Mud from Tire

A car is stuck in the mud. In his efforts to move the car, the driver splashes mud from the rim of a tire of radius R spinning at a speed v, where v > gR Neglecting the resistance of the air, show that no mud can rise higher than a height R+ v² /2g+ gR²/2v² above the ground.

SOLUTION

Mud flying from different points on the tire will rise to different heights, depending on the initial height and angle of ejection. Introducing an angle α and the height h of the point of ejection O above the equator of the tire (see Figure 1.1), we can write using energy conservation

where h₀ is the height to which the mud rises above O, and v = ωR is the speed of the rim of the wheel. The height H above the ground will be

Figure 1.1

Now find the maximum height by setting the derivative of H with respect to α equal to 0:

There are two solutions of this equation. First,

(1)

This case yields a maximum only when v² ≤ gR, and the highest point of the wheel is the maximum height. But here v² > gR, so we will consider the other case:

 (2)

The height becomes

We can check that