Hi, just saw this thread and years ago i provided the full lession in a somewhat different context (
http://www.racedepartment.com/threads/r ... ost-349119 ) but still true:
Power versus Torque
Or
When a car does accelerate best?
The acceleration (a) of a car is proportional to the force (Fw) which is coming from the driven wheels and therefore proportional to the torque at the wheel (Tw), which depends on the torque of the engine (Te) and on the transmission ratio (ne/nw), which is the rpm of the engine (ne) divided by the rpm of the driven wheels (nw) and thus a summary of everything in between engine and driven wheels like the radius of the wheels (rw), the diff-settings and the gear ratio. Introducing Power (P) as P=T*2*PI*n - thus meaning Power is proportional to the product of Torque and rpm; PI is that 3,14… - we can examine the acceleration using the general relations F=m*a and T=F*r as
a = Fw / m = Tw / (rw*m) whereas m is the mass of the car of course.
Now we use the transmission ratio to switch to the torque of the engine with Tw=Te*(ne/nw) and get
a = Te * (ne/nw) * 1/(rw*m) and name that equation with (I).
Needless to say in a racing forum that Te varies with ne. Now it’s time for the Power of the engine and we can use Pe=Te*2*PI*ne to replace the torque of the engine
a = Pe/(2*PI*ne) * (ne/nw) * 1/(rw*m)
and sorting things in a different way leads us to
a = Pe/(2*PI*rw*nw)* 1/m
which now gives us the idea to detect the actual velocity v=2*PI*rw*nw of the driven wheel rolling on the track which is under assumption of full grip the velocity of the car. So we finally have
a = Pe/v * 1/m and name that equation with (II).
Needless to say that Pe varies with ne, too, of course.
Now we have everything together.
Examining equation (I) we’ll see that for a fixed transmission ratio the acceleration would be at maximum for a maximum torque, but a fixed transmission ratio means we must stay in a fixed gear and we are not allowed to gear up or down! The assumption of having only one allowed gear is not very practical. Much more practical is the question: Having the same actual velocity at present, maybe we could reach a better acceleration with another gear?
Examining equation (II) which is independent of any transmission ratio we’ll see that for a given velocity the best acceleration is just where the Power is highest and so we choose our gear and rush away! This can’t always be the absolute maximum of power, because we are examining a given velocity and a somehow given gearbox and that must not fit.
[...]So in general the acceleration is at its best for the absolute maximum of power! Just to confuse you but true: The absolute maximum of acceleration is always reached using the 1st gear (or often even the reverse gear) at torque maximum - cause here we can't choose an "old" gear which could hit the absolute maximum of power; think about it...
[...]But we wanna drive racecars and they must be adjusted to the track, so:
(_) Setting up a racecar to a track therefore means just take the last gear to reach the highest speed in the maximum of power, take the first gear to the slowest corner and spread out the gears in between equal or maybe in respect to some corners – and don’t care about torque.
(_) Shifting with that gearbox always will mean that you have to shift up if the new gear will have a better power than your present one and that must be after the maximum of power with your old gear, of course, and anyway if it’s a drag-setup or a track-setup! – Shifting at the absolute maximum of power always means a lack of acceleration.
But don't forget endurance in longer races.
