Two ships $P$ and $Q$ move with constant velocities $\vec{u}$ and $\vec{v}$. At the initial moment,

Question

Two ships $P$ and $Q$ move with constant velocities $\vec{u}$ and $\vec{v}$. At the initial moment, their position vectors are $\vec{p}$ and $\vec{q}$ respectively. Which of the following defines the strict condition for the two ships to collide?

Professor PC Thomas Classes · Answer

(d) : Let the two ships collide after time $t$. At the instant of collision, their position vectors must be identical. Therefore, $\vec{p} + \vec{u}t = \vec{q} + \vec{v}t$, which rearranges to $\vec{p} - \vec{q} = (\vec{v} - \vec{u})t$. Since time $t$ is a positive scalar, the vector $(\vec{p} - \vec{q})$ must point in the exact same direction as $(\vec{v} - \vec{u})$. Hence, their unit directional vectors must be perfectly equal, which gives $\frac{\vec{p}-\vec{q}}{|\vec{p}-\vec{q}|} = \frac{\vec{v}-\vec{u}}{|\vec{v}-\vec{u}|}$.