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To understand motion is to understand nature - Leonardo Da Vinci

Physics is the study of the universe in terms of its basic constituents (what is it made of?), and the rules of its operations (why do things work the way they do?). The first true advances in the modern era of science began with observation and experimentation on motion. The reason is quite simple. Those things which are of interest in science are the things which undergo change. To understand how something works, you have to see it in action. The workings of the universe include anything in the universe which experiences change according to some repetitive pattern. Change could be in the form of a chemical reaction, an increase or decrease in the population of butterflies, etc. The easiest changes to observe are those of motion. An object is moved from one position in space to another. The motion generally leaves the object itself unchanged and thus simplifies the observation.

Lecture material

Although it may seem trivial to discuss an object moving from one spot to another, the quantitative description of motion quickly advances one's understanding of observations of the operation of the universe, first on a descriptive level, then on the deeper level of explaining why things occur. In the short span of time between Galileo and Newton, mankind went from understanding the motion of falling bodies to understanding the motion of the moon around the earth. The study of motion explained the regularity of the pendulum clock and predicted the position of the planets in the nighttime sky. Much of the reason for this rapid success is the relative simplicity of the mathematics associated with motion as well as the ready availability of experiments to clarify ideas. For this reason, it is inevitable that studies of physics also begin with the study of motion.

To begin any serious study, we first need to find a method for quantifying data. In the case of motion, the evidence that motion has occurred is a displacement of the object that moved or changed position. The position is quantified by measuring it relative to some fixed position (usually referred to as the origin) in terms of an internationally agreed upon standard. The standard unit of distance is the meter. In mathematical terms, we use a symbol, like x, to specify the position of an object relative to the origin. Subscripts are used to specify more than one position. For example, we might say that an object is first located at position x₁ and then later at position x₂ , still later at position x₃ and so forth.

Displacement is defined to be the change in position or distance which an object has moved. Hence, we quantify how much something has moved by measuring its displacement in units of meters. The larger the displacement, the more motion that has occurred.

In order to be useful, we also need to specify something about time. After all, a motion which causes a displacement of 10 meters can be large or small depending on whether it took a second or a thousand years to do it. The international standard unit of time is the second. Hence, time and space are inextricably linked in physics since we need both to explain motion and motion is fundamental to all areas of physics. In the language of mathematics, we describe the changes in position or time as

where the 0 and f subscripts depict initial and final, respectively.

This use of symbols requires some patience if you are not already comfortable with it. The normal tendency of students new to physics is to immediately replace symbols with numbers as soon as possible. Experience will show you that this is generally not a good thing to do. Why? The most important reason to you is the very practical problem of mistakes: you tend to make more algebra errors with numbers than with symbols (despite what your instincts may tell you, this is always true!). What's most important to a scientist, however, is that the symbols represent the essence. The numbers are hardly ever the important point in understanding what's going on. Numbers can be changed by situation, choice, or any number of unexplained reasons, but the mathematical description of what happens to the symbols is what represents the underlying truth. In other words, if you derive an equation by mathematical rules that correctly describe the way nature works, its the equation that is always true. The numbers that go into the equation can vary by large amounts, but whatever their values, they must always satisfy the equation! Our hope for the course is to make the language of the equations second nature to you so that the essence of the science represented by them is clear.

As you have seen, visualization is vital to understanding what equations tell us. We will commonly make use of graphs to understand the behavior of the equations we derive. For motion, the Cartesian graph is the standard means of looking at position (and hence displacement) and time. An example of a such a graph for a person running a 100 meter dash is shown below.

<IMG ALIGN=absbottom SRC="http://dept.physics.upenn.edu/courses/gladney/mathphys/images/hundred_meter_dash_graph.gif">

The power of graphs like this is in giving a concise description of what happened during the race. To see this, all we need to do is to give a description, in words, of what the graph depicts as a picture. For example, we see that the runner starts at the origin, moves 5 meters in the first 1.6 seconds, moves 5 meters more (to a position 10 meters from the origin) in the next .4 seconds, etc. From this kind of direct information, we can infer all kinds of other information. For example, we can quantify how long it takes a runner to get out of the starting blocks, how effective the runner's late kick at the end of the race is, and so forth. We also have some notion of how fast the runner is moving. If we compare this runner's motion to that of another runner in the race as in the graph below, we find that we can compress even more information into the picture.

