# Does mass affect distance travelled by a toy car?

Say, I conduct an experiment with a toy car and a ramp (Fixed angle). If I were to use the same toy car but place different weights on it (assuming the same size and shape), would the car with more mass travel further after reaching the bottom of the ramp? Or would it be the other way around and the car with less mass travels further after reaching the bottom of the ramp?.

What would the theoretical result be and what would cause this observed result?

Depending on how realistic a model one wants to construct, one should take into account the following:

• Rolling friction - note that a car is not a block, so instead of sliding friction we have rolling friction The good news is that rolling friction, just as sliding friction, is proportional to the normal force - thus, the calculations given in the other answers apply both on the ramp and on the inclined surface, meaning that in the first approximation the distance traveled by the car does not depend on its weight.
• Air resistance Air introduces a drag force, mentioned already in the answer by @MarcBarcelo. One thing to keep in mind about a drag force is that it has two contributions: one proportional to the velocity, and the other proportional to the velocity squared (see here, here, and here). In either case the drag coefficient depends on the car shape, whereas the acceleration is inversely proportional to the car mass. Thus, if loading the car does not alter the drag coefficient drag coefficient (e.g., it is a truck with load hidden inside), the heavier car would travel further.
• Random effects If the surface of the ramp and the path down the ramp is not very smooth, the bumps on the car path could affect its movement. This mainly has to do with altering the friction coefficient or possibly breaking from time to time the contact between the wheels and the surface. My guess is that heavier and/or bigger car is less sensitive to such randomness, and would likely travel further.

Assuming both cars are completely identical with only exception of having different mass, then ideally both cars should have the same final velocity and hence travel the same horizontal distance. I say ideally because there might be some other effects, such as change in coefficient of friction with mass, but simple friction models do not account for this effect.

The free-body diagram of the car on the ramp would result in

$$ma = m g \sin\theta - \mu m g \cos\theta = mg (\sin\theta - \mu\cos\theta)$$

where $$a$$ is acceleration along the ramp, and $$\mu$$ is coefficient of friction. Note that rotational motion of wheels has been neglected here. As you can see, the mass appears on both sides of the equation and cancels out, i.e. the car acceleration does not depend on its mass

$$a = g (\sin\theta - \mu\cos\theta)$$

Both cars should have the same final velocity at the bottom of the ramp. Once the car gets horizontal, the free-body diagram would result in

$$ma = -\mu m g$$

and the horizontal displacement is

$$\Delta x = -\frac{\mu}{2} g t^2 + v_f t$$

where $$v_f$$ is velocity the car had at the bottom of the ramp. There could be some mass-related effects at the point where the car transitions from the ramp to the horizontal surface, but it is difficult to comment without specifics.

• Would momentum have anything to do with how far the cars can travel after reaching the end of the ramp? Does a higher momentum (mass) equate to a tendency for the car to want to continue on its path? I feel like final velocity is one thing, but distance traveled post ramp could depend on more than that. Commented Apr 8, 2022 at 7:53
• @bleuofblue Once the car is horizontal, the momentum does not affect the displacement (distance): $\Delta x = -\frac{\mu}{2} g t^2 + v_x t$. There could be some momentum-related effects at the place where the car is transitioning from the ramp to the horizontal surface. Commented Apr 8, 2022 at 7:58