# Physics Homepage

## Graphing Motion

We will introduce motion through graphing. This allows us to get familiar with different types of motion without having to rely on equations and math too much. Have the “Graphing Motion” power point open at the same time you are looking at these pages.

When graphing motion we are looking at two changing variables. A driving car covers more and more distance the more time passes. Therefore the two variables that we graph are time and distance covered and the graph is called a “distance versus time graph”. Sometimes the distance axis is labeled position. This is misleading as it always refers to position with respect to a specific starting point. Taking both the position and the starting point into account it is clear that this is nothing but the distance. The same is actually true for the time axis, it displays the time passed from an initial starting time, which is in most cases zero. This means we start our clock at zero. Time is the independent variable (since you have no influence over it) and goes on the x-axis. Distance the dependent variable (it depends on how much time has passed) and therefore goes on the y-axis.

 When graphing motion you will either be given a complete graph or data that you will have to put into the graph. Check your workbook for examples. When you are given data you will notice that each point on the graph has two “coordinates”, one for distance and the other for time. Once you have input all your data into your graph you can then proceed to draw the line of best fit, that is the line that represents your collection of points best. Remember never to connect the points, but instead draw the best-fit line. Since we are trying to gain more insight into motion we will now look at the graph closer. If the best-fit line is straight you can find the slope of this line. In order to find the slope you will need to pick two points. In the power point I talk a little about what mistakes are being made if you were to pick only one point.

The slope is found by taking the ratio of the rise and the run of the best-fit line. This ratio represents how much the graph goes “up” versus how much it goes “over”. If the line is straight then the slope of this line must be the same everywhere. This is in fact the definition of a straight line - a line that has the same slope everywhere. The run in a distance versus time graph represents the time interval you are looking at, the amount of time that has passed. The rise represents the distanced covered. Therefore the slope, which is the ratio of the rise and the run is in fact the ratio of the distance covered and the time interval. We call this ratio the speed. This makes sense because with a steeper slope you cover more distance in the same amount of time, therefore you are going faster.

When the line is straight the slope of this line is the same everywhere. If the slope is the same everywhere that means that the speed never changes, since the speed is defined as the slope of a distance versus time graph. If the speed is the same or constant we are talking about uniform motion.

 Looking at distance versus time graph allowed us to find the operational definition of speed, namely that it is the distance covered in a certain time interval. In the power point I talk about what would happen if the axis were flipped and we were discussing a time versus distance graph. It is after all just a convention to put the independent variable on the x-axis. In this case we could also find the slope, but it would mean something very different; it would be “how long it takes to go a certain distanced”. Does this make sense? Does it have a meaning? Certainly. For instance, how long does it take you to run a mile? If the answer would 6 minutes 10 seconds, then this is how long it takes you to run a mile. Somebody running a mile in 6:40 is running slower. So it also makers the speed somehow. But it can’t be the same, because the greater this number the slower you are and the smaller your speed. You would have to make up your own name for this number, but it is useful nevertheless, when splits are given in cross-country for example.

One thing to keep in mind is that a graph does not represent a picture of the motion, which is evident in the example of the picture below. Describe the motion of biker in this graph. I am sure you will have said that he is going uphill fieriest, then staying on top of the hill and then turning around. However, this is a graph not a picture. All we can tall form the graph is the speed of the biker not where he is in France or at the North Pole, or whether he is going uphill or downhill. We see that his starting point is noted with “0”. He proceeds at a constant speed for 600s since the line is straight. . Calculate the speed by finding the slope. Then he continues at a constant speed (the line segment is straight again) for 400 s. However, the slope now is zero, therefore the speed is zero and the biker is resting. Then he comes back. The distance goes closer to zero, his starting point. Again he does so at a constant speed and you are able to find this speed by finding the slop of the line. Keep in mind that a graph does not represent a picture of the path the biker takes.