Collision points

Question

Find all collision points between the parametric curves $x_1=\cos t$, $y_1=2 \sin t + \sin 2t$ and $x_2=2t$, $y_2= t - \frac12$.

I have no idea of how to proceed from the starting step of setting the $x$ values equal to each other, and the $y$ values equal to each other.

I also don't know what the answer should be.

Thanks!!

Answer 1

$x_1 = x_2 \,\Rightarrow \cos t=2t \,\Rightarrow t=0.450184$

$y_1=y_2 \, \Rightarrow 2\sin t + \sin(2t) = t-\frac{1}{2} \,\Rightarrow t=-1.547, -0.169352, 1.863$

They are only ever at the same x-coordinate when $t=0.4501284$ and only at the same y-coordinate when $t=-1.547, -0.169352, 1.863$, so since the parameter ($t$) of $x$ and $y$ is never the same when they are in the same coordinates the two particles never collide.

Answer 2

Let $t$ and $s$ be two values for which \begin{cases} \cos t=2s,\\ 2 \sin t + \sin 2t=s - \dfrac12 \end{cases} we have from first $s=\dfrac12\cos t$ and with the second $2 \sin t + \sin 2t=\dfrac12\cos t - \dfrac12$. After solve this we find $t=0$ and $\tan^3\frac{t}{2}+\tan \frac{t}{2}+8=0$. The last give $\tan \frac{t}{2}\sim-1.07$, with $t_1=0$ and $\tan \frac{1}{2}t_2\sim-1.07$ we find $s_1=\frac12$ and $s_2=-\frac18\tan \frac{1}{2}t_2-\frac12\sim-0.633$. So the collision points between two curves are $(1,0)$ and $(-1.266,-0.133)$.