Position
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Kinematics is the study of motion and comes from the Greek word κι′νηση (ki'ne-se), which simply means motion. As you do more physics and we use the Greek alphabet, you will likely learn to read Greek as a side effect. You won't know what you're saying, but you will learn the alphabet and basic pronunciation with just a little effort. I personally found out by accident that I could read Greek some time during graduate school in physics when I met a Greek guy at school and read his T-shirt.
To study motion we need to first establish some definitions. We need to be able to describe where things are, how they move, etc. The position of an object is a vector that describes the object's location with respect to some agreed-upon coordinate system. Both the location of the origin and the orientation of the axes must be known to make any sense of a position vector.
A position vector in 2 or 3 dimensions is denoted . An example of a position vector is: . Since this vector only has an x and y component, it means that the object is in the xy plane (z=0). If a component isn't present, as in this case, it means that you can assume it's zero. To leave the xy plane, we'd have to have a non-zero component.
In this particular case, an object is located 3 meters along the positive x-axis from the origin and 4 meters along the y-axis from it. As mentioned above, in order for this to make any sense at all, the location of the origin and the orientation of the axes must be known.
The total distance from the origin is the magnitude of the position vector, which is obtained just like we find the hypotenuse of a triangle - by taking the root of the sum of the squares of the side lengths:
Realize that in a two-dimensional case, this is the familiar Pythagorean theorem, or
Here the x and y components can be seen as the lengths of the sides of the right triangle and the vector's magnitude as the hypotenuse.
When an object moves, it is said to undergo displacement. In other words, the displacement is just the change in position. It is denoted . The upper case Greek delta (pronounced like our 'd') is always understood to imply the later (or final) condition minus the earlier (or initial) condition.
Create Vectors
This textbook is not only hosted by GeoGebra, but we will also be using the software to do calculations and to visualize things like vectors this semester. I will lead you through the process of creating vectors in GeoGebra below.
The Input bar is on the top left just under the 
icon. In the input bar, to create a vector named "a", all you have to type is
icon. In the input bar, to create a vector named "a", all you have to type is a=(3,4) and press enter. Try this below. It is a GeoGebra convention to name vectors with lower case letters and to name points with upper case letters. For instance, try typing A=(3,4) instead and see what happens. If the on-screen keyboard shows up it may obscure your view of the new vector or point. Close the keyboard and you should see them. GeoGebra will calculate the magnitude of a vector. The function is called "length". After defining a vector named "a", on the next line type length[a]. Time-Dependent Position
If something is in motion with respect to the origin of the coordinate system, its position will naturally be a time-dependent vector. A few examples are , or . Since computer languages generally don't like units, we are always assuming that the units are SI if they are not specified. Thus, we see that if the position is in meters and time is in seconds, that the first vector should rightfully be written such that each component will have units of meters, but I will not always be so explicit. Take note that GeoGebra, and every other computer language that I can think of, expects radians as the input arguments to trigonometric functions. That means the trig functions above should be seen, for instance, as , and not just . With this in mind, can you see in your mind's eye what these time-dependent vectors would look like as time (represented by the t) ticks on? Let's try to make these time-dependent vectors in GeoGebra.
Making Time-Dependent Position Vectors
- In order to make anything depend on time, we need a variable to represent time. This always tends to be t in physics. In the input area, type
t=0. You'll notice a slider is automatically created. We can change the range of values by clicking on it.
- Once t is defined, we can define the vectors above just as before. Let's call the first one 'a', and the second one 'b'. Type
a=(2*t,1)and press enter. The * stands for multiplication. To create the second vector, typeb=(3+cos(t),2+sin(t))and press enter.
- Once the vectors are created, there are two things you may be interested in doing: Animating the vectors and showing what's called a trace of them. First, right-click on t, and you'll see the option to turn the animation on. See what it does. Then right click on one of the vectors and see what happens if you turn the trace on. Using trace will allow you to see the shape of the path that the second vector follows if you didn't recognize it when you saw the function.