# Detached wheel from a moving vehicle speed

If a wheel is detached from a moving vehicle, would it speed be higher than the vehicle? Why?

• Assuming its not a trick question of the car screeching to a halt due to the missing wheel. – Ed Yablecki Jan 8 '16 at 21:50

Assume the car travels at speed $v$ and the wheel has a radius $R$. Assume also that the wheel was rolling without slipping, then:

$$v=\omega R,$$

where $\omega$ is the angular speed of the wheel.

When the wheel detaches from the car and assuming no torques or forces act on it from then on, then Newton's Laws tell us the state of motion of the wheel will not change: $v$ and $\omega$ will be conserved, 'forever'.

In the real world, forces like air drag, bumps in the road and rolling resistance do cause the wheel to decelerate somewhat and eventually come to a halt.

If the wheel is detached while traveling on a level plane, the wheel would travel at the same speed as the vehicle. In a frictionless environment on a level plane it would continue to travel at that same speed.

If the vehicle was traveling uphill, the wheel would slow down at a lower rate than the vehicle and would move ahead of it. This is due to the fact that the wheel and the car both possess the same translational kinetic energy per mass unit, but the wheel possesses more rotational kinetic energy per mass unit and thus more total energy per mass unit. Thus, the wheel will advance in front of the vehicle in an uphill climb because it possesses more total energy per mass unit at any particular speed and it can gain more potential energy for any given change in speed than the vehicle can. Said another way: for the same gain in vertical position and the same gain in potential energy per mass unit, the wheel will have to give up less speed because it is converting more rotational energy per mass unit into potential energy for the same change in speed.

Why should it be faster? If you instantaneously remove the axle and let the tire roll freely (without the car crashing on it), what kind of forces would act on the wheel? There is just the rolling friction of the tire against the street.

The angular velocity of the tire would be such that the tangential velocity is the same as of the car. It just has to be that case as the car would slide otherwise.

As there are no significant forces on the tire it will just keep its motion. It would roll with the car (assuming that the car would still drive) and slowly slow down due to the friction.

No, although it's tempting to think it might because it's lighter.

Before being detached, all of the vehicle (including wheels) is moving at the same velocity. At the instant the wheel is detached, the wheel is still moving at the same speed (assuming no forces acted in the detachment).

This is true because momentum is conserved $$\rho_{before}=\rho_{after} \\ (m_{vehicle} + m_{wheels})v_{vehicle} = m_{vehicle}v_{vehicle} + m_{wheels}v_{wheels}$$ Minus $m_{vehicle}v_{vehicle}$ from both sides $$m_{wheels}v_{vehicle} = m_{wheels}v_{wheels} \\ v_{vehicle}= v_{wheels}$$ Therefore the speed of the wheel(s) is the same as the speed of the car.

In reality, the wheel will quickly slow down after detaching due to frictional forces.