Velocity and Acceleration — Question 8

Question 8 – Solution

Strategy Path intersection may use different parameter values; collision requires equal positions at the same time.

Path intersection Write the first path as $y=x$. On the second, $x=2-u$ and $y=u^2-1$. Thus $$u^2+u-3=0,\qquad u=\frac{-1\pm\sqrt{13}}2.$$ The full paths intersect at $$\boxed{\left(\frac{5\mp\sqrt{13}}2,\frac{5\mp\sqrt{13}}2,0\right)}.$$ If both paths are restricted to nonnegative times, only the point with $5-\sqrt{13}$ remains.

Collision test At the same time $t$, equality of $x$-coordinates requires $t=2-t$, so $t=1$. But $$\vec r_1(1)=\left\langle 1,1,0\right\rangle,\qquad\vec r_2(1)=\left\langle 1,0,0\right\rangle.$$ Therefore $\boxed{\text{the particles do not collide}}$. The geometric intersections occur at different parameter times.