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Calculus Primer 2: A Moving Experience
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Copyright © 2004, P. Lutus

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Imagine you are riding in a car, moving North, and you look at the car's speedometer. What is the speedometer telling us? To be very precise, the speedometer shows a rate of change in the car's position with respect to time, or the car's "velocity" (read this if you prefer "speed").

As we travel along the road, we can determine two properties of the car's motion, both with respect to time:
  • The car's position along the road at different times.
  • The car's velocity, how the car's position changes with respect to time.
We can determine the car's position by noting where the car is at different times, and we can determine the car's velocity by looking at the speedometer. The car's velocity is the rate of change in the car's position with respect to time.

Position and velocity are so closely connected that we only need one of them to get the other. This is because position and velocity are two ways of describing the same event. If we have a record of the car's velocity at different times and a starting position, we can use this record to get the car's position for any time. Or, if we have a record of the car's position at different times, we can establish the car's velocity at any time.

Here is a formal statement of the relationship:

  • Velocity is the instantaneous rate of change in position with respect to time.
  • Position is the summation of all the velocities our car has experienced over time.
Here is an example using a table of times, positions and velocities. We start at position zero, in front of our favorite drugstore, and we press the car's accelerator pedal. The car begins to move forward in an improbably precise way, and we (being uncharacteristically observant) take down all the position and velocity information. The table on this page shows our progress.

Getting velocity from position
0 0 0
1 2 1
2 4 4
3 6 9
4 8 16
5 10 25
6 12 36
7 14 49
8 16 64
9 18 81
10 20 100
11 22 121
12 24 144
13 26 169
14 28 196
15 30 225
16 32 256
17 34 289
18 36 324

Table 1
Now look carefully at the table. Let's try to derive the velocity column using only the position and time columns of our table. How shall we do that?

On reflection we realize the difference between two adjacent positions is equal to the average velocity for that time interval. We see that, between 10 and 11 seconds, the position changed by 21 meters, then we notice the velocity value for 11 seconds is 22 meters per second. Is this an error? No, not really — by subtracting the positions for 10 and 11 seconds, we acquired the average velocity for that time interval. We therefore should take the average of the listed velocities for 10 and 11 seconds. In so doing, we will obtain the same velocity value acquired by subtracting positions:

(1)    20 + 22 = 21

A quick review: at 10 seconds, the car was moving at 20 meters per second, and at 11 seconds, it was moving at 22 meters per second. The distance covered will normally be equal to the average of these two velocities, or 21 meters per second.

Let's try to write a general equation to describe this relationship between position and velocity:

(2)    p2 - p1 = v
t2 - t1

Where p1 and p2 are two positions and t1 and t2 are times corresponding to the two positions. This equation says that velocity is equal to a difference in position divided by a difference in time. Let's try an example that covers a larger span of time, using the positions for 5 seconds and 15 seconds:

(3)    225 - 25 = 20
15 - 5

Now let's verify this result by taking the average of the velocities for 5 and 15 seconds:

(4)    30 + 10 = 20

It is important to point out that this rather coarse way of determining a velocity isn't particularly useful in real-world calculations — its primary value is in clearly showing the relationship between position and velocity.

Getting position from velocity
Now let's do the reverse of the above example — let's try to obtain positions by using the numbers in the velocity column. Let's think about this.

  • The car's position is increased by a velocity value each second.
  • If we add a velocity to a position for a particular time, we will get the position for the next time.
  • We can repeat this procedure for each second of the desired time interval.
  • Therefore, to get the position for any time, we need only take a starting position and time, then add all the velocity numbers from that time to the ending time.
Let's perform an example using our table. Remember, we want to acquire a position by using only the numbers in the velocity column plus a starting position. Let's say we want to compute our position at a time of ten seconds, using as a starting point the position at three seconds. We learned above that, since positions arise from average velocities on a time interval, we should use averaged velocities in computing a position. Here are the numbers:

Starting position at 3 seconds: 9 meters.

Sum of average velocities for 3 through 10 seconds: 7 + 9 + 11 + 13 + 15 + 17 + 19 = 91 meters

Position at 10 seconds: 9 + 91 = 100 meters.

What have we learned?
  • About velocity:
    • We can find a velocity knowing only positions and times.
    • A velocity is computed by taking the difference between two positions and dividing by the difference in time.
  • About position:
    • We can find a position knowing only velocities and a starting position.
    • A position for a specific time is computed by summing all intermediate velocities to a starting position.
  • About both velocity and position:
    • You can acquire either one from the other.
    • If you get a result by applying the velocity -> position method, then apply the position -> velocity method to that result, you will get back the original value.
    • This means the two operations (v -> p and p -> v) are reciprocal — if you apply both in sequence, you return to where you started.

Some new terminology
  • The process of acquiring a velocity from positions, with some important refinements to be presented later, is called "derivation", or "taking a derivative".
  • The process of acquiring a position from velocities, again with some refinements to be presented later, is called "integration".
  • Integration and derivation are reciprocal operations, and (very important) if you apply both operations in sequence, you get back what you started with. This is called the Fundamental Theorem of Calculus.
  • You can apply these operations repeatedly. For example, if you take the derivative of position, you get velocity. If you take the derivative of velocity, you get acceleration, which is the rate of change in velocity with respect to time.
  • To sort out a sequence of related derivatives, we use ordinal numbers (first, second, third ...). Velocity may be described as the first derivative of position, acceleration the second.
  • The repetition property applies in both directions — one can also repeatedly integrate. If one integrates acceleration, one gets velocity, and if one integrates velocity, one gets position. The same ordinal numbering scheme is used — velocity is the first integral of acceleration and position is the second.

This example used a table of numbers to clarify the computation methods, but in typical calculations, one is much more likely to use a mathematical function as a source of data. The following pages will show how to obtain derivatives and integrals using functions rather than tables of numbers.

"Velocity" footnote:  Now that I've received over 100 e-mails complaining about the distinction between velocity and speed, I want to say that this is a beginner's Calculus tutorial and I selected the most commonly used terms for position, velocity and acceleration, all of which are more likely to be vectors than scalars in practice. I did say the car was moving North, but this seems not to have made any difference to the volume of mail. Thanks for not contributing to this onanistic debate.


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