JEE MAIN - Physics (2021 - 16th March Evening Shift - No. 6)
A mosquito is moving with a velocity $$\overrightarrow v = 0.5{t^2}\widehat i + 3t\widehat j + 9\widehat k$$ m/s and accelerating in uniform conditions. What will be the direction of mosquito after 2 s?
$${\tan ^{ - 1}}\left( {{\sqrt {85} } \over 6}\right)$$ from y-axis
$${\tan ^{ - 1}}\left( {{5 \over 2}} \right)$$ from y-axis
$${\tan ^{ - 1}}\left( {{2 \over 3}} \right)$$ from x-axis
$${\tan ^{ - 1}}\left( {{5 \over 2}} \right)$$ from x-axis
Explanation
$$\overrightarrow v = (0.5{t^2}\widehat i + 3t\widehat j + 9\widehat k)$$ m/s
At t = 2 s
$$\overrightarrow v = (2\widehat i + 6\widehat j + 9\widehat k)$$
Direction cosine along y-axis,
$$cos\theta = {{(v.\widehat j)} \over {\sqrt {{9^2} + {6^2} + {2^2}} }} = {6 \over {\sqrt {121} }} = {6 \over {11}}$$
$$ \therefore $$ $$\sin \theta = {{\sqrt {85} } \over {11}}$$
and $$\tan \theta = {{\sqrt {85} } \over 6}$$
$$ \therefore $$ Mosquito make angle $${\tan ^{ - 1}}\left( {{\sqrt {85} } \over 6}\right)$$ from y-axis.
At t = 2 s
$$\overrightarrow v = (2\widehat i + 6\widehat j + 9\widehat k)$$
Direction cosine along y-axis,
$$cos\theta = {{(v.\widehat j)} \over {\sqrt {{9^2} + {6^2} + {2^2}} }} = {6 \over {\sqrt {121} }} = {6 \over {11}}$$
$$ \therefore $$ $$\sin \theta = {{\sqrt {85} } \over {11}}$$
and $$\tan \theta = {{\sqrt {85} } \over 6}$$
$$ \therefore $$ Mosquito make angle $${\tan ^{ - 1}}\left( {{\sqrt {85} } \over 6}\right)$$ from y-axis.
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