What's the Big Deal Anyways?

The concepts of horsepower and torque are probably the most misunderstood basic concepts in the auto world. The problem is a lot of people are very adamant about their opinions but haven't really done the research to back up what they're saying. I hear a lot of this, especially on the forums. In truth, though, horsepower and torque are NOT that intuitive of concepts. Remember that guy Isaac Newton? Well, it took him years to come up with this stuff and he was definitely no slouch when it came to mental capabilities.

So, why does any of this matter anyways? For starters, understand the truth behind horsepower and torque will give you a better understanding of engines and why certain types of engines are good for some things and not others. Second, you won't shift at the wrong point just because your friend says: "shift now, you'll get more torque in the next gear." It's also just satisfying to know exactly what makes your car move.

All right then, I've blabbered too much so now on to the good stuff. I'll try to keep it as brief and simple as possible but, remember, these are tough concepts to REALLY grasp.

Definitions
First, lets get some definitions in place before I try to relate the two concepts.

Term	Definition
Force	Simply put, a push or a pull. Typically measured in pounds (lb) or newtons (N).
Work	Force over a certain distance, expressed as force * distance. Typically measured in foot-pounds (ft-lb), newton-meters (N-m), or joules (J). Note that a force only counts as towards work done if it is in the same direction as the distance being traveled.
Power	Work done over a certain amount of time, expressed as force * distance / time. Typically measured in watts (W).
Newton's Second Law of Motion	Force = Mass * Acceleration, or F = ma. This law says that when a force is applied to an object, the object's acceleration will be equal to the applied force divided by the object's mass.
Torque	A twisting or turning force. Typically measured in pound-feet (lb-ft) or newton-meters (N-m). Note that torque is NOT work, even though it is measured in the same units. The difference is that the force in torque is not in the same direction as the distance (it is perpendicular), so it does not count as work. Notice also that lb-ft is used instead of ft-lb; engineers do this as a convenient way to quickly tell the difference between a torque and work measurement.
Horsepower	A measurement for power that is commonly used in the U.S., abbreviated as "hp". James Watt invented this unit after observing that a horse could lift at a rate of about 550 foot-pounds per second for about an 8-hour shift.

Deriving the Relationship
To be clear, torque is a force that causes something to rotate. That something is usually circular, but it doesn't have to be. The distance part of torque is also called a lever-arm (or moment arm). The lever-arm acts to multiply the force being exerted. Think of opening a door, if you try pushing near the hinge, it takes a lot more force than pushing form near the handle. The distance from where you push to the hinge is the length of the lever-arm. Again, don't confuse torque with work. Think of torque as an instantaneous push on the door, where as work would be pushing the door over a distance.

Horsepower is how quickly you can apply torque over a given distance. Watt's definition says that 1 hp = 550 ft-lb/s. Now, how do we convert horsepower into something that relates to torque and engine rpm? Well, keep in mind that horsepower is force times distance divided by time, and our distance is going to be in a circle (i.e. in revolutions), so we can do the following conversions (I'll explain everything after the result):

1 hp = 550 lb * 1 ft / 1 sec

1 hp = 550 lb-ft * 1 rad / 1 sec

1 hp = (550 lb-ft * 1 rad / 1 sec) * (1 rev / 2pi rad) * (60 sec / 1 min)

1 hp = 5252 lb-ft * 1 rev / 1 min

1 hp = 5252 lb-ft * 1 rpm

-The first line is the original equation written out to better show the units and values.

-In the second line we want to convert the straight-line work to its rotational equivalent. This involves using radians instead of feet, and torque (lb-ft) instead of just pounds. Radians are actually a unitless measurement of a circular distance, and are abbreviated as "rad".

-The third line converts radians to revolutions and seconds to minutes. There are 2pi, or about 6.28, radians in one revolution and 60 seconds in 1 minute.

-The fourth line multiples the constants and cancels out some of the units, leaving us with a simpler equation.

-The last line just substitutes in the common abbreviation "rpm" (revolutions per minute).

