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and so on. This is modified by the resistance of the air, which becomes important at higher speeds and usually sets an upper limit ("terminal velocity") to the fall velocity--a much smaller limit for someone using a parachute than one falling without. The number g is close to 10--more precisely, 9.79 at the equator, 9.83 at the pole, and intermediate values in between--and is known as "the acceleration due to gravity." If the velocity increases by 9.81 m/s each second (a good average value), g is said to equal "9.81 meters per second per second" or in short 9.81 m/s2. Adding an Initial VelocitySuppose the ball was given an initial velocity u upwards or downwards. If distances down are chosen as positive, the velocity due to gravity will always be positive, while u will be positive if directed downwards, negative if upwards. With this provision, observations show that u must be added everywhere to the velocity due to gravity, making the velocity in consecutive seconds (as in the list above) and in general, after t seconds, with gt meaning "g times t" The Distance CoveredThe distance covered by the ball can be shown to be the same as the one covered if the ball has a constant "mean" velocity v(mean), equal to half the sum of the velocities at the start and at the finish. For the falling ball in the preceding example, after t seconds that would be v(mean) = (1/2)[u + (u+gt)] making the distance covered distance = t v(mean) = ut + (1/2) gt2 Galileo's ExperimentThe book "The God Particle" by Nobel prize winner Leon Lederman with Dick Teresi (a good book, if you tolerate its irreverent attitude and corny humor) tells how Galileo showed that the downward motion of a ball due to gravity had a constant acceleration, and that starting from rest (u = 0) it covered distances which increased (as shown in the formula above) in proportion to the square of the time that has passed.
Since a vertical fall was too fast for Galileo to measure accurately, he slowed it down by making the ball roll down an inclined board. Across the board, along to its surface, he strung a number of taut horizontal wires, making the ball sound a click whenever it jumped over one of them. Galileo then moved the wires up and down the board, until the clicks sounded evenly spaced. If the acceleration now was a (much less than g) and the time t was measured in clicks, by the formula above, starting with the ball at rest (u= 0) one gets for the distance S covered After one click S = (a/2) 12= a/ 2 and so forth. The ratio between the distances should be that of the square numbers 1, 4, 9, 16, 25. . . and that was what Galileo confirmed. If one could convert the clicks into seconds, take into account the angle of the board and also the fact that the ball was rolling (a theory that was not available in Galileo's day), one could even in principle calculate g from a. An experiment similar to Galileo's, easy to perform, is described at the end of this file Waterfalls and BaseballsIf in addition the ball is also given at the start a horizontal velocity w, its horizontal motion continues unimpeded, advancing a distance w each second, even while the ball is falling (air resistance is now ignored). The two motions take place simultaneously, making the ball follow a curve, which gradually becomes steeper because the downward velocity keeps increasing while the horizontal one does not. That is how droplets move in a waterfall, which gives the cross-section of a waterfall its characteristic shape (left). 
The situation one sees again and again in "Road Runner" cartoon films, where Wily Coyote runs past the edge of a cliff, hangs motionless in space (as the realization dawns on him), then plummets down--that is a fanciful invention, quite contrary to the laws of physics. Actually two motions like that always take place simultaneously. Likewise, a bullet fired from a rifle at a target range begins falling the moment it leaves the muzzle. Adjusting the sights on the rifle for the distance shifts the aim to a point above the bull's eye, calculated so that at the appropriate distance, the fall of the bullet brings it down to the level of the target. For a stone thrown upwards, u is negative, as are any vertical distances above the throwing point. In that case, let û = -u be a positive number. Then after t seconds velocity = (-û + gt) m/s At first t is small and the velocity is negative, showing the stone travels upwards. At the time when gt = û the sum is zero and for the briefest instant the stone is at rest, at the top of its trajectory. At that time t = û/g and if that value is used in the formula for the distance, the height at which this happens is found. And finally, if one adds to this motion a constant horizontal speed w, the motion of a baseball or cannonball is derived. About air resistanceThe above formulas only hold for ideal cannonballs that experience no air resistance. In actual practice, air resistance modifies the motion and may have to be taken into account, especially in the fall of a light object, such as a feather. A science demonstration popular for centuries had a coin and a feather dropped simultaneously inside a glass tube from which air had been pumped: the two are always seen to drop equally fast. Such a demonstration was also made on the surface of the Moon by David Scott, one of the Apollo astronauts. Not only has the Moon no atmosphere, but its gravity is several times weaker, making the fall slower and easier to observe. In front of a TV camera, the astronaut simultaneously let go of his geology hammer and a feather, and his audience on Earth, watching their TV screens, observed the two falling together.
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The idea of the experiment is to attach the weights to the string at gradually increasing separation distances. Then holding up the end of the string on the side where the separations are largest (perhaps doing so while standing on a chair, but be careful), let it hang down from your hand, with the bottom weight just resting on the floor. We will call that "weight 0" and number the other weights consecutively from the bottom as "weight 1," "weight 2" and so forth. Then let go: you will hear "clack-clack-clack..." the sounds of the impacts of the weights on the floor. If top and bottom weights are separated by a distance of 1.25 meters, the duration of the fall will be 1/2 second, and with 5 weights as drawn, your ear should distinguish 1/4 of that interval. A longer string of course extends the fall time. The clacks replace the sounds of balls jumping over the wires in Galileo's experiment. They should sound evenly spaced: if they do not, adjust their distances (see further below) until they do (it may help having two separate experimenters, one dropping the string, the other listening to the clacks). Then measure the distance of each weight from "weight 0". When the clacks sound evenly spaced, the impact of each is separated from its neighbor by the same time interval, which we shall denote T. Weight 1 hits the floor time T after you release the string, weight 2 after time 2T (2 times T), weight 3 after time 3T (3 times T) and so forth. If the distance covered by an object falling from rest is proportional to the time squared (the formula is [1/2](g t2), then the distances of the weight from the end should be proportional to
(2T)2 = 4 T2 (3T)2 = 9 T2 and so forth, that is, they should have ratios 1:4:9:16... That can be checked: the distances of weights 1, 2, 3, 4... divided by the squares of these numbers, that is by 1, 4, 9, 16..., should all give the same value. In attaching the weights, you loop the string around twice around each nut, or pair of nuts--that is, if you thread the string from side "A" to side "B", take its end back to side "A" and thread it again, and again. With fishermen's weights, threading once should be enough. That way, if you want to move the weight, all you need to do is loosen the loop, then haul in string from one side and let it out on the other. Loosening the loop may be helped by a very thick needle, or a thin nail.
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Exploring FurtherA college study unit on motion down an inclined plane, showing that gravity is "diluted" by a factor sin A, where A is the angle of the slope. This point is also discussed in the next section Article "The Leaning Tower of Pisa" by Paolo Heiniger, p. 62, Scientific American, Vol. 273, No. 6, December 1995. Questions from Users: Speed of a toy car rolling of an inclined ramp |
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Author and Curator: Dr. David P. Stern
Mail to Dr.Stern: stargaze("at" symbol)phy6.org .
Last updated: 9-21-2004
