The speed at which grip hands over to power

A car is sold on its power. The number on the brochure is in kilowatts or horsepower, not newtons, and not joules: nobody advertises how hard an engine pushes or how much energy it holds, but how fast it can deliver energy. The energy that depends on the observer showed that the energy a car gains is a number about the road as much as about the car. Power is what decides how quickly it gains it, and the relation between power and push is where the confusion starts.

Power is force times speed, $P = Fv$. Read one way, that says a given engine can push hard at low speed and only gently at high speed. Read the other way, it says that at a standstill a finite power could deliver an infinite force — which no car does, because something else caps the force long before. That something is the tyre, and the speed at which the cap changes hands is the most useful single number about how a vehicle accelerates.

Two ceilings on one force #

A driven wheel pushes the car forward by pushing the road backward, and the road pushes back only as hard as friction allows. The force that takes what it needs describes static friction as a response rather than a value: it supplies whatever is asked of it, up to a ceiling of $\mu N$, and then gives way. For a car with all four wheels driven the load is the whole weight, and for these tyres the ceiling is $0.9 \times 1{,}500 \times 9.81$ newtons — 13.2 kN. No engine, however large, can make the wheels push harder than that without spinning them.

The engine sets the other ceiling. If the power reaching the wheels is $P$, the force they can deliver at speed $v$ is at most $P/v$, a hyperbola that is enormous at low speed and small at high speed. The force actually available is the lower of the two curves, and the figure draws it solid: flat, then falling.

The corner between them is at

$$v^* = \frac{P}{\mu m g},$$

7.6 m/s for this car, about 27 km/h. Below that speed the car is a friction problem and above it a power problem, and the distinction is practical. Below it, a bigger engine changes nothing and stickier tyres help. Above it, stickier tyres change nothing and more power helps. Asking which upgrade makes a car faster is asking which side of $v^*$ the speed range that matters lies on.

A straight line, then a square root #

On the grip side the force is constant, so the acceleration is constant and the speed rises in a straight line: 8.83 metres per second every second, which is 0.9 of the acceleration of free fall, reached in the 0.86 s the car takes to get to $v^*$.

On the power side it is not the force that is constant but the rate at which energy arrives. Ignoring resistance, the kinetic energy rises by $P$ joules every second, so $\tfrac12 mv^2$ grows in proportion to the time and the speed grows as its square root. The figure’s integration of the equations of motion reproduces that closed form to better than a millimetre per second, which is the check on the integration rather than a result.

The square root has a consequence that surprises people who think in forces. Without resistance, this car takes about 1.9 s to reach 50 km/h and a further 4.3 s to get from 50 to 100, because the second 50 km/h carries three times the kinetic energy of the first and arrives at the same rate. The hill that gives it back makes energy the natural currency of motion, and here is the price of that currency in time: equal steps of speed cost more and more of it as the speed rises.

Resistance then adds its own drag on the process. Rolling resistance is nearly constant, about 180 N for this car; air resistance grows as the square of the speed and by 100 km/h is already 300 N more. The run with both passes 100 km/h at 6.6 s rather than 6.2, and by 30 s it has reached 186 km/h and is still climbing slowly towards its top speed. The kinetic energy at 100 km/h is 579 kJ, which at 100 kW is 5.8 s of full power: the 6.6 s is that, plus the time lost to the grip at the start, plus the work done on the air.

The sprint power stops buying #

Plotting the time to 100 km/h against power turns the crossover into a law of diminishing returns. At low power the car spends almost the whole sprint on the power side, and doubling the power roughly halves the time: 13.7 s at 50 kW, 6.6 s at 100. At higher power the grip-limited start takes up a growing share of the sprint and the returns fall. At 368 kW the crossover speed $v^*$ has risen to 100 km/h, the car is grip-limited for the entire run, and the time settles on a floor of 3.21 s. The figure checks that the integrated time at that power equals the floor; beyond it, extra power buys nothing at all in this particular measurement.

