Dividing the equations N sin θ = mv^{2}/rN cos θ = mg yields the equation tan θ = v^{2}/rg. Solving for v produces the desired resultv_{critical} = √ (rg tan θ).

N sin θ = mv^{2}/rN cos θ = mg

tan θ = v^{2}/rg.

v_{critical} = √ (rg tan θ).