Linear Kinematics

    Let us consider the motion of an object. For simplicity, let us take an ideal object, which does not take up any real space, or interact with the rest of the world. To begin with, let us describe its position. In order to say where an object is, we must first define a coordinate system. This is known as a frame of reference. Frames of reference are wholly man made. They are artifacts that allow for easier description of the problem. Since we live in three dimensions, we describe the position of an object by stating how far away the object is from three separate coordinate axes. Again, for simplicity, let us consider one dimension only. Then in order to describe the position, we need only know the point on a number line where the object is at. The easiest way of seeing this is to look at a graph of the position as a function of time. If the object is not moving, then the graph would look like
graph

We see that the slope of an object at rest is zero. What if the object were moving at a constant speed? Then a graph of the speed as a function of time would be

graph

Similarly, intuition tells us that if something moves with constant speed, then it is changing position at a constant rate, and a graph of the position as a function of time looks like

graph

Here the slope of the line is a constant, and in fact is equal to the speed. Lastly, we can also ask what would happen if we let the speed change at a constant rate? A change in the speed of the object is called its acceleration. From before, we see that graphs of the speed and acceleration verses time would look like

graph
graph

Notice that again the acceleration is the slope of the graph of the speed verses time. What would a graph of the position verses time look like? We have seen that the speed is the slope of the position, but now that slope is increasing at a constant rate. Thus, we see that the graph of the position must look like

graph

    We see that for many cases, the easiest way to analyze one dimensional motion is to look at it graphically. However, if the object has a complicated motion, or it moves in more than one dimension, it is not always easy to graph it.

Algebraic Equations for Motion

    Now let us denote the object's position by the vector x. We associate the unit meters with x. Now look at the graph of distance verses time. Since the y coordinate is the distance, and the x coordinate is the time, the slope can be written as
 
formula (1)
We call vave the average velocity of the object. If we let the time difference t2 ­ t1 go to zero, then we get the definition of the instantaneous velocity
 
formula (2)
 
This last term may be unfamiliar to many of you. This is the definition of a derivative. In this course we will only use the average velocity.
    Can we come up with a similar definition for acceleration? Looking at the graph of the velocity verses time, we recall that the slope of that line was the acceleration, so we can follow the same approach as before to define the average acceleration
 
formula (3)
Similarly, we can define the instantaneous acceleration as
 
formula (4)
Notice from (2) and (4) that if position is in meters, then velocity is in meters per second and acceleration is in meters per second per second, or meters per second squared.
    For now, lets work in one dimension. Suppose that t = t, v = v - v0 and x = x - x0. Then we can rewrite (3) as
 
formula (5)
and (1) as
 
formula (6)
Now recall that v in (6) is the average velocity. In order to determine the average, remember that the average is defined as
formula

Substituting (5) in for v, we get

formula

If we substitute this into (6), we get the desired result
 

formula (7)
where I have dropped the average on a since we will usually take it to be a constant.
    We can eliminate t by using (7) and (5). Solving (5) for t we get
formula

Substituting this into (7), we get

formula

or
 

formula (8)
Example:
    A car is traveling east at 45 m/s when the streetlight turns red. If the car decelerates at 5 m/s2, how long does it take to stop? How far away from the streetlight must the car start breaking in order to stop in time?
drawing

We can find the time from (5)

formula

We can get the distance from either (7) or (8). From (7) we get

formula

From (8) we get

formula

as before.

Ballistic Motion

    An important example of this is when the acceleration is due to gravity. Galileo was the first one to realize that two objects will accelerate as they fall, and that the acceleration was independent of the composition of the object. By performing experiments, we can determine the amount that gravity accelerates an object. This is
g = 9.8 m/s2

in the downward direction. Also, notice that this result is an idealization. If we were to drop a baseball and a feather, we would see that the baseball reached the ground first. Was Galileo wrong? No. In this case we also have the effect of the air on the object, and it does depend on the composition of the object. If we were to repeat the experiment in a vacuum we would find that they fall at the same rate. The motion of an object due to gravity is known as ballistic motion. This is because it also describes the motion of a cannonball that is fired.

Example:
    A student wants to play a practical joke on his poor physics professor. He climbs to the top of a 5 story (15.25 m) tall building and prepares to drop a water balloon on the professor as he walks underneath. If the professor is walking by at a speed of 2.0 m/s, how far from "ground zero" should the professor be when the student drops the balloon?

drawing

    In order to determine where the professor should be, we need to know how long it takes for the water balloon to fall. We get this from the position equation

formula

Since v0 = 0. Using y(t) = 0 and solving for t, get

formula

To find the professor's position, we use the fact that his motion is uniform. Thus

formula

or, since x(t) = 0,

formula