That floor is why cars built to sprint spend so much effort on the other ceiling. Driving all four wheels uses the whole weight as the load rather than half of it. Wider, softer tyres raise the effective coefficient, at a cost in wear and in rolling resistance. Weight moves backwards under acceleration, which is why a rear-driven car launches better than a front-driven one of the same power. Wings that push the car down raise the load without raising the mass, but only once the car is moving fast enough for the air to push. And an electric motor controlled wheel by wheel, holding each tyre at the slip where it grips hardest — the peak that the grip that needs a little slipping finds a few per cent from rolling — gets closer to the floor than a driver’s foot can.

Road cars quoted at under three seconds to 100 km/h have tyres better than $\mu = 0.9$, and most of them have exactly the engineering in that paragraph. None of it is about the engine.

Railway engineers draw the first figure routinely and call it a tractive-effort curve. A locomotive’s pull is capped at low speed by adhesion — steel wheels on steel rails, with a coefficient of about a quarter to a third, times the weight resting on the driven axles — and at higher speed by power. A heavy freight train is started on the flat part of that curve, with sand dropped onto the rails ahead of the wheels to raise it, and a line that climbs a long grade at low speed is worked by locomotives chosen for their weight on the driving wheels rather than for their power. The weight that a road vehicle carries as a burden, a locomotive carries as an asset: it is what the grip ceiling is made of.

A belt drive is the same machine turned inside out. A belt pulls a pulley round with a force equal to the difference in tension between its tight and slack sides, and friction caps that difference in the way the wrap of a rope round a drum shows: the tight side can carry at most $e^{\mu\theta}$ times the slack side before the belt slips. The power it transmits is that capped force times the belt’s speed. So a belt, like a tyre, has a force ceiling it cannot exceed and a power it can raise only by running faster, which is why belt drives are geared to keep their belts moving quickly and why a slipping belt squeals at start-up, when the demand for force is highest and the speed lowest.

The cube at the top #

At the other end of the speed range the ceiling that matters is neither grip nor power but the air. Air resistance is a force proportional to the square of the speed, $\tfrac12\rho C_dA\,v^2$, and holding a speed against it takes that force times the speed — power growing as the cube. Between 180 and 250 km/h the car’s total requirement grows with an exponent of 2.77, close enough to three that the rule is worth carrying: to double a top speed takes about eight times the power, and doubling the power buys about a quarter more speed. The figure’s four engines, each double the last, give top speeds rising by factors of 1.29, 1.28 and 1.27.

The cyclist’s figures make the same point at a scale a person can feel. A rider putting out 150 W, a comfortable effort, holds 31 km/h on the flat; 250 W, hard but sustainable for an hour by a fit amateur, holds 37.5; 400 W, which a professional can sustain for about that long, holds 44.4. Sixty per cent more power, 18 per cent more speed. The same arithmetic run the other way is why riders crouch and draft: the power needed is proportional to the drag area, and reducing it by 30 per cent, which a tight tuck or a wheel in front can do, takes the 250 W rider from 37.5 to about 42 km/h without a watt more. On a bicycle, as on a car at speed, the shape is worth more than the engine.

Going uphill changes which ceiling is near. On a steady climb most of the power lifts the rider and the bicycle, at a rate of weight times vertical speed, and the air hardly matters at climbing speeds. Two riders each producing 250 W with the same drag area, one weighing 70 kg with the bicycle and the other 90 kg, hold 37.7 and 37.3 km/h on the flat, within a kilometre an hour of each other; up an 8 per cent grade they hold 14.8 and 11.8 km/h, the lighter one faster by nearly the ratio of the masses. On the flat the figure of merit is power per unit of drag area and on a mountain it is power per kilogram, which is why the riders who win time trials and the riders who win on mountain stages are rarely built alike.

The angle that drag moves meets the same square-law force deciding where a thrown ball lands; here it decides where acceleration ends, which is the speed at which every watt the engine delivers is being handed to the air.

Gears lay a peaked engine along the hyperbola #

The first figure assumed the car could deliver its full power at every speed. A combustion engine cannot. It produces power over a range of crankshaft speeds, rising from idle to a peak and falling towards the red line, and at any one road speed the crankshaft speed is fixed by the gear. A gear ratio stretches the same engine curve along the road-speed axis: low gears put the peak at low road speed, high gears at high.

