Ramp Racer

Which is fastest down a ramp: a cart on wheels, a block of ice or a wooden block? Set the angle, pick two racers and race them side by side.

Racer 1 (front lane)
Racer 2 (back lane)
Racer 1 time
0.00 s
Racer 2 time
Stuck
Racer 1 speed
0.00 m/s
Racer 2 speed
0.00 m/s

The maths, live

Racer 1 a = g(sin θ − μ cos θ)a = 9.81 × (0.342 − 0.01 × 0.940) = 3.26 m/s²
Racer 2 slides if tan θ > μstan 20° = 0.36 ≤ 0.5, friction holds it
Time to the bottom t = √(2L ÷ a)t₁ = 1.11 s, t₂ = never
Speed at the bottom v = a × tv₁ = 3.61 m/s, v₂ = 0.00 m/s

Find the steepest angle where the wooden block still stays put. Is tan θ at that angle close to μs = 0.5?

How it works

On a ramp, part of the racer's weight pulls it down the slope (g sin θ) and part presses it into the ramp (g cos θ). Friction depends on how hard it presses and how grippy the surfaces are (μ), so friction pushes back with μ g cos θ.

Friction can hold a racer still as long as tan θ is no bigger than the static friction coefficient μs. Once it slides, the acceleration is a = g(sin θ − μ cos θ). Less friction means more acceleration, a shorter time to the bottom and a faster finish.

Try this

  1. Find the smallest angle where the wooden block starts to slide.
  2. Race the cart on wheels against the ice block. Which wins, and by how much?
  3. Make the ramp steeper. Does the time to the bottom halve when you double the angle?