Completely stumped on this one -- Kinematic Conceptual problem

[Ray Vickson]

I'm just going off an assumption here. Could it be because Xo is prior to your starting point? So anything g behind your initial position is negative?

Of course. Surely you must have seen a "number line" before? East = positive $x$, west = negative $x$.

 

[kuruman]

Now that the problem has been marked "solved", I can contribute a quicker way to get to the answer. The required distance $x_{min.}$ will depend on two independent parameters, the constant speed $v_0$ of the trailing car and the maximum acceleration $a$ of the leading car. The leading car must reach speed $v_0$ over the minimum distance $x_{min.}$. If it moves slower than that, it started accelerating too late; if it moves faster than that, it started accelerating too soon. Therefore, from the "fourth" kinematic equation $$2ax_{min.}=v_0^2 - 0^2 \rightarrow x_{min.}=\frac{v_0^2}{2a}$$

 

[haruspex]

Now that the problem has been marked "solved", I can contribute a quicker way to get to the answer. The required distance $x_{min.}$ will depend on two independent parameters, the constant speed $v_0$ of the trailing car and the maximum acceleration $a$ of the leading car. The leading car must reach speed $v_0$ over the minimum distance $x_{min.}$. If it moves slower than that, it started accelerating too late; if it moves faster than that, it started accelerating too soon. Therefore, from the "fourth" kinematic equation $$2ax_{min.}=v_0^2 - 0^2 \rightarrow x_{min.}=\frac{v_0^2}{2a}$$

C.f. post #23.

 

[kuruman]

C.f. post #23.

Sure, but it's not necessary to run time backwards. In either car's reference frame the other car drops its speed from initial speed $v_0$ to zero over distance $x_{min.}$.

 

[haruspex]

Sure, but it's not necessary to run time backwards. In either car's reference frame the other car drops its speed from initial speed $v_0$ to zero over distance $x_{min.}$.

Yes, it is not necessary to run time backwards; I was just trying to make it exactly like the commonest example of the equation.