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Apologia Physics Sample

Module 2: Motion In One Dimension

Introduction

As I said in the introduction to Module #1, the science of physics attempts to explain everything that is observed in nature. Now, of course, this is a monumentally impossible task, but physicists nevertheless try to do the best job they possibly can. Over the last 3,000 years, remarkable advances have been made in explaining the nature of the world around us, and in this physics course, we will learn about many of those advances. This module will concentrate on describing motion.

If you look around, you will see many things in motion. Trees, plants, and sometimes bits of garbage blow around in the wind. Cars, planes, animals, insects, and people move about from place to place. You should have learned in chemistry that even objects which appear stationary are, in fact, filled with motion because their component molecules or atoms are moving. In short, the world around us is alive with motion.

In fact, St. Thomas Aquinas listed the presence of motion as one of his five arguments for the existence of God. He said that in all of our experience, humans have found that motion cannot occur without a mover. In other words, in order for something to move, there must be something else that moves it. When a rolling ball collides with a toy car, the car will move because the ball gave it motion. But, of course, the ball would not have been rolling to begin with if it had not been pushed or thrown. Thus, Aquinas says that our practical experience says that any observable motion should be traceable back to the original mover. When the universe began, then, something had to be there to start all of the motion that we see today. Aquinas says that God is this "original mover."

While philosophers can mount several objections to St. Thomas Aquinas' argument, it nevertheless shows how important motion is in the universe. Thus, it is important for us to be able to study and understand motion. In this module, we will attempt to understand the most basic type of motion: motion in one dimension. Remember from geometry what "one dimension" means. If an object moves in one dimension, it moves from one point to another in a straight line. In this module, therefore, we will attempt to understand the motion of objects when they are constrained to travel straight from one point to another.

Distance and Displacement

When studying the motion of an object, there are a few very fundamental questions you can ask yourself. You can ask "where is the object," "how fast is it moving," and "how is the object's motion changing?" In physics terms, we say that the answers to these questions are the object's displacement, velocity, and acceleration. We will examine each of these concepts individually, so for right now, let's concentrate on the concept of displacement.

Displacement - The position of an object relative to a fixed point

Before we study this concept in detail, you need to be aware of a very important distinction that physicists make. Notice the definition of displacement. It says that in order to determine displacement, you must determine the position of an object compared to some fixed point in space. Although this might sound pretty basic to you, it is actually one of the fundamental concepts in this module. The best way to show you the importance of the concept is by example.

Suppose you are sitting on the sofa reading a book (maybe even this one), and you suddenly decide that you want to go to the refrigerator for a drink. You get up, and you move to the refrigerator which is 10 meters away from the sofa. You get your drink and then walk 10 meters back to the sofa. How much distance did you travel in your quest for liquid refreshment? Well, you walked 10 meters there and 10 meters back, so you walked a total of 20 meters. After everything was finished, what was your total displacement? It was zero meters!! You see, before everything began, you were at the sofa. Since you started there, we can define it as the fixed point. You moved to the refrigerator, at which point you were 10 meters displaced from the sofa. However, when you turned around and came back, you ended up at exactly the same point from which you started. In the end, then, you were 0 meters from the fixed point, thus your displacement was 0.

You see, then, that the concept of displacement carries with it some information about direction, whereas the concept of distance does not. In the situation we just imagined, you walked a distance of 20 meters, but your displacement was 0 because you walked 10 meters in one direction and then another 10 meters in precisely the opposite direction. Since the displacement in one direction canceled the displacement in the opposite direction, your total displacement was zero. When a physical quantity carries information concerning direction we call it a vector quantity. When the physical quantity does not carry information concerning direction, we call it a scalar quantity.

Vector Quantity - A physical measurement that contains directional information

Scalar Quantity - A physical measurement that does not contain directional information

Thus, distance is a scalar quantity and displacement is a vector quantity.

When dealing with displacement, we must find some mathematical way to denote the direction that is inherent in the measurement. The way we will do this is to label displacement in one direction positive and displacement in the opposite direction negative. That way, when you add displacements together, motion in one direction will cancel motion in the opposite direction. Thus, we could say that in the situation above, your displacement was +10 meters when you moved from the sofa to the refrigerator and -10 meters when you moved the opposite direction from the refrigerator to the sofa. Your total displacement, then, was +10 meters plus -10 meters, which is zero.

What's really nice about this mathematical way of noting direction is that it doesn't really matter which direction you label as positive or which you label as negative. We could just have easily said that your displacement when you arrived at the refrigerator was -10 meters. That would mean that your displacement when you moved from the refrigerator to the couch was +10 meters. The total displacement would still be zero. Thus, it doesn't matter which direction you label as positive, as long as you keep it consistent. Study the following example and solve the "on your own" problem after it to make sure you understand this important principle.

EXAMPLE 2.1

A child is standing 5.0 meters away from a wall and rolls a balls towards it. The ball hits the wall and bounces back, rolling 3.3 meters before coming to a halt. What is the total distance covered by the ball? What is the ball's displacement?

The total distance is easy to calculate. The ball rolled 5.0 meters to reach the wall and 3.3 meters in the other direction after bouncing back. The total distance then, is simply:

Total Distance = 5.0 meters + 3.3 meters = 8.3 meters

Calculating the displacement is a bit more difficult, however. To do this, we must first define directions. I will say that motion from the child to the wall represents positive displacement while motion from the wall to the child is negative displacement. Thus, the ball first had a displacement of +5.0 meters and then a displacement of -3.3 meters. The total displacement, then, is:

Total Displacement = 5.0 meters + -3.3 meters = 1.7 meters

This is a positive displacement, which means that the ball is 1.7 meters away from the child, in the direction of the wall.

Alternatively, I could have said that motion from the child towards the wall represented negative displacement. In that case, the ball would have had a -5.0 meters displacement followed by a +3.3 meters displacement. This would indicate a total displacement of -1.7 meters. You might think that this is a different answer than the one I got previously, because this one is negative. Remember, however, what negative displacement means in this case. It means displacement from the child towards the wall. Thus, my answer is still 1.7 meters away from the child, in the direction of the wall. As long as you stay consistent, your answer will be the same regardless of which direction you say is positive and which is negative. The trick is to give your answer in relation to a fixed point, not with just a positive or negative sign.