Today, let's think about falling. The University of Houston's College of Engineering presents this series about the machines that make our civilization run, and the people whose ingenuity created them.

The concept of acceleration is hard to see clearly without calculus and graphs. Yet acceleration is with us every waking moment. We all swim in the same sea of uniform gravitational acceleration. We feel it all the time. Every time we drop or toss an object, gravity acts upon it in the same way. Jump from a height of five feet, and you'll strike the earth at eighteen feet per second. From a ten-foot wall, that becomes twenty-five feet per second.

So when you double the height, you don't double the speed you reach. Speed rises only as the square root of the height of the fall. By the way, you start endangering your limbs at about twenty feet per second (depending on your age and physical condition).

Gravity will accelerate any object at a rate of 32 feet per second per second. But what do we do with that number? What it means is that if we fall for one second we'll reach a speed of 32 feet per second. After two seconds we reach 64 feet per second. The speed rises as the square root of height, but in direct proportion to time.

So acceleration is trickier than it might first seem. Nothing accelerates until a force acts upon it. Yet we feel no force as we fall. The force of gravity is there, acting on every molecule in our bodies -- but the force is unopposed, so we feel nothing. Not until we stand on a solid floor do we feel the force of gravity. The floor is what resists gravity, and it acts only on our feet.

So an orbiting astronaut, who feels no gravity, is in a perpetual free fall, constantly accelerating toward Earth and hurtling forward at the same time. The Space Shuttle keeps falling away from a straight path, but just fast enough to stay a constant height above Earth as it falls -- and falls, and falls.

Swing a rock on a string, and it follows the same kind of circular path as the Space Shuttle does. But there's no significant force of gravity to attract the rock toward you. That's why you had to replace gravity with a string. Now you feel just how much force it takes to accelerate the rock away from straight flight.

Of course most accelerations don't have the uniformity of gravity. A rising elevator accelerates at first, and we feel our weight increase by a few pounds. When we decelerate at the 18th floor, our weight drops just a tad. (That can be a nice feeling.)

But too many people don't get it -- like motorists who tailgate or don't slow down for a curve on an icy road. Acceleration can deceive us. That's why Isaac Newton, who first explained how force and acceleration are related, was also an inventor of calculus -- that special language for explaining how things change in time and space. Acceleration is so much clearer when we have that new language to describe it. And I hear echoes of a fine old saying about the language of math: "Mathematics lets fools do what only geniuses could do without it."

I'm John Lienhard, at the University of Houston, where we're interested in the way inventive minds work.

(Theme music)

I include no reference material with this episode, since the ideas in it may be found in any beginning physics book at the high-school or college level. Some useful expressions for the motion of a body that starts out stationary and is acted upon by a uniform gravitation, a, for a time, t, are:

The distance traveled is s = at^2/2

And the speed it reaches is v = sqrt(2as) = at

For these formulae to work properly, the units must be consistent. Express everything either in feet and seconds or in meters and seconds. The acceleration of gravity is 32.17 ft/s^2 or 9.807 m/s^2.

Astronaut Mary Ellen Weber, weightless and falling within a KC-135 aircraft. By flying in a ballistic parabola, the aircraft moves as a projectile would.
(Photo courtesy of NASA)