Interesting Links (example problems):

An interesting problem from the University of Oregon

A falling elevator - can the rider save herself?

Does the dog bite the jogger?

The constant acceleration equations can be derived:

graphically
using calculus

The slope of the velocity vs time graph at any point is the instantaneous acceleration. If the graph is a straight line, the slope (and therefore the acceleration) is constant.

The area under the graph is the sum of the two areas (rectangle and triangle) shown:

Since the acceleration is the slope...

I substitute this into the first equation:

The first mini-lecture showed that the area is the displacement. I have used s-s_o for the displacement since s is the final position vector and s_o is the initial position vector. To put this in English (Example 1): Suppose a car is 1.5 miles north of a certain intersection which I will use as the origin of coordinates. It travels 27.3 miles north, ending up 28.8 miles north of the intersection. Then

s = 28.8 miles north
s_o = 1.5 miles north

B) Calculus

I first write the differential equation

Separating variables, recognizing that a is constant, and integrating, I get

As you can see, this is the same as the slope equation in the graphical section. Since v = ds/dt,

C) Summary

The equations are

The last equation was obtained by algebraically eliminating t between the first two equations. You should verify this equation for yourself by doing the algebra.

One-Body Applications:

Vector notation is not being used here because everything is one-dimensional and signs can completely include the vector nature. But you must not forget that s, a, and v are vectors and assign the proper signs!

Another step that must not be skipped is the explicit determination of an origin in the problem. Consider the car example above. The choice of an origin does not affect the displacement, but a choice of a different origin would mean that the two position vectors would be different. And you are often asked for a position, not the displacement! Remember, too, that distance traveled is not displacement!

Example 1 - A stone thrown upward from a bridge

Example 2 - A stone thrown upward from the ground

Two-Body Applications

Two-body (or multi-body) problems involve at least two bodies moving with their times and displacements related. For example, they may start at the same time or at different times, thus implying a relationship between the times. Or, they may begin at different points relative to some origin.

Example 3 - The Troll and the Billy Goat

Example 4 - Speedy Sue and Dopey Dan