So one horsepower is a force (a torque to be exact) of 5252 lb-ft @ 1 rpm. Now it is easy to come up with a formula for horsepower given engine rpm and torque. Say we have X lb-ft of torque @ Y rpm and we want to convert that amount of power to horsepower. The last equation tells us we can convert from (lb-ft * rpm) units to horsepower units by multiplying torque and rpm and then dividing by 5252:

power = X lb-ft * Y rpm
power = X lb-ft * Y rpm * (1 hp / (5252 lb-ft * 1 rpm))
power = (X lb-ft * Y rpm / 5252) hp

So,

Now you know exactly how horsepower and torque are related. There are no exceptions to this rule; they will always be related by this formula. In fact, it might be better to see them as just flip sides of the same coin. One result of this formula is that below 5252 rpm, torque will always be more than horsepower, at 5252 rpm they will be equal, and above 5252 rpm torque will be less. Note that a dyno never measures horsepower; it can only measure torque and then use the above formula to get horsepower.

Acceleration
So we now know that horsepower and torque are basically two different views of the same thing - change one and the other most also change. What we are really interested in is how they work together to affect a car's overall acceleration. Using Newton's Second Law, we know that F = ma. So we just rearrange the formula to solve for acceleration:

a = F / m

So acceleration is just the force at the wheels pushing the car forward divided by the mass (measured in kilograms) of the car. The force at the wheels is directly proportional to the torque exerted on the wheels by the driveshaft. Of course, the wheels don't actually move the car forward, the friction from the tires does, and we have to assume in all of this that we're not peeling out like in the picture.

So, our acceleration at any given moment is only dependent on the torque at the wheels (again, assuming that traction is not broken). That means the acceleration of the car is constantly changing to match the torque curve, and the car accelerates hardest at the torque peak. The name of the game is to keep your car producing as much torque as possible, for the longest amount of time possible. That will give you the best acceleration and fastest quarter mile times.

The Transmission Effect
Now comes the tricky part: the transmission. The fact that our engines cannot rev indefinitely means we must have transmissions to allow our wheels to keep spinning faster while keeping the revs under redline. A transmission essentially makes engine torque meaningless because it can multiple your flywheel torque to any amount it pleases, and is only limited by physical gear size. This is because gears can be arranged to increase or decrease their lever-arms and thus change the final torque output.

However, don't go out and buy yourself a giant gear set just yet: transmissions cannot amplify horsepower. This would violate the law of conservation of energy and change the world as we know it. Instead, as we increase the torque output for a certain gear, we decrease the maximum speed that we can have in that gear. Would you want your Mini Cooper Hardtop to have 600 lb-ft of torque in 1st gear but only be able to go 8 mph in that gear? Probably not. The opposite is also true; as you decrease the torque output of a certain gear, you increase maximum speed, assuming you can reach it. That is the main reason why you accelerate slower in higher gears.

The Verdict
So what advantage does a higher horsepower engine have compared to a lower one, assuming they both have similar torque curves? In short, the higher horsepower one will perform much better. The car with the high-reving, high-horsepower engine can run a more aggressive set of gears while still being able to spend the same amount of time in each gear as the slower car. This will give it better overall acceleration because of increased torque to the wheels. If both cars run the same gears, then the high-power one will be able to spend a longer time in each gear, giving it better overall acceleration, especially in the lower gears. Either way, it is an advantage. If you take away anything from this article, let it be that gearing is extremely important in determining a car's overall performance.

As far as shifting goes, always shift to maximize transmission output torque. It turns out this is exactly the same as saying shift to maximize engine power. Never shift at torque peak, even if your best friend tells you he swears it's faster that way. You will lose the overall higher torque of the current gear and it will also put you in a worse spot in the next gear. Most cars (Mini Cooper Hardtops included) will obtain their best quarter mile times by shifting at redline, but it is not true for every car. For some cars, it is necessary to look at a dyno graph to really see where the best point is.

And that's it. Now if someone claims your Mini Cooper Hardtop can't go fast because it doesn't have enough torque, you can prove them wrong!