The centripetal acceleration can be inferred from the diagram of the velocity vectors at two instances. In the case of uniform circular motion the velocities have constant magnitude. Because each one is perpendicular to its respective position vector, simple vector subtraction implies two similar isosceles triangles with congruent angles – one comprising a base of Δ Δ v and a leg length of v , and the other a base of Δ Δ r (position vector difference) and a leg length of r : | Δ Δ v | v = | Δ Δ r | r | Δ Δ v | = v r | Δ Δ r | Therefore, | Δ Δ v | can be substituted with v r | Δ Δ r | : a c = lim Δ Δ t → → 0 | Δ Δ v | Δ Δ t = v r lim Δ Δ t → → 0 | Δ Δ r | Δ Δ t = ω ω lim Δ Δ t → → 0 | Δ Δ r | Δ Δ t = v ω ω = v 2 r The direction of the force is toward the center of the circle in which the object is moving, or the osculating circle (the circle that best fits the local path of the object, if the path is not circular). The speed in the formula is squared, so twice the speed needs four times the force, at a given radius.