The most basic math has to do with horsepower. To begin, horsepower is a rate, a measure of force divided by time. The word comes from 18th-century Scottish inventor James Watt, who determined through observation that a typical horse could lift 550 pounds one foot off the ground in one second. This gives us the standard formula, 550 foot-pounds/second = 1 horsepower.
But Watt's horse was raising its weight in a straight line, straight up. Crankshafts in engines rotate, so we use a unit of force measurement called pounds-feet, the rotational variation, by which we define a twisting force, or torque. Watt went through some gyrations of his own--the stuff of first-year physics classes--to produce the following formula:
Horsepower = rpm x torque/5252.
The essential fact to note is that maximum horsepower is a product of both torque and rpm, so an increase in either will result in higher net horsepower and higher top speed.
Torque is always proportional to engine displacement; that is, a larger engine will always produce more torque than a smaller one. This is why vehicles used to tow boats and trailers (trucks and SUVs) typically are equipped with high-displacement V8s and even V10s.
Big engines have been around for a long time. Cadillac made a V16 engine in the 1930s with a displacement of 452 cubic inches. This monster motor produced a generous 320 pounds-feet of torque, about what a modern Corvette V8 produces. However, because the Cadillac V16 had such a low rpm limit (around 3600), it could generate a mere 175 hp, half of a Corvette's power.
It only goes to show that a smaller engine can produce more horsepower than a larger one by spinning faster. It is here we see the most significant gains in ordinary, four-cylinder mom-mobiles. Twenty years ago, a fast four-cylinder engine might redline at 5,000 rpm, redline being the upper limit at which the engine can safely spin. Today, most four-cylinder engines can turn up to 7,000 rpm, and some, like the Honda Civic Si, can turn up to 7,800 rpm, producing an amazing 160 horsepower out of a tiny, 1.6-liter engine, without benefit of exotic boosting apparatus like superchargers or turbochargers. Honda's unassuming runabout peaks at 127 mph.
Since more of the engine's torque is available to be multiplied by high rpm, more horsepower is produced. Remember the formula.
At moderate speeds on level pavement, the power required to move an automobile is only a portion of the power the engine is capable of developing in its upper range of speeds. Thus, the engine may operate at an uneconomically light load unless some means is provided to reduce its speed and power output.
This is what transmissions do. Transmissions are sets of gears that work exactly like the sprockets in a 10-speed bicycle. The larger the gear, the larger the multiplication of engine torque; engine speed, meanwhile, is divided by exactly the same numerical factor.
In the 1960s, most vehicles had three forward gears--the familiar "three-on-the-tree," for instance. The top gear was typically a 1.1 ratio--that is, for every turn on the crankshaft, the driveshaft turned one revolution.
By the mid '70s, four-speed transmissions began to appear, with "overdrive" gears, that is, with ratios under 1:1. These allowed cars to travel at higher speeds with lower engine speeds, and therefore less gas consumption. By the mid-'80s, European imports with five speeds, including very long-legged fifth gears designed for high-speed Autobahn travel, became common in America. The 1985 Audi Coupe, for example, arrived with a tall overdrive gear ratio of 0.64:1.
Today, most five-speed manual transmissions have an overdrive ratio of around 0.70:1, meaning that for every seven rotations of the crankshaft, the driveshaft connected to the driven wheels turns 10 times. Most automatic transmissions have similarly geared overdrives.
To see how it all works together, take for example the humble Dodge Neon ES. At 4,600 rpm, the engine produces 130 pounds-feet of torque. Multiplied by the Neon's first gear, with a ratio of 3.54:1, that amounts to 460 pounds-feet of torque, plenty to get the little car moving. Meanwhile, the engine's 4,600 rpm has been divided by the same factor, coming out to 1,299 rpm.
Now, if the wheels were directly connected to the transmission, they would also turn at 1,299 rpm. Measuring the circumference of the Neon's tire, about 72 inches, and multiplying it by the rpm gives you a total rolling distance of about a mile-and-a-half per minute, or 90 mph. Not bad for first gear!
Obviously, another gear intervenes, and this is the axle ratio, or in muscle-car lingo, the rear-end gear (found in the medicine ball-shaped differential case under a rear-wheel-drive car). In the muscle-car '60s, rear-end ratios in cars like the Pontiac GTO and Olds 442 were high, anywhere from 3.90:1 to 4.33:1 or more. These rear ends multiplied torque handily for smoky burnouts and quarter-mile drags but limited top speed.
As fuel-efficiency became more of a priority, axle ratios, like overdrive ratios, became smaller, lowering engine speeds at highway speeds. This gave engines more room to operate at higher rpm. As they say in NASCAR, new cars don't run out of gear.
With the Dodge Neon ES example again, take the available torque (130 lbs-ft. @ 4600 rpm), multiply it by the overdrive 5th gear (0.81) and multiply the result by the axle ratio (3.55). Now you have, theoretically, 373 pounds-feet or torque at the front wheels.
(130 lbs-ft) x (0.81) x (3.55) = 373 lbs.-ft of torque.
At the same time, the engine rpm of 4600 has been divided by the same gear numbers, so that the wheels are turning at 1,600 rpm, or 109 miles per hour. Aerodynamic drag finally stops the little car at a top speed of 118 mph.
And that is how cars go.