If i'm not completely mistaken, your speed can gain up to your acceleration in a combat turn, so you would need to wait the second combat turn to be at full speed. As for the distance needed, if you use the speed*30kph = actual speed for groundcraft vehicle, I think you would need approximately 120 meters to get to speed 5 (that would take the first combat turn), and another 100 meters to get from speed 5 to 8 (that would take about half a combat turn).
Now, if someone is right in front of the car (I will assume less than 10 meters), and you want some realism, you will be at speed 1 when you hit him,
If you don't want to calculate all those distances (took me 2 minutes to do those in my head, but i'm a freaking engineer), you can always say that you are instantly from 0 to 5, or that your starting speed is determined by a pilot test with the limit being your acceleration. Then your speed for the rest of the turn would be that, and at the end of the turn you will reach that 5 speed. Or you can use whatever rule might be in the book that I don't remember right now but may be there.