Cars - For Car Enthusiasts

People with fast street cars like to put them through their paces at the quarter-mile track. One way to get your quarter-mile time is to just buckle up and put the pedal to the metal. But if your car's design is suboptimal, you won't be taking home the bragging rights.

So here's this week's question: Can automotive engineers predict a car's quarter-mile time using physics? And could the physics suggest some tricks to make a car faster? Yes and yes! Let's see how.

When a car launches off the start, its increase in speed is described by its acceleration (the rate of change of velocity). But according to Newton's second law, to increase velocity, you need a force pushing in the direction of travel.

We can model the motion of a car with just three forces. There's the downward-pulling gravitational force (= mass, m, times the gravitational field, g). There is also the interaction between the car and the road. It's useful to split this into two forces: One, perpendicular to the ground, is called the “normal force” (FN). It's the resistance of the ground to gravity—what keeps a car from plunging to the center of the Earth. The other force, friction (Ff), acts parallel to the ground. Here's a picture:

It's the forward-pushing frictional force that makes the car accelerate. (Yes, this is powered by the engine, but the force is exerted where the rubber hits the road.) This frictional force depends on two things: the types of surfaces interacting, captured by a friction coefficient, μ, and the normal force (FN) pushing these surfaces together. With that, we get the following maximum frictional force:

The frictional force that drives the car ahead depends on the specific materials in contact (captured by a friction coefficient, μ) and the normal force. We can quantify this by looking up the friction coefficient (around 0.7 for tires and asphalt) and calculating the normal force. Since the car stays on the ground, the acceleration in the vertical direction is zero. This means the net force in the vertical direction must be zero. So the normal force and the gravitational force must be equal.

This gives us a value for the maximum frictional force (Ff). Since this is the only force in the horizontal direction, Newton's second law says it must equal the product of the car's mass (m) and its acceleration (a).

So if we know the mass of the car (m) and the coefficient of friction (μ), we can calculate the acceleration. This is going to be useful. First, however, there's another factor we have to look at—engine power.

Constant-Power Model

Suppose you have a nice sports car. It's red and has a nice sound system, but most importantly it's a Porsche 911 Carrera with a 379-horsepower engine. We can use this power figure to model the motion of a car on the track. Power is something everyone thinks they understand, but most people can't define it. So here you go: Power is the rate of change in energy.

A car has kinetic energy (K) by virtue of its being in motion. The amount of kinetic energy depends on the mass of the car and its velocity (v).

Let's say we have a tiny car with a power of 100 watts (1 horsepower = 735 watts). If this car starts from rest, after one second it'll have a kinetic energy of 100 joules (since 1 watt = 1 joule per second). If the car has a mass of 2 kilograms, that would give it a velocity of 10 meters per second after one second. If you double the mass (to 4 kg), it would be 7.1 meters per second.

What happens to the velocity of the 2-kg car during the next second if we assume the power is constant? It will again increase in kinetic energy by an amount of 100 joules. However, since it started at 10 m/s and this starting velocity is squared, its new velocity will be 14.1 m/s (an increase of only 4.1 m/s) at the two-second mark. And so on. With constant power, the gain in speed with each successive second gets smaller. That's how power works.

If we apply this idea to a real car (I'm sticking with the Porsche 911), we can see how long it would take to run the quarter-mile track. I'm going to convert the power to watts and the distance to meters—it's just easier that way. The Porsche's mass is 1,493 kilograms (3,291 pounds), which gives me the following plot of velocity vs. time. As you can see, at 10 seconds the 911 is going 63 m/s (140 mph).

This constant-power model looks nice. It shows that the car increases in speed but not to infinity and beyond. As it goes faster, it's acceleration (the slope of the velocity curve) decreases. Seems real.

Not so fast! There's a problem with this model: At the start of the race, the slope of the curve is essentially vertical. That means it would have an infinite acceleration. That's just not possible. Looking back at our equation for acceleration based on friction, the highest acceleration would be:

You can see that it's limited by the materials in the tires and track (captured by the frictional coefficent) and the gravitational field (so, what planet you're on). Notice that the mass has canceled out. It doesn't matter if you have a more massive vehicle. Yes, you get more friction, but it's also harder to accelerate.

Constant-Friction Model

Since constant power doesn't work, what about a constant acceleration due to the friction between the tires and road? Let's say the coefficient of friction is 0.7 (reasonable for a dry road). In that case we would get the following plot of velocity versus time for the quarter-mile run.

I've included the constant-power curve just for comparison. You can see that with this friction model, the car will just keep increasing in speed forever with the same acceleration. That doesn't seem correct either.

A Better Model of Acceleration

How about this? The car increases in velocity—however, the rate of increase (the acceleration) is the lower of the two models. So, at the beginning of the run the acceleration is limited by the friction between the tires and road. Then, when the acceleration using the constant power model is lower, we can use that method.

Before we test this out, we need some real data for comparison. Since I don't own a Porsche 911, I'm going to use the data from this MotorTrend race between a 911 and a Tesla Cybertruck. Here is a plot of the actual position of the Porsche over the quarter-mile track along with the combo power-friction model. (That's now distance on the vertical axis—a quarter-mile is just about 400 meters.)

It's not a perfect fit, but it could be. I lowered the power of the Porsche a bit in my model because not all of those 379 horses go into the kinetic energy of the car—it's not a perfectly efficient system. But still, this fairly powerful car hits a quarter-mile in 12.2 seconds with a final speed of 116.4 mph (from MotorTrend). Is it possible to get a lower time? Yes. With physics.

There you go, bud!