Toy Car Ramp Speed Calculator

A toy car rolling down a ramp converts gravitational potential energy into kinetic energy. The exit speed depends on ramp length, angle, and how much of that energy the wheels and axles give away on the trip down.

The formula:

v = sqrt(2 x g x L x (sin(theta) - mu x cos(theta)))

Where g = 9.81 m/s squared, L = ramp length in meters, theta = ramp angle in degrees, and mu is the effective resistance coefficient for the surface.

Resistance coefficients by surface:

• Smooth wood (sanded, finished): mu = 0.05
• Carpet or felt surface: mu = 0.20
• Rough wood (unfinished): mu = 0.35

A 1-meter ramp at 30 degrees on smooth wood gives 2.99 m/s (10.8 km/h). The same ramp at 15 degrees gives 2.03 m/s. Halving the angle takes about a third off the speed, which is more than people expect, and it is why a track that works at 25 degrees can stall completely at 12.

What the formula leaves out, and why that is the right call for a car

Strictly, a rolling object has to spin its wheels up as well as move forward, so the honest version divides the inside of the square root by (1 + I/mr²), where I is the rotational inertia of the wheels. For a solid ball or a rolling disc that term is large: a disc comes out about 18% slower than this page would say. For a car it is small, because four light plastic wheels are a few percent of the mass of the body they are carrying. Ignoring it costs you maybe 1 to 3% on a typical die-cast, which is well inside the error on the friction coefficient anyway.

So the number here is the sliding-block answer, and for a car that is close enough to be useful. Roll a marble down the same ramp and it will be noticeably slower than this page predicts.

The stall angle

Below a certain angle the car does not move at all, and that angle is arctan(mu). On smooth wood it is under 3 degrees, on carpet about 11, and on rough unfinished wood about 19. That last one catches people out: a 15-degree ramp in unsanded pine is not a slow ramp, it is a ramp the car sits still on.

For toy design

If you want a car to reliably trigger a loop or hit a target, you need predictable ramp speed. Too steep and the car flies off the track. Too shallow and it stalls. The sweet spot for wooden track toys is usually 20-35 degrees. Sanding and finishing the running surface buys you more speed than adding 5 degrees of angle does, and it costs nothing.

Note: this assumes the car rolls without slipping. If the wheels lock and skid, the resistance is far higher and the car is much slower than any figure here.