Horsepower - Part 1

One of the main performance characteristics of a car is its “Horsepower”. In this article we’ll dive into what this means exactly, starting with basic concepts of mass, force, and acceleration.

Force

Force corresponds with how hard you push something. We often confuse force and mass. For example, my new M5 weighs in at a hefty 4313 lb ('lb' is short for pounds because it was based on the Roman libra), according to the big scale.

Lighter cars are considered sportier because they accelerate faster and turn quicker. But the M5 needs air conditioning and power seats, so compromises must be made!

But if that scale and my new car were both moved to the moon, she would weigh in at only 720lb. 

Does that mean my car is sportier on the moon? Unfortunately not. My car’s acceleration would be no better at all because the M5 has an inherent tendency to resist being accelerated. This tendency to resist being accelerated is called Mass, and corresponds to the “amount of stuff” in something (the mass of all the atoms added up). Clearly the amount of stuff in my M5 does not change between the earth and the moon.

What the big scale measures directly is the force exerted on it. The force exerted on the scale by my M5’s mass is larger on the earth than on the moon, because the gravitational attraction is six times more on earth than on the moon (because the earth is six times more massive than the moon).

In the international unit system, called SI, the Kilogram is the official unit of mass, and the Newton is the official unit of force, so it’s easy to see which is which. In the Imperial system, the unit called the “Pound” is ambiguous, and when we want to be precise we indicate “Pound (mass) - lbm” or “Pound (force) - lbf”, where a Pound (force) is defined as the force exerted by earth’s gravity at sea level on a Pound (mass). Since cars can be considered to be driving at or around sea level on the planet earth, we can use the two units somewhat interchangeably, and you just need to understand which one is meant by the context in which it is used.

Acceleration

There is a relationship between Mass, Force, and Acceleration which is called Newton’s law, after the fellow who figured it out for the first time in 1687. He found that the Acceleration of an object (how quickly it changes speed - denoted by the letter 'a') is directly proportional to the Force exerted on the object (F), and inversely proportional to the object’s Mass (m). In the language of mathematics,

• a = F / m

For example, with the same force applied, a car that is only half the mass of another car would accelerate exactly twice as fast. And if we then double the force on the heavier car, they would then accelerate the same. (for a quick little tutorial on the simple math I use here, see my Math and Units blog post).

A common unit of acceleration is the “g”. If a car you are seated in is accelerating forwards at 1g, you would be pushed into the seat with the weight of your own body. In other words, the feeling on your back would be as if you were standing still, and the car was stood on its tail like a rocket ship.

A “g” is the equivalent of speeding up by about 22 miles per hour each second. So, accelerating at 1g, then after 3 seconds you would be travelling at 66 mph. That’s supercar territory for sure! 

Using g’s you can game lbf and lbm in a lovely way such that

• Acceleration (in g’s) = Force (in lbf) / Mass (in lbm).

So if a force of 4313lb were applied to the backside of the M5 weighing 4313lb, it would accelerate at exactly 1g. The lesson in this is that if you pick the right units, you can avoid messy conversion constants in your math.

Traction

In practice, 1g of acceleration would never happen with the M5 because its road tires just aren’t that sticky. Generally a good rule of thumb for a decent road tire is that it can push forward with about the same force as the weight pushing down on it.

Since the M5 is a rear wheel drive car, and about half the weight of the car is on the rear wheels, it can only accelerate at about 0.5g to start. It gets a bit better than this in practice because as the rear wheels start accelerating, the weight goes more onto the back wheels resulting in an effective acceleration of about 0.7 g’s with good tires on good pavement.

An extreme example of "squat"!

Formula 1 race cars are rear wheel drive cars as well with roughly even front and back weight distribution, but have incredibly sticky tires that allow them to accelerate at more like 1.45 g’s, getting to 60 mph in under 2s. Mind you, those tires are massively wide, are crummy on anything but great pavement, suck in the rain or snow, and will disintegrate every 50 miles or so!

Four wheel drive cars have an advantage from a standstill, as the full weight of the car is available for acceleration and hence can theoretically hit 1g on street tires (0-60 in under 3s), assuming the torque is perfectly directed, but things don't usually work out so well in practice and, besides, straight line acceleration isn't everything, as we shall discuss later.