# A mosquito is moving with a velocity

Question:

A mosquito is moving with a velocity $\overrightarrow{\mathrm{v}}=0.5 \mathrm{t}^{2} \hat{\mathrm{i}}+3 \mathrm{t} \hat{\mathrm{j}}+9 \hat{\mathrm{k}}$ m/s and accelerating in uniform conditions. What will be the direction of mosquito after $2 \mathrm{~s}$ ?

1. $\tan ^{-1}\left(\frac{2}{3}\right)$ from $x$-axis

2. $\tan ^{-1}\left(\frac{2}{3}\right)$ from y-axis

3. $\tan ^{-1}\left(\frac{5}{2}\right)$ from $y$-axis

4. $\tan ^{-1}\left(\frac{5}{2}\right)$ from x-axis

Correct Option: , 2

Solution:

Given :

$\overrightarrow{\mathrm{v}}=0.5 \mathrm{t}^{2} \hat{\mathrm{i}}+3 \mathrm{t} \hat{\mathrm{j}}+9 \hat{\mathrm{k}}$

$\overrightarrow{\mathrm{v}}_{\mathrm{at} t=2}=2 \hat{\mathrm{i}}+6 \hat{\mathrm{j}}+9 \hat{\mathrm{k}}$

$\therefore$ Angle made by direction of motion of mosquito will be,

$\cos ^{-1} \frac{2}{11}($ from x-axis $)=\tan ^{-1} \frac{\sqrt{117}}{2}$

$\cos ^{-1} \frac{6}{11}$ (from y-axis) $=\tan ^{-1} \frac{\sqrt{85}}{6}$

$\cos ^{-1} \frac{9}{11}$ (from z-axis) $=\tan ^{-1} \frac{\sqrt{40}}{9}$

None of the option is matching.

Hence this question should be bonus.