Maximum Acceleration: Peak Power Vs Peak Torque
Peak power versus peak torque - which one will deliver the highest acceleration?
If you buy a nice lively car like the Hyundai i30 SR Premium - nothing outrageous, but a definite step up from boredom on wheels, the chances are you’ll want to make it perform one day - really perform. Not all the time - just occasionally.
You don’t have to wear your cap sideways and incorporate the word ‘yo’ within your vernacular.
You’ve got seven gears, down there, in the dual-clutch transmission, and many of them can be used at most legal speeds. Your mission is to pick the one that delivers the maximum performance. So basically there are two options.
Need a primer on power and torque: Here's my Idiot's guide to power & torque >>
Power V Torque
Some people say you get maximum acceleration at any speed with the engine at peak power, and some people say it’s peak torque from the engine that delivers the maximum possible acceleration.
As hypotheticals go - I can see they’ve both got merit. But they can’t both be right. So let us unleash some righteous jihad on bullshit using the hi-tech miracle of physics.
First I want to be very clear what I mean by ‘maximum acceleration’. Obviously cars operate through a great many speeds, and the maximum possible acceleration obviously varies with speed. You typically get higher accelerations at lower speeds.
What I’m talking about is taking a snapshot in time, with the car at some arbitrary speed - and then working out how to get the maximum acceleration out of the car at that speed. I’m going to use 75 km/h here (that’s about 45 mph) - but you can use any particular speed - the actual amount of acceleration possible will change, but the revs you need to achieve it will not.
Which Revs for Peak Acceleration?
Look at it like this:
You clip the apex some speed, and start winding on the power. To get the most performance out of the car, you need to get maximum acceleration. There’s only one set of revs that will achieve this. Our mission today is to find out what revs will achieve that.
I’m also going to assume the tyres keep gripping the road - meaning we don’t spin the wheels. And we use wide-open throttle, because obviously engine performance is only available if you crack the throttle wide-open.
Acceleration depends on only two things: the mass of the car, which is generally pretty constant, and the accelerative force the driveline is capable of applying. That’s Newton’s second law of motion.
TRACTIVE EFFORT MINUS DRAG
And of course you need to subtract any resistances the car is battling against - like rolling resistance of the tyres, and aerodynamic drag, which is hugely speed dependent.
There’s also gravity, which helps if you’re running downhill, and hurts if you’re slogging uphill.
So: How do we maximise the force the driveline applies to the road?
The shove forwards comes from the wheels, obviously. Specifically the wheels doing the driving - in this case, the front two.It’s actually the torque driving the wheels around, like literally wrenching the car forward. (You can move torques around anywhere on rigid bodies - it doesn’t make any difference to their behaviour.) The wheels are torqueing the car forwards.
And that force forwards - tractive effort - depends on only two things: First there’s the torque at the wheels, which comes from the crankshaft, but gets tuned up via the gearbox and final drive reduction.
Second, you need to factor in the rolling radius of the tyre. 319 millimetres here.
So for maximum acceleration of the car at any snapshot in time, we need to maximise the torque at the driving wheels. Hold that thought and let’s think about the engine.
ENGINE OUTPUTS: POWER, TORQUE & REVS
Three things come out of your engine: Power, torque and revs. Power is conceptually complicated, because you can’t see or touch it, but it doesn’t have to be. Power equals torque times revs. It’s that simple - but you have to get the units right.
I’ve done a whole video on this - link in the description - but in brief, you balance everything up by getting the units right. It’s easy. Kilowatts equals Newton-metres times rpm divided by 9549. If you work in imperial units in your neck of the woods, horsepower equals lb-ft times rpm divided by 5252.
People get bogged down by power and literally coming to grips with it. It’s the time rate of change of energy. Force times velocity. The time derivative of mass times acceleration times distance. Literally it’s the rate at which the car acquires kinetic energy. Minus losses.
These glib explanations don’t help.
You don’t need to understand power like some physics lab Yoda - you just need to use it. Thankfully, if you can figure out how to paint a wall, you can work with power.
To paint a wall, you need to know the area, and the area depends on the length and the width. Area equals length times width. Easy. Even a politician would get that.
Power’s kinda the same. It equals torque times revs (with a numeric fudge to get the revs right, discussed earlier). It’s that simple.
Power, torque and revs go into the gearbox and then the final drive, and then they get to the wheels.
You can look up the specs and multiply the gear ratio by the final drive ratio and get an overall gearing reduction at the wheels, for each gear.
Bear in mind the dual-clutch i30s have two different final drive ratios - there’s trap for young players.
Gearing? Really simple.
If you’ve got a two-for-one reduction gear, shaft speed halves on the output side, and torque doubles. And - even simpler, power stays the same - because power equals torque times revs.
Three-to-one: one-third the speed, three times the torque. But there are losses.
