Making more progress

My last musing returns to the issue of making progress. If it seems a bit technical at times, then it only shows that mathematics and the laws of mechanics have a life beyond the classroom in which they were taught.

Why should we bother about keeping an eye on the speedometer? Try this one. We set out to maintain an average speed of 40 mph over 1 mile, but we slacken to only 30 mph in the first half-mile. How fast must we go in the second half to regain the average of 40? The answer is not 50 but 60 mph! At 40 mph a mile takes a minute and a half. At 30 mph half a mile takes 1 minute, leaving half a minute to cover the other half-mile.

Maths is wonderful. Try this one. You are stuck behind a lorry travelling at 60 mph on a motorway. Just as the 1 mile signpost for your exit comes up you have an opportunity to overtake. Can you do it without exceeding 70 mph? This equates in time taken to travelling at 10 mph past a stationary lorry, starting from a safe braking distance behind (73 metres), past the lorry (say 15 metres), to a safe position in front (80 metres if you do not trust the lorry brakes). 10 mph is just under 4.5 metres per second, so this takes nearly 38 seconds. Travelling at 70 mph, you cover nearly 3/4 of a mile in this time. You can do it and impress your examiner with the progress you make, as long as you accelerate smartly from 60 to 70, but do not try it at the half-mile post!

This is now be kind to engine and gearbox time. We know our engines have a maximum power, which out of kindness we would not want to use all the time. What about saying we will regularly use only 80% of the power, and keep the rest in reserve for emergencies? A nice thought, but meaningless. To go twice as fast, you do not need only twice the power, but something like the square or the cube of two (between four and eight) times the power. Let us suppose that maximum power is developed at a maximum speed of 100 mph, a not uncommon aximum even in small cars these days. The surprise is that at 80% power you are still doing 90 mph. Alternatively, at the legal maximum of 70 mph you need less than half the maximum power. Anyone who uses 80% power is already a naughty boy (or girl).

But, you say, I use more power because of accelerating. No, this is the confusion between power and torque. When a mass such as a car is moved against a resistance (wind resistance), energy is consumed. The ability to provide that energy in a given time is power. What happens during acceleration is that extra force is required to push the car faster. You could do this yourself by pushing a broken down car off the road; quite a lot of force is needed to get started, but when it is going it somehow seems easier. You are using less force but more power. When the engine is running, the push comes via the push of the piston on the connecting rod, the push of the con-rod on the crankshaft and, in the gearbox, the push of one cog tooth against another. Because this force is applied in a circular direction round the gearshaft, it is called torque.

(Politically incorrect joke - Why do women accelerate better than men? Because they torque more.)

Power equals top speed, torque equals acceleration, and both vary in different ways, depending on engine speed, and on the engine itself. The motorcycle I currently ride comes in two stages of tuning. One has high torque and high power, the other even higher power but less maximum torque. The high power version does not accelerate as well. Humans can apply quite a lot of force even from a standstill, but these infernal combustion engines cannot do that at all.

We are now back to watching the rev counter, as the engine speed varies between idle and a red line or area. I have already mused on avoiding the red line, or at least consulting the vehicle handbook before doing so. What is not always clear is that increasing the revs will increase the power available up to a maximum, but after that the power available is actually less. The torque available behaves in the same way, reaching a maximum at fewer revs/min than even the maximum power. In engineering-speak, the torque curve is not the same as the power curve. This is, however, why continually revving up to the red line is pointless; the upper range of engine speeds is to be kept in reserve.

Let us put in some numbers. In general terms, if the maximum revs are 6000 or so, the maximum power will be developed at 5000 to 5500, and maximum torque will be at 4000 to 4500. (All engines are different, but these numbers are representative and will do to start with.) In my car 4000 revs in top gear correspond to 70 mph. It is curious that at these engine revs I am nowhere near maximum power (and indeed the engine could go on all day), but I am near the point of maximum torque, meaning that I will get maximum acceleration if I rev up to this sort of number in each gear before changing. This is why the brigade intent only on first gear to 10 mph, second to 20, third to 30, will not get on very fast: it is mechanical nonsense if one is seriously interested in making progress.

It is certainly more difficult to maintain the revs at these levels when changing gear, but the skill is worth acquiring. Running at these revs does not stress the engine or the gearbox in the slightest, and the performance gain is spectacular. Consider waiting at a T-junction to slot into a small gap in the traffic. Take first gear, move off, make the turn, straighten the wheels, and then change into second. This can all be performed in first gear as part of a continuously accelerating progress and is far safer than changing into second during the course of the turn. It is unlikely that you will get anywhere near maximum revs.

Finally, what to do with the surplus revs at the top?. You are stuck behind a 30 mph bus on a country lane when a half a chance to overtake presents itself. Now I know that my car will get almost to 60 mph in second gear before the needle gets to the red line. I can therefore call up some suddenly electric performance (by comparison with third gear), knowing that I still do not have to change gear during the overtake manoeuvre, which would be unsafe. This is only possible by virtue of the extended revs, which bring extended speed, not extra power or extra torque, and are available for this very occasional use. It is therefore well worth while finding out what the safe maximum speeds in each gear are.

Postscript :

Direct current electric motors exert their maximum torque at zero revs and do not need multi-speed gearboxes in consequence. If only one could store as much energy in batteries as in the same volume and weight of petrol...

Post-postscript :

Well, I passed. My thanks to all my observers for taking the time to sort me out, to the editors of Steering Column for allowing me to air my views, and to the Chief Observer for replying in such a tolerant fashion. I would like to repay all these debts in due course. If I may offer a final suggestion, the one element that I found lacking in my instruction was a demonstration drive with an observer, to give me an idea of how rules in the book were implemented in practice. Personally I would be happy to offer a drive to an associate, especially with a senior observer sitting in the back to comment on negative as well as positive features.

Correction: in my previous musing I suggested "when accelerating through the gears towards the current limit make the last gear change at least 5 m.p.h. below the limit." This was not meant as "always change up at 5 m.p.h. below the limit" but as "if you have the intention of changing up, do not do it too close to the speed limit currently in force". The reason for this is that the first accelerator pressure after changing gear may take you over the limit.

Copyright PHP Harris 1999

Chief Observer's reply

The Chief Observer was happy to let this one go past him. I am grateful to him for allowing me to publish his responses to my previous musings on this website. I hope both observers and associates will benefit from the points raised.

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