If you pitted your car’s engine against your car’s brakes, which would win?

As you’ve almost certainly heard, Toyota is under a tremendous amount of scrutiny at the moment due to reports of “unintended acceleration” in its cars.   My personal belief is that the problem is driver error attributable to mass delusion, since you’re no more likely to encounter such a situation in a Toyota than in any other brand.  One tends to find what one is seeking to find.   It hasn’t helped that Toyota’s initial handling of the situation will go down as a textbook case of what not to do.

Okay, but what about the reports where drivers claimed to be pressing the brake pedal to no effect?  Can the engine really overpower the brakes in a modern car?  Let’s do some back-of-the-envelope math to find out.

Assume that a car (a simplified model of the Lexus ES 350 in this case) and its passengers weighs 4000 lbs, has a 270 hp engine operating at the peak of its powerband, is going 70 mph, and can stop from 70 mph in 176 ft.   In metric units, that’s 1818 kg, 201 kW, 31 m/s, and 53.6 m.  Assuming constant deceleration, we see that the acceleration and time to decelerate is

v^2 = v_0 ^2 + 2 * a * (d-d_0)

0 = 31.3^2 + 2 * a * (53.6)

a = -9.13 m/s^2

t = (v – v_0) / a

t = 3.42 s

This means that the car is capable of decelerating at just under 1 g.

Now, let’s  look at how much power the brakes are dissipating during that stop.  First, we need to find the energy of the car at speed. Again assuming constant deceleration,

Ek = 0.5 * m * v^2

Ek = 0.5 * 1818 * 31.3^2

Ek = 890 kJ

Next, we solve for power:

P = E/t

P = 890 kJ / 3.42 s

P = 260 kW

Since the brakes are capable of dissipating more power than the engine is capable of producing, the brakes will win over the engine — though obviously the stopping distance and time will be increased.

Now, this margin might not appear to be much above the power output of the engine, but it ignores a few things:

• The engine produces peak power only in a limited rpm band, so the power output at any given time is likely to be less than the theoretical maximum
• There are significant drivetrain losses, so the actual power at the wheels is about 15% lower than that at the engine’s crankshaft
• Torque, and not power dissipation, is a strong factor when considering the actual stopping capabilities of a braking system

That last point is probably quite important, but I will leave exact calculation of its significance as an exercise for the reader.

There are some other caveats.  Brake effectiveness decreases with increased heat (a condition known as “brake fade”), so continued or repeated application of the brakes might severely hamper their ability to stop the vehicle.  The situation is further degraded by the loss of vacuum assist after repeated applications of the brake pedal while at speed.  Engine vacuum decreases with engine speed, so at full throttle, if the vacuum reserve has been depleted, the driver might find it very difficult to fully apply the brakes.

As a further aside: what about the parking/emergency brake?  Although it operates using an entirely separate system, it is very weak.  My BMW had no problem overpowering the parking brake from a standstill — something that was so easy and so common that it warranted an idiot light on the dash — so I am quite doubtful that it could stop the car in a reasonable distance from highway speeds even without the extra force from the engine.  (No numbers for analysis here, sorry.) Stick with the main brakes.

The idea that a car’s braking system can stop a car from highway speeds, even at full throttle, is supported by empirical evidence.  Car and Driver did a series of tests of the matter, in which they had no problems stopping various cars.  For example, the V6 Camry they tested stopped from 70 mph in 174 ft under normal conditions.  When the test was repeated with the throttle wide open, the same stop required 190 ft — an increase of less than 10%.

If anything, it seems like poor user interface design or poor user training is at fault here.  How many people know that power brakes offer less power assistance after being pumped?  Who besides techies familiar with ATX-style power supplies would guess that one must hold down the start/stop button on a modern car for about three seconds to force it to turn off?  How many drivers have practiced panic stops or evasive driving?

But I suppose new floormats are a good enough placebo.