Question

Find the velocity v(t)v(t) and speed ∥v(t)∥∥v(t)∥ of a particle whose motion is described by x=6t3−9t2,y=3,z=2t2−4t+2

Answer 1

Let $\mathbf p(t)=(x(t),y(t),z(t))$ denote the particle's position. Then its velocity is given by

$\mathbf v(t)=\dfrac{\mathrm d\mathbf p}{\mathrm dt}=(x'(t),y'(t),z'(t))$
$\mathbf v(t)=(18t^2-18t,0,4t-4)$

The speed is then given by the absolute value/magnitude/norm of the velocity function,

$\|\mathbf v(t)\|=\sqrt{(18t^2-18t)^2+(4t-4)^2}$