## Related questions with answers

The position vector r describes the path of an object moving in space. (a) Find the velocity vector, speed, and acceleration vector of the object. Position Vector: r(t) = ti + t²j + ½t²k Time: t = 4

Solutions

VerifiedIf $f,\,g$ and $h$ are twice-differentiable functions of $t$ and $\mathbf{r}$ is a vector-valued function given by $\mathbf{r}(t)=f(t)\mathbf{i}+g(t)\mathbf{j}+h(t)\mathbf{k}$, then the velocity vector at time $t$ is:

$\begin{align} \mathbf{v}(t)=\mathbf{r}'(t)=f'(t)\mathbf{i}+g'(t)\mathbf{j}+h'(t)\mathbf{k}. \end{align}$

The acceleration vector at time $t$ is defined as:

$\begin{align} \mathbf{a}(t)=\mathbf{r}''(t)=f''(t)\mathbf{i}+g''(t)\mathbf{j}+h''(t)\mathbf{k}, \end{align}$

and the speed at time $t$ is defined as the magnitude of the velocity vector $\mathbf{v}$:

$\begin{align} ||\mathbf{v}(t)||=||\mathbf{r}'(t)||=\sqrt{\left[f'(t)\right]^2+\left[g'(t)\right]^2+\left[h'(t)\right]^2}. \end{align}$

$v(t)=<1,2t,t>$

$s(t)=\sqrt{1^2+(2t)^2+t^2}=\sqrt{1+5t^2}$

$a(t)= <0,2,1>$

For velocity, take the first derivative of each component of r(t). $\\$ For speed, take the magnitude of the velocity vector. For acceleration, take the derivative of each component of v(t).

## Create a free account to view solutions

## Create a free account to view solutions

## Recommended textbook solutions

#### Thomas' Calculus

14th Edition•ISBN: 9780134438986 (11 more)Christopher E Heil, Joel R. Hass, Maurice D. Weir#### Calculus: Early Transcendentals

9th Edition•ISBN: 9781337613927 (3 more)Daniel K. Clegg, James Stewart, Saleem Watson## More related questions

1/4

1/7