Making the most of your powerband
What is a powerband?
A powerband is the interval of RPMs at which either the engine produces the most power, or where the most power is put to the wheels in each gear.
Okay, so when do I shift?
Now an engine will always have the same powerband, but where you want to shift may differ slightly based on the gearing of a car. Lets say your engine's power curves peak around 5.6k. First instict would make it seem that you want to shift right at the peak of power. However, with a little reason, one can see that it makes more sense to shift somewhat after the peak, so that when the revs drop, they are still near the peak potential of the engine. Ideally you want the engine's peak power to be roughly in the middle of the rev range you are utilizing in each gear.
Now what's that you said about gearing though?
Due to factors such as gearing, and velocity dependent factors such as drag, the best place to shift may be slightly higher or lower than where you would considering engine revs alone. The best way to find this would be to start with what appears to be the best based on the engine's power curve, and experiment on a dyno. Generally you will want to shift at slightly higher revs in the higher gears, but there is no tell-all for this. You need to experiment on a dyno, and ideally on a timed 1/4 to figure out the best shift points for each gear. If you can, change one gear at a time. For example, once you find the best time to shift out of first, then start playing with second. You will get better results working up from 1st than back from 5th or what have you.
Launching
Just as with shifting, it may seem that the most effective launch would be at the peak of the torque curve. Not the case at all. The most effective launch is generally the one at which you produce the most torque that the limiting components of your car can handle. Those limiting components are generally the tires, and in less common cases, the clutch. If your tires are spinning, you're probably launching too high, and you're wasting time and power spinning the tires instead of the car. Its also not good to dump the clutch at high revs. This will no doubt strip the clutch if you make a habit out of it.
On stock tires less than 1k is probably ideal, maybe up to 15 or so, I don't know. The better grip your tires have on the surface, the higher you can launch at, and thus the faster. Higher grip tires enable you to launch faster, and mitigating circumstances such as balding tires, wet/uneven pavement, or anything else that reduces friction, will lower your ideal launching point.
