Two particles start simultaneously from the same point and move along two straight lines, …

Question

Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity $\vec u$ and the other from rest with uniform acceleration $\vec f$. Let $\alpha$ be the angle between their directions of motion. The relative velocity of the second particle with respect to the first is least after a time:

Accepted Answer

Let $\vec v_1=u\hat i$, $\vec v_2=(ft\cos\alpha)\hat i+(ft\sin\alpha)\hat j$. Relative velocity $\vec v=\vec v_2-\vec v_1=(ft\cos\alpha-u)\hat i+ft\sin\alpha\,\hat j$

$$|\vec v|^2 = f^2t^2+u^2-2uft\cos\alpha$$

Minimizing: $\dfrac{d|\vec v|^2}{dt}=2f^2t-2uf\cos\alpha=0 \implies t=\dfrac{u\cos\alpha}f$ (and the second derivative confirms a minimum).

Solutions · 1