Question

Velocity and acceleration vectors: Find the velocity and acceleration vectors for the given vector functions.

$\mathbf{r}\left(t\right)=(\cos 2t,\sin 3t)$

Short answer

Ans:$v\left(t\right)=⟨-2\sin 2t,3\cos 3t⟩\mathrm{and}a\left(t\right)=⟨-4\cos 2t,-9\sin 3t⟩$

Step-by-step solution

Step 1. Given information: 

$\mathbf{r}\left(t\right)=(\cos 2t,\sin 3t)$

Step 2. Denoting the velocity and acceleration of the vector:

The velocity vector, $v\left(t\right)$ is given by

.

That is, we take the derivative of each component function of $r\left(t\right)$.

The acceleration vector, $a\left(t\right)$ is given by

$a\left(t\right)=\frac{d}{dt}v\left(t\right)$.

That is, $a\left(t\right)$is obtained by taking the derivative of each component function of $v\left(t\right)$.

Step 3. Finding the velocity and acceleration of the vector: 

Consider $r\left(t\right)=⟨\cos 2t,\sin 3t⟩$

thus

$$\begin{gathered} v\left(t\right)=⟨-2\sin 2t,3\cos 3t⟩ \\ \mathrm{and}a\left(t\right)=⟨-4\cos 2t,-9\sin 3t⟩ \end{gathered}$$