Before I try to bore you with science, I'm going to begin with a hypothetical experiment. Imagine you have a 2.2 liter ECOTEC engine out of a Saturn ION. (I picked this because I happen to drive one, but the principles here will apply to any engine.) According to General Motors, this version of the ECOTEC develops 145 horsepower at 5600 rpm and 150 ft-lb of torque at 4200 rpm. I recreated their dynamometer ("dyno") chart here. Torque is shown in dark blue, and horsepower is shown in purple.
Now, let's say I have a frictionless gear set (2:1) that I can put between the engine and dyno to step up the torque while I step down the speed. Some motor head guys would love this new engine. It would develop twice the torque at half the speed. Let's see what happens to horspower in the process, though.
As expected, the new engine develops 300 ft-lb at 2100 rpm. Why did the horsepower remain the same at 145hp, now developed at 2800 rpm? Let's try a 5:1 stepdown.
Now we have 750 ft-lb of stump-pulling torque (out of a 2.2 liter gas engine!) developed at a lazy turnover of only 840 rpm. (The original engine idles around this range.) Horsepower stayed the same, but now we develop it at 1120 rpm. Now let's flip our gear around and spin the output up while we lower the torque, again 2:1.
A tuner might love this engine. It redlines at 12900 rpm. Maximum power is developed at a sport-bike 11200 rpm, but why is it still 145 hp? Max torque is 75 ft-lb at 8400 rpm. How are we getting all this power with no torque? Notice, just like the stock engine, that our torque and horsepower curves cross at 5252 rpm. What's going on here?
We need to talk physics for a bit here to know what's really going on. We need to understand some key terms. If we don't know them and understand them, we won't stand a chance at understanding how an engine makes a car go.
First, we have force. A force is a push or pull on an object. You can't see a force; you can only see what it does to things. You can push against a wall with nothing happening, but you still apply a force. You can multiply or divide a force with levers and other machines. According to Newton's Second Law, we can define force as the acceleration that it causes on an object: F=ma (Force = mass * acceleration). If we want to accelerate an object that weighs one kilogram by one meter per second per second, we need to apply a force of one newton. If you want a detailed explanation, feel free to look at the Wikipedia page.
Second, we have torque. A torque is a turning force that you get when you multiply a force by a lever arm. A breaker bar has a long arm to get a lot of torque with comparatively little force, but a screwdriver acts on a small arm, making bolts hard to turn. We can use pulleys or gears to multiply or divide a torque just like we can a force.
Remember when I said you could multiply forces and torques? You sacrifice the distance you move in the process. Let's say you have an engine lift that uses rope and pulleys. To lift an engine that weighs 1000 pounds up by 1 foot, you'll have to pull on your rope with a force of 100 pounds over a distance of 10 feet. (You'll need to set up your system for 10:1 operation, but that should be understood here.) Any product of these two (force * distance) will give you the same results. Superman might want to pull 10,000 pounds over 0.1 feet to speed things along, but a patient old lady might pull 10 pounds over 100 feet of rope.
Energy is a very fundamental property of the universe. You can't create it or destroy it. Whatever energy you put into a system is the maximum amount you'll be able to pull out. You can't win. This is the First Law of Thermodynamics.
Whenever you see the word "work" just remember that work is just the mechanical form of energy. Energy can come in many forms. There is a certain amount of energy in a given amount of fuel, and a battery can store a certain amount of energy, but we're only concerned with the mechanics of a car, so we're interested in work.
Work is basically a force over a distance. Work = force * distance. In the engine lift example, we needed to accomplish 1000 ft*lb of work. Wait a minute... doesn't that unit (ft*lb) look familiar?
Let's say we have a winch instead of a pulley system. When we rotate the winch, we apply a force over a distance. The distance we apply the force over is actually the circumference of the circle formed by the radius arm.
Energy = Force * Distance = Force * 2 * pi * Theta
(Theta is the angle in radians. 360 degrees = 2 * pi radians.
1 radian ~ 57.3 degrees.)
Thus, we can not only define energy as force in a straight line, but we can also define it as torque through an angle. For example, if we need to rotate the shaft three times against a 100 ft*lb force, there are many combinations of force and lever arm that will get us there, but we're using the same energy regardless.
Power is just work over time. If I expend 10 ft*lb of energy in a straight line for 5 seconds, I develop 2 ft*lb/s. Horsepower is just a unit of power. If I expend 33000 ft*lb in one minute, I develop one horsepower.
Let's say I have an engine that outputs 5252 ft*lb of torque through one revolution for one minute. Let's pick a lever arm of 1 ft. (This is arbitrary.) Through this one foot, I get a force of 5252 lbs. Around the circle, this lever arm swings through 2*pi*1 feet, or approximately 6.28 feet. Multiplying the force by the distance, we get roughly 33000 ft*lb, and the time is still one minute. Thus, 5252 ft*lb for one revolution per minute is one horsepower.
