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JEE Main & Advanced JEE Main Paper Phase-I (Held on 08-1-2020 Evening)

  • question_answer
    A particle moves such that its position vector \[\vec{r}\left( t \right)=cos\omega t\,\hat{i}\text{+}sin\omega t\,\hat{j}\] where \[\omega \] is a constant and t is time. Then which of the following statements is true for the velocity \[\vec{v}(t)\] and acceleration \[\vec{a}(t)\]  of the particle [JEE MAIN Held on 08-01-2020 Evening]

    A) \[\overrightarrow{v}\]and \[\overrightarrow{a}\]both are perpendicular to \[\overrightarrow{r}\]

    B) \[\overrightarrow{v}\] is perpendicular to \[\overrightarrow{r}\] and \[\overrightarrow{a}\] is directed towards the origin

    C) \[\overrightarrow{v}\] and \[\overrightarrow{a}\] both are parallel to \[\overrightarrow{r}\]

    D) \[\overrightarrow{v}\] is perpendicular to \[\overrightarrow{r}\]and \[\overrightarrow{a}\] is directed away from the origin

    Correct Answer: B

    Solution :

    \[\vec{r}=cos\omega t\text{ }\hat{i}+sin\omega t\,\,\hat{j}\] \[\vec{v}=\frac{d\vec{r}}{dt}=\omega \left( -\sin \omega t\,\,\hat{i}+\cos \omega t\,\,\hat{j} \right)\] \[\vec{a}=\frac{d\vec{v}}{dt}=-{{\omega }^{2}}\left( \cos \omega t\,\,\hat{i}+\sin \omega t\,\,\hat{j} \right)\] \[\vec{a}=-{{\omega }^{2}}\vec{r}\]    \[\therefore \,\,\vec{a}\,\,is\,\,antiparallel\,\,to\,\,\vec{r}\] Also \[\vec{v}\cdot \vec{r}=0\]               \[\therefore \,\,\,\,\,\vec{v}\bot \vec{r}\] Actually particle is in state of uniform circular motion.


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