Question

The position of a particle is given by
$r=3.0t\hat{i}+2.0t^2\hat{j}+5.0\hat{k}$
where $t$ is in seconds and the coefficients have the proper units for $r$ to be in metres.
(a) Find $v\left(t\right)$ and a $\left(t\right)$ of the particle.
(b) Find the magnitude and direction of $v\left(t\right)$ at $t=1.0s$

Answer 1

The position of the particle is gievn by,

$\overrightarrow{r}=\left(3.0t\hat{i}+2.0t^2\hat{j}+5.0\hat{k}\right)m$

Differentiating with respect to $t$ , we get

$v\left(t\right)=\left(3.0\hat{i}+4.0t\hat{j}\right)m/s^2$

Differentiating with respect to $t$ again , we get

$a\left(t\right)=\left(4.0\hat{j}\right)m/s^2$

At $t=3\sec$

$v\left(3\right)=\left(3.0\hat{i}+12.0\hat{j}\right)m/s$

Direction is calculated as,

$\theta ={\tan }^{-1}\left(\frac{v_y}{v_x}\right)$

$={\tan }^{-1}\left(\frac{12}{3}\right)$

$={76}^{\circ }$

Thus, the direction is ${76}^{\circ }$ above the positive x-axis.