A particle moves such that its position vector where w is a constant and t is time. Then which of the following statements is true for the velocity and acceleration of the particle:

Engineering

Physics

Horizontal Circular Motion

Question

A particle moves such that its position vector $\vec{r} (t) = cosωt \hat{i} + sinωt \hat{j}$ where w 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:

$\vec{v}$ is perpendicular to $\vec{r} and \vec{a}$ is directed towards the origin.

$\vec{v} and \vec{a}$ both are parallel to $\vec{r}$

$\vec{v} and \vec{a}$ both are perpendicular to $\vec{r}$

$\vec{v}$ is perpendicular to $\vec{r} and \vec{a}$ is directed away from the origin

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$\vec{r} = cosωt \hat{i} + sinωt \hat{j}$

$\vec{v} = \frac{d \vec{r}}{dt} = ω (- sinωt \hat{i} + cosωt \hat{j})$

$\vec{a} = \frac{d \vec{v}}{dt} = - ω^{2} (cosωt \hat{i} + sinωt \hat{j})$

$\vec{a} = - ω^{2} \vec{r} ∴ \vec{a} is antiparallel to \vec{r}$

$\vec{v} \cdot \vec{r} = ω (- sinωtcosωt + cosωtsinωt) = 0$

So $\vec{v} ⊥ \vec{r}$