This graph shows two runners (red and blue) which are racing each other. They start at the same origin, but blue is faster out of the blocks, so that red is left behind (i.e. blue gets to each 5 meter marker at less time than red). Eventually, red's late kick for the finish line blows past blue and red wins the race. The picture sums up all of this and much, much more in a very neat fashion.

In looking at such graphs, we need to be mindful of some shorthand notations as well some additional mathematical aspects. First, even though we have shown a graph which deals only with positive displacements, there is nothing to prevent us from displaying negative ones as well. Therefore, we need to be mindful of the sign we get from . For time, the convention is usually to formulate our plots and our equations so that is always positive (note that this a convention only; the equations of physics are, with the exception of thermodynamics and some exotic parts of particle physics, just as true for negative time changes as for positive!). For this reason, we usally don't bother with writing out , but instead, assume that whenever we start looking at the situation it is zero time. We are free to start our observations whenever we want since the important aspects of a problem usually do not depend on when we start, but simply how much time passes during our observations.

To continue our rather legal-sounding definitions, we note that there are two other aspects of motion which are important. The first is that we would like to quantify the amount of motion taking place. Therefore, we define the average velocity or speed of an object. As we said above, the idea of quantifying motion involves both the distance traveled and the time it took to travel it. For that reason, it makes sense to define the average velocity, as follows: This equation has the right characteristics for what we desire: if the time interval is small or the displacement is large, we have a large value for , i.e. a lot of motion is taking place. On the other hand, if the time interval is large or the displacement is small, then the average velocity is small, i.e. not much is happening as far as motion is concerned. Notice that on the graphs like the ones we showed above, the average velocity is just the slope of the x vs. t data.

The other useful definition is the average acceleration, . By definition, acceleration is just the change in velocity per time, so we have We could go on. For example, the change in acceleration per time is called the jerk (can you see why?) and finds use in automobile crash-testing. By and large, though acceleration, velocity, and displacement are all we usually need to consider to describe almost any motion we see. You should remember that each of these quantities can be positive OR negative (or zero) and that the sign of one of them does NOT tell you the sign of the other two. It's perfectly OK to have an object which undergoes a positive displacement even though it has a negative acceleration and vice versa! Let's practice getting information from graphs which describe motion, by checking the following:

The fact that changes in acceleration do not usually need to be considered is usually exploited when considering the mathematical description of the motion by noting that if the change in acceleration is small or non-existent for the motion we are observing, then the acceleration for the problem must be constant. Under this condition, we can drop the subscript and just write a for the acceleration. Not much of a gain is it? In fact, it is a tremendous gain on any occasion in which you can assume something is constant in physics (or any other science for that matter) because the mathematical description of the situation becomes considerably easier to handle. For the case of constant acceleration, we can view the power in the simplification by noting that we now have relationships between displacement, velocity, and acceleration which allow us to make an even more compact description of motion than we get from graphs. Let's start with the definition of acceleration:

Remember that implies a change from initial to final (expressed as final - initial) and that, by convention, we usually express as . An alternative way of expressing the average velocity is simply to note that an average is just the sum of quantities divided by the number of quantities in the sum. Therefore, as long as we are dealing with a situation in which the acceleration is constant and we know the initial and final velocities we can write The advantage of having two different ways of expressing the velocity is that we can deal with problems in which some piece of information is not known. If the acceleration and time are known, for example, then we can calculate the change in velocity by our first formula. If the acceleration and time are not known, then we can still calculate the average velocity as long as the initial and final velocities are known.

We can also relate average velocity and displacement by definition: We can even relate acceleration, velocity, and displacement by using these expressions:

Finally, if we have a problem in which the change in time is not known, then its good to have a relationship between velocity, acceleration, and displacement which does not explicitly depend on time. Fortunately, we can get such an expression by noting that time can be replaced with the following equation: again assuming constant acceleration. Now, going back to displacement, we get

Finally, let's list each of our expressions in one place as a way of gathering things together.

where we now follow the conventions of most physics textbooks by dropping the f subscript for the final quantities.

