What is relative distance and how can you tell?

Question

In velocity vectors (Apparent path), the time taken for 2 objects to intercept is:

$$\frac{\text{Relative distance}}{\text{Relative velocity}}$$

What does Relative distance mean, and how do can you tell?

Answer 1

In this case, relative distance just means distance. If the two objects are described as $x(t),y(t)$, the distance is $$ rd(t) = \|x(t)-y(t)\|_2 $$ were $\|.\|_2$ is the euclidian norm on $\mathbb{R^3}$. It is more or less defined by the pythagorean theorem.

$$\left\|\left(\begin{array}{lll}x_1\\x_2\\x_3\end{array}\right)\right\|_2 = \sqrt{x_1^2+x_2^2+x_3^2}$$

Denote $\dot{x}$ as the derivative of $x$ with respect to the time $t$, then the relative velocity can described as $$rv(t) = \|\dot{x}(t)-\dot{y}(t)\|_2.$$ Since $x$ and $y$ are vectors it is necessary to take the norm of both expressions. Otherwise, the division in your formula makes no sense.

Answer 2

When 2 objects collide, the initial relative displacement vector is parallel to the relative velocity.

To calculate the time taken, you need the initial distance between them (magnitude of initial relative displacement) divided by the relative speed (magnitude of relative velocity).