Explanation:
Imagine a point moving about a circle with a speed of v. At time t = 0, the point is at the bottom of the circle. It has an initial velocity u pointing to the right. The horizontal component is uₓ = v, and the vertical component is uᵧ = 0.
After a short time t, the point has moved a small angular distance of θ. At this position, the horizontal component of the velocity is vₓ = v cos θ, and the vertical component is vᵧ = v sin θ.
When θ is very small, v cos θ ≈ v, and v sin θ ≈ v θ.
The linear acceleration is the change in horizontal velocity over time:
aₓ = (vₓ − uₓ) / t
aₓ = (v − v) / t
aₓ = 0
And the centripetal acceleration is the change in vertical velocity over time:
aᵧ = (vᵧ − uᵧ) / t
aᵧ = (v θ − 0) / t
aᵧ = v θ / t
The time is the distance (in this case, the arc length) divided by speed:
t = s / v
t = rθ / v
Substituting:
aᵧ = v θ / (rθ / v)
aᵧ = v θ × (v / (rθ))
aᵧ = v² / r
