Question

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

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

Answer 1

Answer: (A)

$53^{o}$with x-axis

Answer 2

Give : $\overset{\rightarrow}{r} = 3t\widehat{i} + 2t^{2}\widehat{j} + 5\widehat{k}$

Velocity,

$\overrightarrow{v} = \frac{d\overrightarrow{r}}{dt} = \frac{d}{dt}(3t\widehat{i} + 2t^{2}\widehat{j} + 5\widehat{k}) = 3\widehat{i} + 4t\widehat{j}ms^{- 1}$

At t = 1 s, $\overrightarrow{v} = 3\widehat{i} + 4\widehat{j}ms^{- 1}$

Let $\theta$be angle which the directions of $\overset{\rightarrow}{v}$ makes with the x-axis,

Then

$\tan\theta = \frac{v_{y}}{v_{x}} = \frac{4}{3}or\theta = \tan^{- 1}\left( \frac{4}{3} \right) = 53{^\circ}$