Velocity and Acceleration — Question 5

Question 5

At an instant, a particle has velocity $\vec v=\left\langle 2,1,2\right\rangle$ and acceleration $\vec a=\left\langle 1,2,-2\right\rangle$. Find the scalar tangential and normal components $a_T,a_N$. State what each says about the motion.

Question 5 – Solution

Strategy Use $a_T=(\vec v\cdot\vec a)/\|\vec v\|$ and $a_N=\sqrt{\|\vec a\|^2-a_T^2}$.

Components Here $\|\vec v\|=3$, $\|\vec a\|=3$, and $\vec v\cdot\vec a=0$. Thus $$\boxed{a_T=0},\qquad \boxed{a_N=3}.$$

Interpretation The instantaneous speed is not changing because the tangential component is zero. The full acceleration changes the direction of motion; it is perpendicular to the velocity.