Answer
The particle moves along a parabola described by $y = -x^2 + 1$.
First, it will go from $(0,1)$ until it reaches the x-axis at $(1,0)$. Then it will start to move counterclockwise, until it reaches the x-axis again, at $(-1,0)$ and move clockwise again. The process will repeat until $t = 2\pi$, at $(0,1)$
Work Step by Step
1. Convert the equation to a Cartesian one, so we can identify the type of curve:
$x = sin (t)$
$x^2 = sin^2(t)$
$y = cos^2(t)$
Adding the equations:
$y + x^2 = sin^2(t) + cos^2(t)$
$y + x^2 = 1$
$y = -x^2 + 1$
- As we can see by the equation, this curve is a parabola, and it is oppening to the bottom.
** Notice: since $y$ is determined by $cos^2(t)$, this coordinate can only assume positive values (or 0).
2. Find the initial and final point:
Initial point: $t = -2\pi$
$x = sin(-2\pi) = 0$
$y = cos^2(-2\pi) = 1$
Final point: $t = 2 \pi$
$x = sin(2\pi) = 0$
$y = cos^2(2\pi) = 1$
- For simple trigonometric functions like $sin(t)$ and $cos^2(t)$, the particle represented by $(x,y)$ completes a cycle after $t$ increasing by $2\pi$.
From $-2\pi$ to $2\pi$, there is a total of $4\pi$.
Therefore, the particle does a total of 2 complete cycles along the parabola.
- Due to the fact that $cos^2(t)$ can only assume positive values, the particle will bounce at $y = 0$, and start to move in the opposite direction.
- Since $sin(t)$ determines the $x$ position, and $cos^2(t)$ determines the $y$ position, the particle will start to move clockwise along the ellipse
Firstly, it will go from $(0,1)$ until it reaches the x-axis at $(1,0)$. Then it will start to move counterclockwise, until it reaches the x-axis again, at $(-1,0)$ and return to moving clockwise. The process will repeat until $y = 2\pi$
** The curve reaches the x-axis when $y=0$:
$0 = -x^2 + 1$
$x^2 = 1$
$x=1$ or $x=-1$
