Naturally Accelerated Motion

Michael Fowler  UVa Physics

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 two meters per second.  This means that after two seconds it would be moving at four meters per second, after three seconds at six meters per second and so on until it hits the end of the ramp.  (Note: to get an intuitive feel for these speeds, one  meter per second is 3.6 km/hr, or 2.25 mph.)

To get a clear idea of what’s happening, you should sketch a graph of how speed increases with time.  This is a straight line graph, beginning at zero speed at zero time, then going through a point corresponding to two meters per second at time one second, four at two seconds and so on.  It sounds trivial, but is surprisingly helpful to have this graph in front of you as you read—so, find a piece of paper or an old envelope (this doesn’t have to be too precise) and draw a line along the bottom marked 0, 1, 2, for seconds of time, then a vertical line (or y-axis) indicating speed at a given time—this could be marked 0, 2, 4, … meters per second.  Now, put in the points (0,0), (1,2) and so on, and join them with a line. 

From your graph, you can now read off its speed not just at 0, 1, 2 seconds, but at, say 1.5 seconds or 1.9 seconds or any other time within the time interval covered by the graph. 

The hard part, though, is figuring out how far it moves in a given time.  This is the core of Galileo’s argument, and it is essential that you understand it before going further, so read the next paragraphs slowly and carefully!

Let us ask a specific question: how far does it get in two seconds? If it were moving at a steady speed of four meters per second for two seconds, it would of course move eight meters.  But it can’t have gotten that far after two seconds, because it just attained the speed of four meters 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 at a constant rate, is two meters per second after one second (half way through the period) and four meters per second after two seconds, that is, at the end of the period we are considering.  Notice that the speed is one meter per second after half a second, and three meters per second after one-and-a-half seconds.  From the graph you should have drawn above 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, two meters per second. 

Now, the distance covered in any time interval is equal to the average speed multiplied by the time taken, so the distance traveled in two seconds is four meters—that is, two meters 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 two meters per second.  At the beginning of the second, it was at rest.  At the half-second point, the ball was moving at one meter per second.  By the same arguments as used above, then, the average speed during the first second was one meter per second.  Therefore, the total distance rolled during the first second is just one meter. 

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 too 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.  This is left as an exercise for the reader.