The Drag Force

F = kv^xa^yr^z

[M][L][T]^-2 = [L]^x[T]^-x[L]^y[M]^z[L]^-3z

(a) Equating powers of [M]; 1 = z

(b) Equating powers of [L]; 1 = x+y-3z

(c) Equating powers of [T]; -2 = -x

(a) and (c) give x = 2, z = 1 immediately

Substituting for x and z into (b) gives y = 2

So F = kv²a²r. For the most obvious case, it turns out that if a² is written as area, then k =1, and F=rAv²