Polar and path coordinates

Curvature splits acceleration into two causes

For motion along a known path, one part of acceleration changes speed and another bends the velocity direction. This is more informative than treating acceleration as a single undifferentiated number.

Differentiate the tangent vector

Write v = v t̂, where v is speed and t̂ is the unit tangent. Differentiating gives a = (dv/dt)t̂ + v(d t̂/dt). Along a curve of radius R, the tangent turns at rate v/R, so d t̂/dt = (v/R)n̂. Therefore a = (dv/dt)t̂ + (v²/R)n̂. The tangential part changes speed; the normal part points toward curvature.

A 12 m/s car rounds a 30 m curve at constant speed. Its tangential acceleration is zero, but normal acceleration is v²/R = 144/30 = 4.8 m/s², about 0.49 g. A passenger feels this as lateral acceleration.

Straight paths and changing speed

If R → ∞, normal acceleration tends to zero and the path is locally straight. If speed is zero, the curvature term vanishes instantaneously, but a path can still have nonzero tangential acceleration as motion begins.

Work it: A cyclist increases speed from 4 to 7 m/s at 0.6 m/s² on a circular track of R = 18 m. Find both acceleration components at 7 m/s and the total magnitude.