Collisions in Two Dimensions — Question 10

Question 10

Crossing paths is a separate requirement from vector conservation

Two small pucks move freely on a horizontal plane. Puck A starts at $(-2,0)\,\mathrm m$ with velocity $(2,0)\,\mathrm{m/s}$; puck B starts at $(0,-3)\,\mathrm m$ with velocity $(0,2)\,\mathrm{m/s}$. Treat them as points.

1. Determine whether they collide and find their minimum separation.

2. What speed must B have along $+y$ for collision at the crossing, and why must this be checked before applying collision equations?

Question 10 - Solution

Crossing paths is a separate requirement from vector conservation

(a) Derivation and calculation

The trajectories are $\vec r_A=(-2+2t,0)$ and $\vec r_B=(0,-3+2t)$. Their origin-crossing times are 1 s and 1.5 s, so they miss. The separation satisfies $$d^2=(2-2t)^2+(-3+2t)^2=8(t-1.25)^2+0.50.$$ $$\boxed{t_{\min}=1.25\ \mathrm s,\qquad d_{\min}=1/\sqrt2\ \mathrm m}.$$

(b) Result and interpretation

For B to reach the origin simultaneously with A, $$-3+v_B(1)=0\Rightarrow\boxed{v_B=3\ \mathrm{m/s}}.$$ A collision requires the same position at the same time. Conservation equations for an interaction apply only once this contact condition is satisfied.

Physical check

Intersecting geometric paths alone do not guarantee a collision.