Ah ha! But what about friction? Ok, let me add that in. Suppose I use the common model for kinetic friction that says the frictional force is proportional to the force the track exerts on the car:
This just gives the magnitude of the frictional force. The direction is parallel to the track and in the opposite direction to the motion. Oh, typically the force the track exerts on the car is called the Normal force and labeled with an N. So, suppose the track is inclined (at this part) at an angle θ. This would make the force diagram:
What would the acceleration of this cart be? Let me call the direction the car moves the + x direction and the direction perpendicular to the track the + y direction. The net force in the y would be zero since the car does not change its motion that way. This gives:
See what I did? I replaced Ftrack with FN and I also replaced Fgrav with mg. The cosθ term is the component of the gravitational force that is perpendicular to the track. Ok, what about the x-direction?
Putting in an expression for friction and the force the track exerts on the car, you can see something. The acceleration depends on the coefficient of friction and the angle. It does not depend on the mass of the car.
What about air resistance?
Yes, there is air resistance with these pine derby cars. And yes, for that effect the mass of the car does indeed matter. How much does it matter? First, what model will I use for the air resistance force? Here is a common model for the magnitude of the air resistance force. The direction will be in the opposite direction to the velocity of the car.
Here, ρ is the density of air, A is the cross sectional area, and C is the drag coefficient that depends on the shape of the car. So, why does this make mass matter? Well, all the other forces essentially depend on mass. This means that in the F-net = ma equation, the mass would cancel. The air resistance does NOT depend on mass, so it won't cancel. Simple.
But how much does air resistance make mass matter? Motion of a car with air resistance is slightly tricky (but not impossible). Normally, you calculate the forces and acceleration and then you can find the motion of the car since the acceleration would be constant. In this case, the forces (and thus the acceleration) depend on the velocity. There is a quick and easy solution, a numerical calculation (here is a link to introduction to numerical calculations)
Here is the deal. I will make a model for a car rolling down a straight track. Here are my assumptions:
- The wheels. I am going to assume that there is no rotational energy going to the wheels. Oh, I know this is wrong. However, I don't want to deal with wheels right now.
- Coefficient of kinetic friction is about 0.05 - I mostly just made that up. Poof! Just like that.
- Also for friction, I am going to assume that this is just normal kinetic friction from the axles. I will assume that the rolling friction doesn't do anything.
- The track is a slanted part that is 30 degrees at about 2.5 meters long. After that, it is level for about 4 meters (my guess from looking at images online).
- The mass of the car is 142 grams (actually, I will run one car with a mass of 100 grams).
- For the drag coefficient, if it was an unaltered block of pine, it might have a C around 1. A super nice car might have a drag coefficient around 0.35 (from wikipedia).
