Back to Contents!
Next: Derivatives Up: Describing How Things Change Previous: The Derivative

Velocity

The Thrust SSC: An artist's sketch of the first vehicle to break the sound barrier on land.

When an object moves, its position changes. If the object moves along a line (whether the line is horizontal, vertical or oblique), we can attempt to describe the motion of the object by giving a function x(t) which reports the displacement x of the object in convenient units from some origin at time t. The function might be described by a formula, by a table of values, or by a graph.
. For example, let's think about the vertical motion of a small object (a cantaloupe, say) that is thrown into the air at t = 0 seconds. The general form of the motion of the object is obvious: it goes up, slows down, reverses direction, falls down faster and faster and finally, Splat! This gives a general description of the velocity of the canteloupe, but can we be more precise? For instance, assume that we can measure the height of the canteloupe at any time t. Think of the height x above the ground as a function of time:
t(secs) 0.0 0.5 1.0 1.5 2.0 2.5
x(feet) 95.0 93.5 84.0 66.5 41.0 7.5

Splat! comes sometime between 2.5 and 3 seconds. The numbers don't quite show the behavior noted above because the canteloupe travels upward for less than half a second. But the numbers reveal the acceleration of the canteloupe toward the ground: During the first second, the net displacement of the canteloupe is 84 - 95 = -11 feet (note that we count upward displacement as positive and we place the origin at ground level), but in the second second its displacement is 41 - 84 = -43 feet. Since speed = distance/time, we can say that the average speed over the first second is 11 feet/second and the average speed over the time interval 1 < t < 2 is 43 feet/second. In general, the average velocity of a moving object over the time interval a < t < b is the net change in position x(b) - x(a) divided by the change in the time b - a. Therefore the average velocity of the canteloupe between time t = 0 sec and time t = 1 sec is (84 - 95)/(1 - 0) = -11/1 = -11 feet per second. Similarly, the average velocity of the canteloupe between time t = 1 sec and time t = 2 sec is -43 feet per second. The sign of the (average) velocity has significance! In this case it indicates that the canteloupe is moving downward.


Do exercise 2.1.

The average velocity gives a rough idea of the behavior of the canteloupe, but average velocity over an interval does not solve the problem of determining the velocity of the canteloupe exactly at t = 1, say. To get closer to an answer to that question, we have to look at what happens near t = 1 in more detail.

To do this, "suppose" that the canteloupe incident was videotaped, and we can run the film against a precise stopwatch. We could make some more measurements of the height of the canteloupe at times near t = 1 sec:

t(secs) 0.9 1.0 1.1
x(feet) 86.54 84.00 81.14

If we compute the average velocity of the canteloupe over the time interval [0.9, 1], we find it to be

Over the interval [1, 1.1] we get:

To be even more precise, we could make measurements over shorter time intervals around 1 sec:
t(secs) 0.99 1.00 1.01
x(feet) 84.2684 84.0000 83.7284


You should do exercise 2.2 before proceeding...

What we just did with velocities is the general idea of computing derivatives - we wanted the instantaneous velocity at t = 1, but we could only compute average velocities. To arrive at the instantaneous velocity at a given time, we took average velocities over shorter and shorter time intervals containing that time.

There is no reason to stick to t = 1 - we just happened to have lots of data about the flying canteloupe around that time. In mathematics, one usually begins with a function that is defined for a whole range of values, so it can be sampled as much as you want. For instance, all the data given above was in fact calculated from the formula: x = 95 + 5t - 16t2 (you had already guessed that, right?), so we could compute instantaneous velocities at other times in a similar way.


A summary of the difference between average and instantaneous quantities for velocity can be found here.

To test your general understanding of the relationship between velocity and acceleration, here is an excellent Java applet for you to work with.

Exercise 2.3 leads you through a more accurate representation of the instantaneous velocity using Maple. Also check out Exercise 2.4

Relativity

Perhaps the most famous theory of modern physics is the one put forward by Albert Einstein in 1905, the Theory of Special Relativity. Although the theory itself might seem arcane and counter-intuitive, the essence of it is simple in that it rests on just two postulates, both of which are verified by experiment. The fact that they are postulates means that they are general and, at some level, unprovable. They are:
  • The Relativity Postulate: The laws of physics are the same for all observers as long as all observers are in inertial reference frames (i.e. the observers exist in situations which are not accelerating with respect to one another). Although all frames of reference in the universe seem to have some acceleration with respect to each other, these can often be considered as small enough to constitute no acceleration.
  • The Speed of Light Postulate: The speed of light in a vacuum has the same value, c, in all inertial reference frames and in all directions!
    • The first postulate is easy to accept since it happens to coincide with our everyday life. We, in general, are not often accelerating with any appreciable speed with respect to anything else we are observing. The second, however, is startling and very counterintuitive! If you, as a stationary observer, see two cars moving past you, both with a speed of 30 miles/hour but opposite directions, you naturally think that the drivers of the cars see their relative velocity with respect to each other as 60 miles/hour. We term this type of relativity Galilean since it was Galileo Galilei who first pointed it out. The 60 miles/hour is, in fact, correct to a high degree of accuracy, but, it is not exactly correct. An experiment done in 1898 shows that, for light, the speed you measure is independent of the reference frame you are in. So, if one of the two drivers shines a laser beam at the other while you (the stationary observer) shine a laser beam at the same driver, that driver will measure the speed of the laser beams going past him as having exactly the same speed. Since the speed of light is very high, most of us do not, in everyday experience, come across this most counter-intuitive observation.

    For more on treating special relativity, see the online notes of Dr. Park.


    Next: Derivatives Up: Describing How Things Change Previous: The Derivative

    larryg@upenn5.hep.upenn.edu
    Fri Mar 4 09:58:36 EST 1994
    subsubsection2_1_1_1.html