Projectile Motion

We will use a team of observers to determine the path of a ball projected horizontally. It should look like the figure shown here. We will station observers at regular intervals along the horizontal direction (the x-direction) and each observer will attempt to mark the position of the ball as it goes by. Then we can measure how far the ball has dropped at that x-position. These will be the distances shown in red in the figure. Since these distances are measured along the vertical, or y-axis, we may call them y-values, and make a table of corresponding x's and y's.

Galileo claimed that the path of the ball is a parabola. His reason for thinking so is very ingenious. He imagined that the curve we see is just the combination of two simpler motions. The horizontal motion of the ball is neither increased or diminished by gravity, so it stays constant. Thus the ball passes the observers, who are regularly spaced, at equal intervals, like the ticks of a clock. Tick tick tick tick tick! Now what about the vertical motion? This is just simple falling, and we know how that works: 1, 4, 9, 16, etc. It is uniformly accelerated motion. To test this idea experimentally, we have to see if the y-values are proportional to the x-values SQUARED. This is easy to do. We just square the x values -- which we can represent as the third column in the table that already has a column for x and a column for y -- and then we plot the y versus the x2. If they are proportional, then the points will fall on a straight line. Notice that the way we are measuring, the point (0,0) is the first point. If you don't choose to measure things from the beginning of the horizontally projected trajectory, then everything gets more complicated to describe, needlessly so. The graph might look something like the one below.

You should also try this with a projected rolling object on a ramp. It can be as simple as a wet ball-bearing on a paper towel. The wet track left behind can be analyzed just like the trajectory above. Once again you should measure everything from the highest point of the trajectory, the same point that we called (0,0) in the first figure (upper left corner of that figure).