Question

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

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

Short answer

Ans:

Step-by-step solution

Step 1. Given information: 

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

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

thus