Suppose two cars are racing along a (straight) road at a constant speed $v_{0}$ m/s. At time $t = 0$, Car 2 is ahead of Car 1 by $d_{0}$ meters; or, one could say, Car 1 is losing by $d_{0} / v_{0}$ seconds.
At time $t = 0$, Car 2 speeds up, with a constant acceleration of $a$ m/sec2 for $t_{0}$ seconds, and then proceeds at the new speed $v_{0} + at_{0}$.
My question is this: At time 0, Car 1 is losing the race by $d_{0} / v_{0}$ seconds. How does Car 1's speed need to change in order to maintain this "margin of loss", regardless of the length of the race? That is, how does Car 1's speed need to change so that no matter how long the race, it will always lose by $d_{0} / v_{0}$ seconds?
Is there a closed form for this?
Edit: Here's a way to understand the problem, completely separated from the physical interpretation.
Consider a function $c_{2}(t)$ defined as follows:
- $c_{2}(t) = vt + d$ (for some positive $v$ and $d$) for $t \lt 0$
- $c_{2}(t) = \frac{1}{2}at^2 + vt + d$ for $0 \leq t \lt s$ (for some positive $s$, $a$)
- $c_{2}(t) = (as + v)t + (\frac{1}{2}as^2 + vs + d)$ for $s \leq t$
Now define a second function $c_{1}(t)$ such that for all $t$,
$\frac{c_{2}(t) - c_{1}(t)}{c_{1}'(t)} = d / v$
Is there a closed-form description of $c_1$?
Car 1 just has to make sure that its speed at any point is the same as Car 2's speed at the same point. If you like, Car 1 executes a video replay of Car 2, delayed by $d_0/v_0$ seconds.