Velocity and Acceleration — Question 4

Question 4

For the helix $\vec r(t)=\left\langle 3\cos t,3\sin t,4t\right\rangle$, find velocity and acceleration. Prove that its speed is constant and decide whether acceleration is entirely normal.

Question 4 – Solution

Strategy Compute $\vec v\cdot\vec a$ after differentiating.

Derivatives $$\vec v=\left\langle -3\sin t,3\cos t,4\right\rangle,\qquad \vec a=\left\langle -3\cos t,-3\sin t,0\right\rangle.$$ The speed is $\|\vec v\|=5$, independent of $t$.

Decomposition Since $$\vec v\cdot\vec a=9\sin t\cos t-9\sin t\cos t=0,$$ acceleration is perpendicular to velocity. Hence $\boxed{a_T=0}$ and all acceleration is normal, with $\boxed{a_N=\|\vec a\|=3}$.