How to Win a Hot Wheels Derby on a Moving Treadmill

If the force on the left wheel is greater than the right, it will produce a net torque which will cause the car to turn to the right. However, for some running cars, this is not a problem. Let’s say a car has turned to the left and is moving on the track diagonally (not straight). Now there will be lateral force on the wheels. This will push one wheel on one side of the car into the axle and push the other wheel away from the axle. It is possible that this thrust and traction of the wheels changes the effective coefficient of kinetic friction so that the differential frictional forces cause it to turn in the other direction and directly back down the slope. Lucky cars have the most chances of winning.

What about the wall?

Let’s say a car turns left and moves to the left side of the treadmill until it makes contact with the side wall. He cannot keep moving to the left because there is a barrier there. If it hits at a shallow angle, the wall may exert lateral force to push it back “downhill”. However, if it continues to push against the sidewall, there will be a frictional force between the side of the car and the wall. This frictional force will push up the incline and decrease the net force down the incline. If this frictional force on the wall is just the right amount, the net force will be zero and the car will not accelerate. It will just stay in the same position.

Does the speed of the treadmill even matter?

In the above analysis, none of the forces depend on the speed of the treadmill. And if a car is moving directly on the track, the speed of the treadmill doesn’t matter. But what about a car going down at an angle? Obviously, in a real race with cars that can move in any direction, the speed of the track matters. OK, so let’s just assume we have two cars with the same speed (v) moving on a track. What happens when a car is turning?

Illustration: Rhett Allain

What are these speed labels? It turns out that the speeds are relative to our frame of reference. Both cars have speeds relative to the track. So AT is the speed of car A relative to the track. What about the velocity of the track? This is measured against the soil reference system (TG). But what we want is the speed of the cars over the ground. For this we can use the following speed transformation. (Here is a more detailed explanation.)

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