The position vector of a particle $\vec{R}$ as a function of time is given by $\vec{R}=4 \sin (2 \pi

Question

The position vector of a particle $\vec{R}$ as a function of time is given by $\vec{R}=4 \sin (2 \pi t) \hat{i}+4 \cos (2 \pi t) \hat{j}$ where $R$ is in meters, $t$ is in seconds and $\hat{i}$ and $\hat{j}$ denote unit vectors along $x$-and $y$-directions, respectively. Which one of the following statements is wrong for the motion of particle?\ is the velocity of particle. is the velocity of particle.\

Professor PC Thomas Classes · Answer

(a): Here, $\vec{R}=4 \sin (2 \pi t) \hat{i}+4 \cos (2 \pi t) \hat{j}$ The velocity of the particle is\ $\vec{v}=\frac{d \vec{R}}{d t}=\frac{d}{d t}[4 \sin (2 \pi t) \hat{i}+4 \cos (2 \pi t) \hat{j}]$ $$ =8 \pi \cos (2 \pi t) \hat{i}-8 \pi \sin (2 \pi t) \hat{j} $$ Its magnitude is $|\vec{v}|=\sqrt{(8 \pi \cos (2 \pi t))^{2}+(-8 \pi \sin (2 \pi t))^{2}}$ $$ \begin{aligned} & =\sqrt{64 \pi^{2} \cos ^{2}(2 \pi t)+64 \pi^{2} \sin ^{2}(2 \pi t)} \ & =\sqrt{64 \pi^{2}\left[\cos ^{2}(2 \pi t)+\sin ^{2}(2 \pi t)ight]} \ & =\sqrt{64 \pi^{2}} \quad\left(	ext { As } \sin ^{2} 	heta+\cos ^{2} 	heta=1ight) \ & =8 \pi \mathrm{~m} / \mathrm{s} \end{aligned} $$