Q100 / 180 · JIPMER 2017 · Physics
The $x$ and $y$ coordinates of a particle moving in a plane are given by $x(t)=a \cos (p t)$ and $y(t)=b \sin (p t)$ where $a, b( \lt a)$ and $p$ are positive constants of appropriate dimensions and $t$ is time. Then, which of the following is not true?
- AThe path of the particle is an ellipse.
- BVelocity and acceleration of the particle are perpendicular to each other at $t=\frac{\pi}{2 p}$
- CAcceleration of the particle is always directed towards a fixed point.
- Correct answerDDistance travelled by the particle in time interval between $t=0$ and $t=\frac{\pi}{2 P}$ is a
Solution
$\begin{aligned}
& x=a \cos (p t), y=b \sin (p t) \\
& \therefore \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \text { i.e. equation of ellipse } \\
& \text { Now, } r=x \hat{\mathbf{i}}+\hat{\mathbf{j}}=a \cos (p t) \hat{\mathbf{i}}+b \sin (p t \hat{\mathbf{j}} \\
& v=\frac{d r}{d t}=-p \operatorname{asin}(p t) \hat{\mathbf{i}}+p b \cos (p t) \hat{\mathbf{j}} \\
& a=\frac{d v}{d t}=-p^2 a \cos (p t) \hat{\mathbf{i}}-p^2 b \sin (p t) \hat{\mathbf{j}} \\
& \text { at } t=\frac{\pi}{2 p} \\
& v=-p a \hat{\mathbf{i}} \text { and } \mathbf{a}=-p^2 b \hat{\mathbf{j}}
\end{aligned}$
$v \text { perpendicular to a }$
Also, $a=-p^2 r$, i.e. directed towards a fixed point
Hence, the correct answer is option: 'D'.