## Task $6 / 65$

A cyclist on a rain-soaked road sees drops flying off the highest point of the front wheel. What is the speed of these drops relative to the road?
The cyclist has the speed $v$. This is also the rotational speed of the wheel. For every point on the wheel, the speed in the direction of travel and the circumferential speed of the wheel add vectorially. Since at the highest point of the wheel the tangential direction coincides with the direction of travel, the speed of the droplets in the direction of travel is $2 v$.

A second solution can be obtained using differential calculus: Each point on the wheel describes a cycloid; for points on the circumference of the wheel, this is a common cycloid with the parametric representation

$$
x=r \omega t-r \sin \omega t \quad ; \quad y=r-r \cos \omega t
$$

where $r$ is the radius of the wheel and $\omega$ is the angular velocity of the wheel. $t$ denotes the time. For the speed $v$ of the bicycle, $v=r \omega$ holds (it is equal to the circumferential speed of the wheel). The speed of a point on the circumference of the wheel in the $x$-direction is

$$
\frac{d x}{d t}=r \omega-r \omega \cos \omega t
$$

For the highest point of the wheel, $y_{h}=2 r=r-r \cos \omega t-h$; from this follows $\cos \omega t_{h}=-1$. Substituting this value into equation (1) yields

$$
\frac{d x}{d t\left(y_{h}\right)}=r \omega-r \omega(-1)=2 r \omega=2 v
$$

as the speed of the highest point in the direction of travel. This is the speed of the droplets flying off.