s_0 is dstart. Sorry for using different notation.
t_max is the time when the car nearly touches the dragsters bumper. The question says that the time the dragster sets off is t= 0. Thus we're setting the distances equal at t_max to find the starting distance of the car.
The car starts at some position behind the dragster when the dragster sets off (i.e. t=0). then at t_max it has covered that distance to the dragsters start position and the distance the dragster travels in a time t_max.
That just means that the drag racer waits until he knows the car will just miss his bumper at the later time t = t_max. So when the dragster starts off there is a certain distance (i.e. d_start) between him and the car. If he started just before the car hit him then there would definitely be a crash unless the dragser could accelerate almost instantly.
From the way the question is set up it is in the negative x direction if you like. To make it work in the equations it must have a minus sign that is.
No it looks fine.
Well thats exactly what you have done and is the reason that extra distance term appears in the cars distance equation. I'm just thinking the distance equation for the car might have to subtract the the distance since it starts behind the dragster and the direction they travel in is defined positive.