Need more? Wikipedia's entry on gearing >> is pretty good.
Unfortunately there’s quite a bit of friction in transmissions, and about 15 per cent of power is lost mechanically on the way through. So the speeds are mechanically locked by the gearing but about 15 per cent of the torque, and therefore 15 per cent of the power, goes away.
It’s a cost of doing business with gears and oil.
SIX GEARS TO CHOOSE FROM
I chose the i30 SR as a demonstrator of sorts here for a reason. Because 75 km/h is real interesting in the i30 - first gear is a non-starter, obviously, because the engine would be revving at over 10,000 revs. Good luck with that. But you could easily be in second, third, fourth, fifth, sixth or even seventh at 75 kays an hour.
If you’re in second, the engine’s at 6000rpm making its peak of 150 kilowatts. But third to seventh is interesting. In third, you’re at 4100 revs, and in seventh you’re at 1600 - so the engine’s making its peak torque in all gears above third. 265 Nm. Because the SR makes peak torque from 1500 to 4500rpm.
So then I calculated the wheel revs at 75 km/h. Dead simple - you can do that from the tyre size. They’re 225/40R18. If you’re at 75, those wheels are doing 625 rpm.
Doesn’t matter what gear you’re in. Obviously. If you’re at 75, that’s how fast the wheels are turning.
CRANK POWER V WHEEL POWER
So, we’ve got 150 kilowatts at 6000rpm for second gear at 75, and you can calculate the power out of the crank at all other gears, because power equals torque times rpm divided by whatever units you’re working in - metric or imperial.
So there’s 150 kilowatts coming out of the crank at 6000rpm, 125 at 4500 and just 42 at 1500.
But you lose 15 per cent of that power getting to the wheels so there’s actually 128 kilowatts at the wheel at 6000rpm, 106 kilowatts at 4500rpm and just 35 kilowatts at 1500rpm.
See how the power drops even though the engine torque is the same at 4500 as it is at 1500?
Because we know the power and the revs at the wheels (wheel revs are 625 at 75 km/h) we can calculate the torque at the wheels.
We simply use the same equation as before, only on the wheel, not the engine. And once we know that, we can calculate the force the driveline applies to the car via the wheels (you just divide by the radius of the wheel - 0.319 metres). And then, to make it digestible, you can divide that force by 9.81
FINAL RESULTS: WHICH GEAR DELIVERS MAXIMUM THRUST?
That’s the magic acceleration due to gravity, at least in the imperial world.
What that does is convert Newtons of force into kilograms force, which are just easier for non-physics aficionados to cope with.
This correlates pretty neatly with the experience of driving the car. Get to 75 in manual mode, upshift to seventh, floor the throttle. It’s just bogging down all over town. Downshift to second, repeat the experiment: big difference.
So you can very clearly see that, provided you can find the right gear, maximum acceleration at any particular speed can only ever occur when the engine makes peak power - because peak power at the engine delivers peak torque at the driving wheels. Every time.
Regardless of what speed you’re at, subject to the wheels not spinning.
And, at the upper limit of speed you need to know that aerodynamic drag can overcome the ability of the engine to make the car accelerate at all. So the maximum possible acceleration at the car’s terminal velocity is zero. (The other option is: car bounces off the rev limiter because the gearing is too low.)
Here’s how this plays out in reality, for the i30 at 75km/h.
You can be in every gear except first, but at 75km/h maximum acceleration really depends on you being at those peak power revs. It makes such a huge difference to the vehicle’s performance. Check the tractive effort numbers on the right, above - they’re the ones that count.
You can be in six of the seven gears - it’s up to you - but clearly if performance is the objective, only one of them is right.
When you look at the revs on upshift in the next gear - I took a theoretical upshift at 6500 rpm for these calculations - you can see that the dual-clutch transmission in the i30 SR Premium is really designed primarily for comfort and economy, and occasional fun.
It’s certainly not a track special with a close-ratio gearbox - but trust me: as the owner of (among other things) the last manual RenaultSport Clio, these do get somewhat demanding when you’re not having a red-hot go, which is most of the time for ordinary drivers in big cities.
Basically, high gears rock for cruising comfort and also fuel economy, but when you want to turn hydrocarbons into noise and speed, peak power is absolutely where you need to be. Any time you’re in performance driving mode and you see the tacho on the quiet side of 3500, you really need a downshift if you want to make the scenery go faster.
Maximum acceleration at any particular speed happens when the engine is delivering maximum power. If anyone tells you this occurs at peak engine torque, they are bullshitting you - probably not maliciously.
Of course - you should only ever invoke maximum acceleration in situations where it is safe to do so. The core business of driving on a road is safety and than means managing risk effectively. Thankfully there are still times when it’s OK to have a little fun. If you know what you’re doing.