Question

A particle moves in x-y plane such that its position vector varies with time as $\overline{r}$ = (2sin3t)$\hat{i}$ +2(1-cos3t)$\hat{j}$. Find the equation of the trajectory of the particle.

Answer 1

Comparing r = (2sin3t)$\hat{i}$ + 2(1 - cos3t)$\hat{j}$
with $\overline{r}$ = x$\hat{i}$ + y$\hat{j}$, we have x= 2 sin 3t and y = 2(1 - cos 3t).
This gives sin3t = $\frac{x}{2}$ and cos 3t = 1 - $\frac{y}{2}$.
Eliminating t by squaring and adding the above terms, we have
$\frac{x^2}{4}$ + $(1-\frac{y^2}{2})$ = 1