Consider a particle following the parametric equations
\begin{align*}
x &= 1 + \sin (\pi t),\\
y &= 3 \sin (\pi t).
\end{align*}a) Give a precise description of the graph of these parametric equations if we allow $t$ to be any real number.
b) Describe the path the particle takes from time $t = 0$ to $t = 2$.
c) Find a parametrization such that the graph of this parametrization from $t = 0$ to $t = 2$ matches our graph above, but the motion of the particle is different.
I finished part (a), and I found that the graph covers the line $y=3x-3$ where $x\in[0,2]$. For part (b), I plotted some points out, and I think I know what the answer is. However, I'm having trouble proving the answer. I'm also not sure how to do part (c). I would greatly appreciate it if anyone could help!
Thanks in advance!!!
Best Answer
The path taken by the particle is on the straight line $y = 3x - 3$ from point $(1,0)$ at $t = 0$ to point $(2.3)$ at $t = \frac 12$ and then from $(2,3)$ in opposite direction to point $(0, -3)$ at $t = \frac 32$. Then finally from $(0, -3)$ back to starting position $(1, 0)$ at $t = 2$.
Now what you are looking for is for the particle to follow a parametric curve with the same graph for $t \in (0, 2)$ but different motion than above. For example, $(t, 3t-3), t \in (0,2)$, where the path is from point $(0,-3)$ at $t = 0$ to $(2,3)$ at $t=2$