These equations, called the equations for motion were first arrived at during the time of Galileo, who recognized their tremendous power for describing natural events. For example, the earth's gravity provides an essentially constant downward acceleration for any object that is not supported by something else. In this case, the acceleration is termed g = 9.8 m/s² and is always directed downward.

The power of these equations is only realized if we know when to apply each one. Therefore, we should practice just recognizing which equation is most likely to get us to the answer in the shortest number of steps. With a little experience, this becomes second nature. Until we get that experience, a practical rule of thumb is to list the known quantities - in this case we are translating from English to math. Once the known quantities are listed, ask yourself which unknown is requested by the problem. You should now have enough information to see whether any of the four equations allows you to get the answer directly or whether you will have to use several steps to get to the solution. Let's practice.

First and foremost is the need to take what you learned from the derivation of these equations to predict what will happen for experiments that you've never actually done. The power of Galileo's prediction lies precisely in the fact that you can make such predictions and be very sure that they'll be borne out when you actually try them. A further advantage is that you can predict motion in more than one dimension for certain constrained cases. For example, Galileo studied balls moving down smooth inclines. Since the gravitational acceleration is only in the down direction and is always constant, it provides a constant acceleration for any object which falls down a constant slope. That acceleration is less than the value of g, 9.8 m/s², we talked about (in fact, the value is g*sin(theta) where theta represents the angle of inclination with respect to the horizontal; we'll discuss this in a later section), but the fact that it is constant in direction (i.e. parallel to the slope of the incline) and magnitude is what's important for predicting the motion. Constant acceleration along the incline can be considered as 1-dimensional since the ball can only move up or down the incline. Here's a Java applet that let's you try to make predictions as Galileo did. Try it out!

Get some exercise with motion graphs by doing the following: 1-7 and 1-8 in the assignment pack.

The initial velocity is assumed to be zero unless we are told otherwise. The displacement is 2100 meters. So listing the knowns and unknowns, we have

The obvious equation to use is .
Using Maple, we can define the equation and trivially solve for the answer.

Note the curve that Maple produces. We now see that the length of the runway has a large effect on the final velocity for small lengths. As the length gets larger, the effect on the velocity reduces. This behavior is the nature of the square root in our velocity expression. If we want, we can use our equation to figure out the minimum runway length for any aircraft. We just plug in the acceleration for that aircraft, make the plot, and noting what takeoff speed it needs, we look along the curve to read off what distance is necessary. One thing we notice right away is that aircraft which have high takeoff speeds need very long runways.

If we want to see the effect of the acceleration, we could try different values of the acceleration and compare plots. We can also just use our intuition.

The most common use of quadratic functions in physics are problems which involve the motion of a body which undergoes a constant acceleration. Any body which falls freely near the surface of the earth under the influence of gravity has constant acceleration as long as we neglect air resistance (we will see later how air resistance can be included). In this case, the magnitude of the downward acceleration is 32 feet/s² or 9.8 m/s². In physics or calculus class, we would use the following formula to show the vertical position of the freely-falling body:

This equation holds IF we assume that positive x is in the upward direction. Note the negative sign for the acceleration term. We have replaced the usual symbol for acceleration, a with -g to reflect the downward direction of the pull of gravity. If the initial velocity, , is in the downward direction, it should also have a minus sign in front of it. Remember that signs are determined by the direction they point in relative to the coordinate axes. If you decide that positive x is in the down direction, then the sign for the acceleration term would be positive instead of negative. You have complete freedom to decide what is positive direction at the start of the problem. Once chosen, you MUST be consistent with your definition throughout the problem. There is no big secret to getting signs correct, yet it is one of the most troubling errors for first-time students. You can practice up on associating signs and magnitudes of velocity, displacement and acceleration with this Java applet.

Do exercises 1-9 through 1-19 in the assignment pack.

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