse wheel to zoom, and use the sliders to change the angles!

Here is a GeoGebra link if you want to edit the file in full-screen mode.