Velocity and Acceleration — Question 10

Question 10

For a regular motion with speed $v=ds/dt$, prove the Frenet acceleration formula $\vec a=(dv/dt)\mathbf T+\kappa v^2\mathbf N$. Identify the geometric role of each term.

Question 10 – Solution

Strategy Write velocity as speed times unit tangent and apply the product and chain rules.

Derivation $$\vec v=v\mathbf T.$$ Therefore $$\vec a=\frac{d}{dt}(v\mathbf T) =\frac{dv}{dt}\mathbf T+v\frac{d\mathbf T}{ds}\frac{ds}{dt}.$$ Since $d\mathbf T/ds=\kappa\mathbf N$ and $ds/dt=v$, $$\boxed{\vec a=\frac{dv}{dt}\mathbf T+\kappa v^2\mathbf N}.$$

Meaning The tangential term changes speed. The normal term changes direction and points toward the local center of curvature; its magnitude grows quadratically with speed.