Do bigger skiers go faster?

If we consider a skier of mass sliding straight down a uniform slope of angle θ degrees, then the magnitude of the total force on the skier (measured down the slope) is:

F = Fg – Ff – Fd

where

Fg = mg sinθ is the force from gravity

Ff = μmg cosθ is the frictional force from the snow for some constant μ

Fd = ½ρv2CA is the drag from air resistance

Now for 2 different skiers sliding at the same speed, is constant and ρ is the density of the air, so is a constant. We further assume that both skiers have the same shape (not size) and are wearing similar clothes, so we can assume C, the drag coefficient, is approximately constant, and hence

Fd = kA

for some constant k, where A is the area of the skier’s body presented to the air in the direction of travel (the cross-section).

Then

ma = mg sinθ – μmg cosθ – kA

so

a = g sinθ -μg cosθ -kA/m

Hence the acceleration of the skier down the slope depends on the ratio A/m. If we assume that the skiers’ bodies have the same constant density ρ, then the acceleration is dependent on A/ρV, where V is the skier’s volume. For a linear increase L in dimension, A goes as the square of L, and V goes as the cube, so in fact, with the assumptions stated, the “bigger” skier will accelerate more, hence go faster, because the retarding force due to drag will be less.

So there you have it – bigger skiers go faster – maybe!