Kinematics

JEE Main level, single correct, medium

306 of 308

Question 306 of 308

A particle moves such that its position vector r⃗(t)=cos⁡⁡ωti^+sin⁡⁡ωtj^\vec{r}\left(t\right)=\cos ⁡\omega t\hat{i}+\sin ⁡\omega t\hat{j} where ω\omega is a constant and tt is time. Then which of the following statements is true for the velocity v⃗(t)\vec{v}\left(t\right) and acceleration a⃗(t)\vec{a}\left(t\right) of the particle:

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Answer: (D) v⃗\vec{v} is perpendicular to r⃗\vec{r} and a⃗\vec{a} is directed towards the origin

r⃗=cos⁡⁡ωti^+sin⁡⁡ωtj^\vec{r}=\cos ⁡\omega t\hat{i}+\sin ⁡\omega t\hat{j}
v⃗=dr⃗dt=ω(−sin⁡⁡ωti^+cos⁡⁡ωtj^)\vec{v}=\dfrac{d\vec{r}}{dt}=\omega \left(-\sin ⁡\omega t\hat{i}+\cos ⁡\omega t\hat{j}\right)
a⃗=dv⃗dt=−ω2(cos⁡⁡ωti^+sin⁡⁡ωtj^)\vec{a}=\dfrac{d\vec{v}}{dt}=-{\omega }^{2}\left(\cos ⁡\omega t\hat{i}+\sin ⁡\omega t\hat{j}\right)
a⃗=−ω2r⃗∴a⃗\vec{a}=-{\omega }^{2}\vec{r}∴\vec{a} is antiparallel to r⃗\vec{r}
v⃗.r⃗=ω(−sin⁡⁡ωtcos⁡⁡ωt+cos⁡⁡ωtsin⁡⁡ωt)=0\vec{v}.\vec{r}=\omega \left(-\sin ⁡\omega t\cos ⁡\omega t+\cos ⁡\omega t\sin ⁡\omega t\right)=0
So v⃗⊥r⃗\vec{v}\perp \vec{r}

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