SCM 541: Describing the Physical World
Lesson 6: Displacement and Average Velocity
Five of the quantities that are used to describe motion are position, distance, speed, displacement, and average velocity. Position, distance and speed are scalars, while displacement and velocity are vectors. These are the topics of Lesson 6.
Working with scalar quantities is quite intuitive:
Position is simply the point in space at which you are located (e.g., point A, x = 0...) Distance is how far you travel in getting from point A to point B (e.g., from x = 0 to x = x1). For example, if you walk in a straight line from your kitchen door (point A) to your pantry door (point B), which is 10 feet away, you walked a distance of 10 feet. If you return by the same path, you walked another 10 feet, so that the total distance you traveled is 20 feet. Yet you ended up where you started, at point A.
Speed is the distance you have traveled per unit time. So if you travel a distance x in a time t, you average speed during the trip is
average speed = x / t.
If, in the above example, it took you 5 seconds to walk from the kitchen door to the pantry, your average speed would be:
average speed from kitchen to pantry = 10 ft / 5 s = 2 ft/s.
If it takes you 10 seconds to return from the pantry to the kitchen, your average speed would be:
average speed from kitchen to pantry = 10 ft / 10 s = 1 ft/s.
Question:
What would be you average speed for the entire trip from kitchen to pantry to kitchen in the above example?
Vectors are a bit more complicated.
Displacement describes the net change in the position that you experience. Since displacement is a vector, we ascribe both magnitude and direction to it. So if we use a coordinate system in which + describes the direction from the kitchen door to the pantry door (e.g., the +x direction), then the displacement you experience in going from the kitchen to the pantry is d1 = +10i ft. When you return, the direction you travel is the opposite of your original walk (the -x direction), but the displacement is the same: d2 = -10i ft. (Note that the unit vector i points in the +x-direction, and that in both walks, the distance is simply 10 ft...a scalar.)
The net displacement is the sum of these two component displacement vectors:
d = d1 + d2 = (+10i ft) + (-10i ft) = 0i ft.
So displacement tells you that no matter how much you walked, there was no net change in your position: you started at point A and you ended up at point A. And it really doesn't matter if the path you took back to the kitchen door was the same as going, or included a walk out the front door, in the rear door and back to the kitchen. In this case, you may walk more returning from the pantry door than you did going to it, but the result was the same: there was no net change in your position, so that your displacement is still 0i ft.
Velocity is the measure of your net displacement in a unit of time. Velocity is also called the time rate of change of the displacement, and its mathematical definition is:
v = d / (t2-t1),
where t1 and t2 are the times at the start of your walk and at the end of your walk, respectively. The difference (final minus initial) is called the time interval.
If, in the example above, it took you 5 seconds to go from the kitchen door to the pantry door, your velocity would be v1 = +2i ft/s, where the + sign denotes the direction towards the pantry (i.e., the +x direction). This is an average velocity because it is computed only on the basis of the displacement between the beginning and the end points and the total time it took to get there. It does not consider any deviation you might have taken from the direct line path from point A to point B (e.g., stopping off at bathroom on the way) or and any change in speed during the trip (e.g., running for 3 seconds, crawling for 1 second and stopping for 1 second).
If it took you 10 seconds to return to the kitchen along the same path, your average velocity would be v2 = -1i ft/s, where - denotes the direction from the pantry to the kitchen (i.e., the -x direction).
The elements of this lesson are contained in the following links. Proceed to each element below in the order listed. You may want to bookmark this page so you can find these elements quickly in the future.
When you're finished the elements of this lesson, go the the "Assignments" section of the course for the assignments for Lesson 6.
© 2002 Barry L Lutz
