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Naturally Accelerated Motion

Michael Fowler, Physics Dept., UVa

Distance Covered in Uniform Acceleration

In the last lecture, we stated what we called Galileo's acceleration hypothesis:

A falling body accelerates uniformly: it picks up equal amounts of speed in equal time intervals, so that, if it falls from rest, it is moving twice as fast after two seconds as it was moving after one second, and moving three times as fast after three seconds as it was after one second.

We also found, from the experiment, that a falling body will fall four times as far in twice the time. That is to say, we found that the time to roll one-quarter of the way down the ramp was one-half the time to roll all the way down.

Galileo asserted that the result of the rolling-down-the-ramp experiment confirmed his claim that the acceleration was uniform. Let us now try to understand why this is so. The simplest way to do this is to put in some numbers. Let us assume, for argument's sake, that the ramp is at a convenient slope such that, after rolling down it for one second, the ball is moving at four feet per second. This means that after two seconds it would be moving at eight feet per second, after three seconds at twelve feet per second and so on until it hits the end of the ramp.

The hard part is figuring out how far it moves in a given time. Let us ask a specific question: how far does it get in two seconds? If it were moving at a steady speed of eight feet per second for two seconds, it would of course move sixteen feet. But it can't have gotten that far after two seconds, because it just attained the speed of eight feet per second when the time reached two seconds, so it was going at slower speeds up to that point. In fact, at the very beginning, it was moving very slowly. Clearly, to figure out how far it travels during that first two seconds what we must do is to find its average speed during that period. This is where the assumption of uniform acceleration comes in. What it means is that the speed starts from zero at the beginning of the period, increases steadily, is four feet per second after one second (half way through the period) and eight feet per second after two seconds, that is, at the end of the period we are considering. Notice that the speed is two feet per second after half a second, and six feet per second after one-and-a-half seconds. If you draw a graph of the speed as it varies in time, it should be evident that, for this uniformly accelerated motion, the average speed over this two second interval is the speed reached at half-time, that is, four feet per second. Thus, the distance travelled in two seconds is eight feet---that is, four feet per second for two seconds.

Now let us use the same argument to figure how far the ball rolls in just one second. At the end of one second, it is moving at four feet per second. At the beginning of the second, it was at rest. At the half-second point, the ball was moving at two feet per second. By the same arguments as used above, then, the average speed during the first second was two feet per second. Therefore, the total distance rolled during the first second is just two feet.

We can see from the above why, in uniform acceleration, the ball rolls four times as far when the time interval doubles. If the average speed were the same for the two second period as for the one second period, the distance covered would double since the ball rolls for twice as long a time. But since the speed is steadily increasing, the average speed also doubles. It is the combination of these two factors---moving at twice the average speed for twice the time---that gives the factor of four in distance covered.

It is not difficult to show using these same arguments that the distances covered in 1, 2, 3, 4, ...seconds are proportional to 1, 4, 9, 16, .., that is, the squares of the times involved.

A Video Test of Galileo's Hypothesis

In fact, using a video camera, we can check the hypothesis of uniform acceleration very directly on a falling object. We simply record it and play it back frame by frame. The camera has a built-in clock---it films at thirty frames per second. Thus, by counting frames, we can see exactly how long the ball takes to fall through a certain distance. By using a pre-measured distance, we can figure out its average speed during that time interval, and by doing this for different distances, check that it is gaining speed at a uniform rate.

The simplest thing to do with the video camera, though, is just to repeat more directly Galileo's ramp discovery that in twice the time the ball falls four times the distance. Checking this for a series of distances ensures that the rate of increase of speed is uniform, and furthermore we can find out what it is.

The Motion of Projectiles

We follow fairly closely here the discussion of Galileo in Two New Sciences, Fourth Day, from page 244 to the middle of page 257 .

To analyze how projectiles move, Galileo describes two basic types of motion:

(i) Naturally accelerated vertical motion, which is the motion of a vertically falling body that we have already discussed in detail.

(ii) Uniform horizontal motion, which he defines as straight-line horizontal motion which covers equal distances in equal times.

This uniform horizontal motion, then, is just the familiar one of an automobile going at a steady speed on a straight freeway. Galileo puts it as follows:

"Imagine any particle projected along a horizontal plane without friction; then we know...that this particle will move along this same plane with a motion that is uniform and perpetual, provided the plane has no limits."

This simple statement is in itself a substantial advance on Aristotle, who thought that an inanimate object could only continue to move as long as it was being pushed. Galileo realized the crucial role played by friction: if there is no friction, he asserted, the motion will continue indefinitely. Aristotle's problem in this was that he observed friction-dominated systems, like oxcarts, where motion stopped almost immediately when the ox stopped pulling. Recall that Galileo, in the rolling a ball down a ramp experiment, went to great pains to get the ramp very smooth, the ball very round, hard and polished. He knew that only in this way could he get reliable, reproducible results. At the same time, it must have been evident to him that if the ramp were to be laid flat, the ball would roll from one end to the other, after an initial push, with very little loss of speed.

Compound Motion

Galileo introduces projectile motion by imagining that a ball, rolling in uniform horizontal motion across a smooth tabletop, flies off the edge of the table. He asserts that when this happens, the particle's horizontal motion will continue at the same uniform rate, but, in addition, it will acquire a downward vertical motion identical to that of any falling body. He refers to this as a compound motion.

The simplest way to see what is going on is to study Galileo's diagram on page 249, which we reproduce here.

Imagine the ball to have been rolling across a tabletop moving to the left, passing the point a and then going off the edge at the point b. Galileo's figure shows its subsequent position at three equal time intervals, say, 0.1 seconds, 0.2 seconds and 0.3 seconds after leaving the table, when it will be at i, f, and h respectively.

The first point to notice is that the horizontal distance it has travelled from the table increases uniformly with time: bd is just twice bc, and so on. That is to say, its horizontal motion is just the same as if it had stayed on the table.

The second point is that its vertical motion is identical to that of a vertically falling body. In other words, if another ball had been dropped vertically from b at the instant that our ball flew off the edge there, they would always be at the same vertical height, so after 0.1 seconds when the first ball reaches i, the dropped ball has fallen to o, and so on. It also follows, since we know the falling body falls four times as far if the time is doubled, that bg is four times bo, so for the projectile fd is four times ic. This can be stated in a slightly different way, which is the way Galileo formulated it to prove the curve was a parabola:

The ratio of the vertical distances dropped in two different times, for example bg/bo, is always the square of the ratio of the horizontal distances travelled in those times, in this case fg/io.

You can easily check that this is always true, from the rule of uniform acceleration of a falling body. For example, bl is nine times bo, and hl is three times io.

Galileo proved, with a virtuoso display of Greek geometry, that the fact that the vertical drop was proportional to the square of the horizontal distance meant that the trajectory was a parabola. His definition of a parabola, the classic Greek definition, was that it was the intersection of a cone with a plane parallel to one side of the cone. Starting from this definition of a parabola, it takes quite a lot of work to establish that the trajectory is parabolic. However, if we define a parabola as a curve of the form y =Cx² then of course we've proved it already!

Written material Copyright © Michael Fowler 1995 except where otherwise noted.