|MadSci Network: Engineering|
Good question. You are right that there is more to how fast a car goes than just knowing the horsepower rating of the engine. First, some basics. Horsepower is a unit of power. Power has units of energy/time (energy per unit time). Physicists use a unit called a joule to measure energy in a system of units known as MKS (Meters Kilograms Seconds, as the units for length, mass and time, respectively). A watt is a unit of power that equals one joule per second. One horsepower equals about 746 watts. James Watt came up with the unit of horsepower by measuring how much power a horse could deliver. [Note that he did not name the watt after himself, that would have been unseemly. If you are a really good physicist you hope that someone else names a unit after you, probably well after you are dead.] The Encyclopedia Britannica says that James Watt must have measured a really strong horse, or maybe he only measured how much power it could put out for a short time. A more reasonable number for a horse working an 8 hour day is about 500 watts. You can get a rough idea of how much power a watt is by knowing that a bright light bulb draws about 100 watts, and that a typical microwave oven draws about 1000 watts. A person who is reasonably fit can put out about 200 watts for times of the order of an hour or more. I believe, though I couldn't confirm this anywhere, that the American cyclist Lance Armstrong has been clocked at 500 watts power output on a stationary bike, but I don't know for how long. You could say that he is as strong as a (typical) horse. Horsepower rating and how fast cars go is pretty complicated. I will simplify things a bit to try and get the essential physics right, and in so doing I will gloss over some important points. First, let's make a distinction between top speed and acceleration. You probably have noticed that large tractor trailer rigs are very slow in reaching their top speed; their acceleration is low. This is due to the fact that they are very massive. However, once you get that mass going at 70 miles per hour you do not need to add any more energy to maintain that velocity, if it wasn't for friction. Friction for a car or a truck is largely due to rolling friction plus air resistance. Neither of these depend directly on mass, so a truck that is empty has a top speed very close to that of a truck that is loaded up. [This is not quite true, as the rolling friction is probably a function of how much the tires are deformed by the load, with higher rolling resistance matching larger tire deformations, corresponding to a heavy load.] So top speed depends just on horsepower and the drag on the vehicle. Now let's consider acceleration. Acceleration has units of velocity/time (velocity per unit time). A car that is accelerating at 10 meters per second every second is going faster and faster. If it starts from rest, then after one second it is going 10 meters/second, and at the end of two seconds it is going 20 meters/second, and so on. Newton's second law helps us out in determining how fast a car can accelerate. To make things simple, we will consider acceleration in the absence of any friction. Then Newton's 2nd law says that force equals mass times acceleration. So acceleration is just the force divided by the mass. So mass counts; the more massive a vehicle is, for the same power, the more slowly it will accelerate. We will run through some typical numbers to see how things work out for a typical car. A car that weighs about 3400 pounds (= 1550 kilograms) and has a horsepower rating of 200 horsepower (149,000 watts) can accelerate to 60 miles per hour (26.8 meters/second) in about 8 seconds. Now if all the energy of the engine went into kinetic energy of the vehicle, we could see how the numbers checked, since we know that the product of the power and the time equals the energy the engine has put out, and we know the kinetic energy of the vehicle equals one half the mass times the velocity squared. Engine energy = Power*time= 149,000*8=1,2000,000 joules. Kinetic Energy= 0.5*1550*26.8*26.8= 560,000 joules. What went wrong? You will notice that a little less than half of the energy showed up as kinetic energy. Where did it all go? I’m not sure where all of it went, but I will hazard guesses as to where most of it went. An engine that is rated at 200 horsepower only puts out 200 horsepower near the top end of its R.P.M range (revolutions per minute of the motor). A car that rapidly accelerates to 60 miles per hour will spend a lot of time at lower R.P.M.s than this, as the cars gears are shifted from low gear to successively higher gears. I think that accounts for most of the discrepancy. Two other factors worth mentioning are that not all the energy that the motor puts out goes into translational kinetic energy of the car, a fairly large fraction is tied up in rotational kinetic energy of the car engine, the drive train and the wheels. I had a physics professor who gave a talk once on “The Physics of Drag Racing,” and I think he put the number at something of the order of 20% for a racing stock car. That is, the ratio of rotational kinetic energy to translational kinetic energy is about 0.2. Another factor we have ignored is friction. At low speeds friction is not a major factor, but certainly above 30 miles per hour we should be taking it into account. I have not done any of these calculations using torque, so I'll just mention briefly what it is. Torque times angular velocity equals the power output, so if you know how fast the engine is turning over and you know the torque you also know the power output. For a given wheel radius, torque is proportional to the force the wheels can exert on the ground, so the torque divided by the mass of the vehicle is proportional to the acceleration. Torque, like horsepower, is also a function of engine speed, with the available torque dropping off at low engine RPM. So you are correct in thinking that horsepower is not the only important thing in determining how fast a car or truck goes. For maximum acceleration you need a lot of power in a light vehicle and a light drive train. You also need a system of gears that allows you to get the most energy out of the engine at both high and low speeds. For high speed you need a lot of power and not much friction, which means a streamlined shape. I hope this helps. Drive safely.
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