Question

3. a. Prove that v . a-v v b. Explain why the above means that if a particle's speed is constant, its velocity and acceleration vector are perpendicular Hint: Differentiate both sides of the equation v2. Note that is not the same as läl; v is the magnitude of the acceleration of the particle along its instantaneous direction of motion.

Answer 1

$\vec{v}\cdot\vec{v}=v^2$

$\frac{d}{dt}(\vec{v}\cdot\vec{v})=\frac{d}{dt}v^2$

$2\vec{v}\cdot\frac{d\vec{v}}{dt}=2v\frac{dv}{dt}$

$\vec{v}\cdot\vec{a}=v\dot{v}$

constant speed (not velocity) means $\dot{v}=0$, which means $\vec{v}\cdot\vec{a}=0$ , precisely, the condition of orthogonality.