Question

The x and y coordinates of a particle moving in a plane are given by $x\left(t\right)=a\cos \left(pt\right)$ and $y\left(t\right)=b\sin \left(pt\right)$ where a, b$(<a)$ and p are positive constants of appropriate dimensions and t is time. Then, which of the following is not true?

• A. The path of the particle is an ellipse
• B. Velocity and acceleration of the particle are perpendicular to each other at $t=\frac{\pi }{2\rho }$
• C. Acceleration of the particle is always directed towards a fixed point
• D. Distance travelled by the particle in time interval between $t=0$ and $t=\frac{\pi }{2P}$ is a

Answer 1

The correct option is D Distance travelled by the particle in time interval between $t=0$ and $t=\frac{\pi }{2P}$ is a

$x=a\cos \left(pt\right),y=b\sin \left(pt\right)$
$\therefore \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, i.e., equation of ellipse
Now, $r=x\hat{i}+y\hat{j}=a\cos \left(pt\right)\hat{i}+b\sin \left(pt\right)\hat{j}$
$v=\frac{dr}{dt}=-pa\sin \left(pt\right)\hat{i}+pb\cos \left(pt\right)\hat{j}$
$a=\frac{dv}{dt}$ $=-p^{2}a\cos \left(pt\right)\hat{i}=p^{2}b\sin \left(pt\right)\hat{j}$
at $t=\frac{\pi }{2p}$
$v=-pa\hat{i}$ and $a=-p^{2}b\hat{j}$
v perpendicular to a
Also, $a=-p^{2}r$, i.e., directed towards a fixed point.