QJEEM20Ph012b: JEE Main 2020 Physics - Class 11 JEEM20Ph012b Solution: A particle moves such that its position vector where is a…

Question

Verified Solution • 2025–26 Session moderate

A particle moves such that its position vector

\overrightarrow r \left( t \right) = \cos \omega t\widehat i + \sin \omega t\widehat j

where

\omega

is a constant and t is time. Then which of the following statements is true for the velocity

\overrightarrow v \left( t \right)

and acceleration

\overrightarrow a \left( t \right)

of the particle :

A $\overrightarrow v$ and $\overrightarrow a$ both are perpendicular to $\overrightarrow r$
B $\overrightarrow v$ and $\overrightarrow a$ both are parallel to $\overrightarrow r$
C $\overrightarrow v$ is perpendicular to $\overrightarrow r$ and $\overrightarrow a$ is directed towards the origin
D $\overrightarrow v$ is perpendicular to $\overrightarrow r$ and $\overrightarrow a$ is directed away from the origin

Accepted Answer

\overrightarrow r \left( t \right) = \cos \omega t\widehat i + \sin \omega t\widehat j

\overrightarrow v = {{d\overrightarrow r } \over {dt}}

=

- \omega \sin \omega t\,\widehat i + \omega \cos \omega t\widehat j

\overrightarrow a = {{d\overrightarrow v } \over {dt}}

=

- {\omega ^2}\cos \omega t\,\widehat i - {\omega ^2}\sin \omega t\widehat j

=

- {\omega ^2}\left( {\cos \omega t\,\widehat i + \sin \omega t\widehat j} \right)

=

- {\omega ^2}\overrightarrow r

\therefore

\overrightarrow a

is antiparallel to

\overrightarrow r

and it's direction towards the origin.

\overrightarrow v .\overrightarrow r =

\omega \left( { - \sin \omega t\cos \omega t + \cos \omega t\sin \omega t} \right)

= 0

So

\overrightarrow v \bot \overrightarrow r

.

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