Each gear’s force curve therefore touches the constant-power hyperbola at exactly one point, where the engine is at its peak, and falls below it on either side. A gearbox is a set of such curves chosen so that their upper envelope stays close to the hyperbola, and five ratios do it well: between the peak of first gear at 47 km/h and the peak of fifth at 206, the best gear delivers at least 90 per cent of peak power over 98 per cent of the speeds. The ratios step by 1.67, 1.50, 1.40 and 1.25 — progressively closer together towards the top, where the power in hand is small compared with what the air takes and a gap between gears would cost the most.

The figure also shows what the ideal curve hid. First gear’s largest force, 9.6 kN at half the peak engine speed, is below the grip line. With this gearbox the tyres never limit the start; the engine does, and the grip-limited launch of the first two figures is available only by slipping the clutch, which turns the difference into heat. Electric motors do not have this problem in the same form. They deliver roughly constant force from standstill up to a base speed and roughly constant power above it, which is the solid curve of the first figure built into the machine, and it is one reason an ordinary electric car can leave a line faster than a petrol car of the same power.

A bicycle’s gears solve the same problem for the same reason. A rider’s legs deliver most power over a narrow band of pedalling rates, and the gears keep the pedals in that band while the road speed changes.

Where the model stops #

The grip is a single coefficient. Real tyres develop their force through slip, and the coefficient itself depends on the slip, the speed, the temperature of the rubber and the road. The grip that is not a coefficient is the account of why; a constant $\mu$ is the peak of that behaviour taken as a flat line.

The load does not move. Under acceleration weight transfers from the front axle to the rear in proportion to the acceleration and the height of the centre of mass, so the grip available at driven wheels changes during the run. The figures drive all four wheels, which makes the total load fixed.

The power is power at the wheels. An engine’s rated output is measured at the crankshaft, and the gearbox, differential and tyres take a tenth or more before it reaches the road. And the engine’s power is itself a small part of what it burns: the ceiling on every engine puts a hard limit on the fraction of the fuel’s energy that can become work, and real engines deliver a quarter to a third.

Grip is shared with cornering. A tyre’s friction is a budget for force in any direction, so a car accelerating out of a bend has less grip for acceleration than one on a straight, and the crossover speed moves up accordingly.

And the air is still. A headwind adds to the relative speed that sets the drag, and the cube makes even a light one expensive at speed.

What the pictures cannot show #

None of the figures draws the energy going anywhere. The kinetic energy the car gains, the heat in the tyres during a clutch-slipping launch, the work done on the air and the rolling losses in the rubber are all flows the integration accounts for and no picture displays, and at a steady top speed every one of the 100 kW is leaving the car as heat in the air and the road while its kinetic energy does not change at all.

Nor do they show the time structure inside an engine. The power curve is an average over many firing strokes, and the smooth hyperbola hides an engine delivering its energy in pulses tens of times a second, smoothed by a flywheel and the elasticity of the drivetrain.

Still open: whether a sprinter is limited like a car #

A human sprinter faces the same two ceilings in a different form. Each footfall must push the body forward and hold it up, through a foot in contact with the ground for about a tenth of a second at top speed, and the muscles can deliver only so much power. Whether the top speed of a runner is set by the force the legs can apply to the ground in that brief contact, by the rate at which they can develop that force, or by how fast the limbs can be swung back into position is a question that measurements have narrowed without closing.

Treadmill studies from 2000 onwards found that faster sprinters do not reposition their legs faster than slower ones; they apply larger forces to the ground in shorter contacts. That points to force application rather than leg speed, but how much of the limit lies in muscle properties, in the stiffness of tendons and joints, or in the coordination that aims the force, and how it changes between the start and top speed, is still argued — and the answer would say whether a runner’s first seconds and last metres are, like a car’s, limited by different things.

The habit worth keeping is the one the crossover teaches. Before asking what would make a machine faster, find which ceiling it is against. Below $v^*$ it is friction and the engine is irrelevant; above it, power and the tyres are irrelevant; near the top, the air, and both are nearly irrelevant. The same car is all three machines in the course of one run.