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

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

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

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

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.

(1) |

(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

(3) |

(4) |

For now, lets work in one dimension. Suppose that

(5) |

(6) |

Substituting (5) in for *v*, we get

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

(7) |

We can eliminate

Substituting this into (7), we get

or

(8) |

A car is traveling east at 45 m/s when the streetlight turns red. If the car decelerates at 5 m/s

We can find the time from (5)

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

From (8) we get

as before.

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?

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

Since *v _{0} = 0*. Using

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

or, since *x(t) = 0*,