This can be rather convenient. Power (hp) = Torque (ft*lb) * Rotational Speed (rpm) / 5252.
Let's look at some implications here. If we want to do the same amount of work but in half the time, we need twice the power. If we want to do the same amount of work in the same amount of time, we can play with force and distance all we want as long as everything works out.
By putting some equations together, we can get another equation for power.
Work = Force * Distance
Speed = Distance / Time
Power = Work / Time
Power = Force * Speed
This last equation is most useful for explaining how a car goes down a track. Ignoring drivetrain losses, at any given instant, the force on the car is the power of the engine (at its current rpm) divided by the vehicle speed. The acceleration is therefore that force divided by its mass.
Power = Mass * Acceleration * Speed
Acceleration = Power / (Mass * Speed)
Whenever you use mass, don't forget the driver. Also, accelerations are easily understood as a multiple of gravity.
g = 9.8 m / s^2 = 32.152 ft / s^2
There is an error in the US Customary unit system that tends to keep Americans from fully understanding physics. A pound is a unit of force but is also often used as a unit of mass. The proper unit of mass is the slug, which is approximately 32.174 pounds. If you want to avoid unit problems, I'd suggest using Google's calculator. Here's an example using the last equation, showing the acceleration (in g's) for my car when shifting gears at 55 mph ignoring drivetrain losses.
So what really matters: torque or horsepower? People say that "torque gets you going and horsepower keeps you going." This is a misconception based on the behavior of their less-than-ideal transmissions. Most transmissions have a finite number of gears, and that means that you have to work within a power band instead of staying at your favorite rpm. A perfectly designed transmission should be able to provide enough wheel torque to spin the tires from a dead stop with any engine, but not all street transmissions can do this because they slip too much or they have gears that are too tall. Remember the formula? When the car speed is zero, the force can be infinite with any amount of power. Drag racers lauch at 6000 rpm because that's where they're developing power. (Anything higher would basically guarantee that they'd lose traction.) The ideal transmission would always keep the engine at max power. Anyone who leans on torque needs to look more closely at their transmission; it's not doing its job if you need a wide power band. They also need to look at what's going on: They're not really relying on torque. Average horsepower throughout the power band is really what's responsible. The torque still doesn't matter. You typically see cars with lots of torque having wide power bands, and that's the only reason why they'd ever beat a car with more peak horsepower. Average horsepower wins races. Waiting 10 seconds to spool your turbo loses races.
Torque is basically a measurement of how much energy an engine develops each time it turns over, but this energy gets translated to power when you develop that energy more often in a given time period (i.e. higher rpms). This is why horsepower graphs can still increase even when torque is decreasing. You can be making less energy per revolution, but as long as you're making it enough times per minute, you can still be making more power.
If you still don't understand this concept, imagine your job is to hammer a nail into a board of wood. It takes a certain amount of work to get this job done, and you want to do it as quickly as possible. You can choose to either use a heavy hammer and do it in five or six blows with a short pause between them ("lots of torque"), or you can use a lighter hammer and whack it many times but more quickly ("high rpms"). If you've accomplished the job in the same amount of time, you've developed the same amount of power. (Yes, I know how friction on a nail isn't this simple, but read the top of the page again before you yell at me about static and dynamic friction, inelastic deformation, and non-Coulomb friction. If you're already to this level of physics understanding than why are you reading this anyway?)
It might be tempting to consider torque to be the fundamental reason why a car moves, but that torque has to go through the gears. The engine has to keep up with the tires through the chosen gear ratio, and this gear ratio is going to change both the speed and torque, preserving the power.
The easiest way to get a naturally aspirated engine to develop serious power is to raise the power band. When you raise the power band, you develop usable torque at higher rpms, which translates to more power, even if max torque falls off. (The power band is raised by opening up the whole gas exchange process to optimize scavenging and ram charging at higher engine speeds.) The limiting factor here is mechanical failure at high engine speeds, and you're also looking at an engine that's difficult to start with decreased performance at lower rpms if your tranmission sucks. (A good transmission should keep you out of that range.)
In a turbocharged engine, we can increase the torque too by increasing boost. Now our limits are mechanical failure, the limits of the turbo, and detonation ("knock"). By increasing both the torque and rpms, we can gain a double benefit over a mild, naturally aspirated street engine.
I plan to expand this with some more illustrations and perhaps more content, but feedback on what I have already is essential and very-much appreciated. If you think anything should be added or subtracted or edited feel free to let me know.