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### Solution to Fermi Problem Friday - Car Week Edition

The question I posed last Friday was "what's the most horsepower a typical street car can use effectively?" (The answer, in case you don't feel like reading the whole post, is in the range 500-600 hp.)

As it turns out, the limiting factor when it comes to acceleration is usually grip. Cars with more power than their tires can handle are fun for making smoky burn-outs, but they don't tend to move very fast when they're doing it.

Street car tires are pushed to their limits during emergency stops. (The video here shows some people testing braking power on a car a lot like the one I drive to work every day.) So if you want to know the maximum amount of power a car can use, you just need to figure out the maximum power it can dissipate when coming to a stop, then imagine launching in the same amount of time and distance as you stopped, and you have a recipe for the quickest a car can possibly accelerate. Once you know that, you can calculate the work required, which in turn tells you how much horsepower you need to accelerate as fast as physically possible.

Here are my assumptions . . .

- A typical street car has a mass of about 1500 kilograms.

- With maximum braking, a car can go from 100 kilometers per hour to a dead stop in about 40 meters. Given enough horsepower, it should be able to launch to 100 kph in about the same distance.

In short, we hope to have just enough power for a car to go from zero to 100 kph in 40 meters. Any more power would only serve to burn rubber.

The work dissipated in such a launch is Work = Force x distance

but Force = mass x acceleration so Work = mass x acceleration x distance

To know the acceleration we need to use an equation to solve for it given the distance traveled and the maximum velocity. The equation is

d = (1/2)(v^2)/a , where v^2 means velocity squared

so a = (1/2)(v^2)/d = (1/2)(100 kph)^2 /(40 meters)

Since I prefer to use standard metric units, kph needs to be converted to meters per second.

100 kph = (100,000 meters)/(3600 seconds)= 27.8 meters per second, which I will round to 28 meters per second

a = (1/2)(28 meter/second)^2/(40 meters) = 9.8 meters/second^2, which happens to equal about 1 g

(This is reassuring because high performance street cars typically can pull 1 g turns on a skid pad, so we're clearly in the right ballpark.)

Plugging things back into the equation, the work performed in launching to 100 kph (or braking from 100 kph) at 1 g is

Work = mass x acceleration x distance = (1500 kg)(9.8 meters/second^2)(40 meters)= 588,000 joules

In order to know the power involved, we have to use the equation

Power = Work/Time, where Time is the amount of time it takes to reach 100 kph.

Now, Velocity = Acceleration x Time, so

Time = Velocity/Acceleration = (28 meters/second),(9.8 meters/second^2) = 2.9 seconds

This is also reassuring because the very fastest sports cars in the world can get to 100 kph (28 meters per second) in about this amount of time.

So, to put it all together, the power required to reach 100 kph at the fastest rate that good street tires can handle is about

Power = Work/Time = 588,000 joules/2.9 seconds= 202,758 watts

Converting that to horsepower, Power = (202,758 watts) (1 hp/745 watts) = 272 horsepower

That's all you need to accelerate as fast as your street legal tires can handle. If it seems low, part of the reason is that cars waste a lot of power in operating the transmission and other moving parts in addition to the wheels. Most car engineers assume that about 25% of the power in an all-wheel drive car is lost, so to make 272 hp at the wheels, you need the engine to be able to generate roughly 360 hp.

Another problem, is that internal combustion engines deliver different amounts of power at various engine speeds. Modern engines typically produce the most power when they are turning at about 6000 revolutions per minute (rpm). If you want 360 hp at 3000 rpm, your peak horsepower will be a good deal higher. For a well designed and tuned car, you may need 50% more power at 6000 rpm than you have at 3000 rpm. That suggests that you need peak horsepower of about 540 hp to go from 0 to 100 kph (62 mph) in 2.9 seconds in a car that drives all four wheels.

Two-wheel drive cars can't make good use of as much horsepower as all-wheel drive cars because they have less grip. Actually calculating the difference is too challenging for me to get into here because of the way weight shifts during hard acceleration, so I'm just going to guess that max useable horsepower for a typical two-wheel drive street is probably about 500 hp.

These numbers about what you can get today buying high performance cars like the Dodge Challenger SRT8, Camaro SS or Nissan GTR. Supercars, typically costing hundreds of thousands of dollars (Lamborghini Gallardo or the Ferrari 599 GTB Fiorano), surpass the 540 hp limit in part because they have much better tires than most street cars.

Few vehicles can hit 100 kph in 2.9 seconds, even with 600 hp, but lots do it in under 4. All in all, I'd say the Fermi problem approach is pretty accurate at estimating how much horsepower you need to reach the very best performance modern tires can manage.

So if anyone tries to tell you why you need a \$1 million, 1000 hp Bugatti Veyron, tell them "No thanks, a \$48 k Mustang Shelby GT500 will do just fine."

### Comments

1. Excellent post! I have another question for you. How much hp can be used effectively to pass on a highway? (Such as accelerating from 90 to 120kph)

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