Question

The position of a particle is given by $\overrightarrow{r}$=3t $\hat{i}$+2 $t^2$ $\hat{j}$+5 $\hat{k}$, where t is in seconds and the coefficients have the proper units for r to be in metres. The direction of velocity of the particle at t = 1 s is:

• A. 53${}^{\circ }$ with x-axis
• B. 37${}^{\circ }$ with x-axis
• C. 30${}^{\circ }$with y-axis
• D. 60${}^{\circ }$ with y-axis

Answer 1

Answer: (A)

. 53${}^{\circ }$ with x-axis.

Given: $\overrightarrow{r}=3t\hat{i}+2t^2\hat{j}+5\hat{k}$

We know that,
Velocity, $\overrightarrow{v}=\frac{d\overrightarrow{r}}{dt}=\frac{d(3t\hat{i}+2t^2\hat{j}+5\hat{k})}{dt}$

$\overrightarrow{v}=3\hat{i}+4t\hat{j}m{s}^{-1}$

Let $\theta$ be the angle in which the direction of $\overrightarrow{v}$ makes with the x-axis.

At $t=1sec:$

Then $tan\theta$ = $\frac{v_y}{v_x}=\frac{4t}{3}=\frac{4}{3}$ or

$\theta =ta{n}^{-1}\left(\frac{4}{3}\right)={53}^{\circ }$.