Rate of Change of Distance

Date: 05/02/2002 at 21:01:32
From: sarah
Subject: Definition of speed

Hello Dr. Math,

Speed is defined as the rate of change of distance moved with time. 
What is meant by "rate of change of distance"?

Best regards.

Date: 05/03/2002 at 09:35:20
From: Doctor Ian
Subject: Re: Definition of speed

Hi Sarah,

Distance is always defined with respect to some location. For example, 
wherever you are now, you are some distance from your refrigerator, 
some distance from the Sears Tower in Chicago, some distance from 
telephone pole number 4328748-11 in Doodle, Kansas, and so on. 

Now suppose you're standing at some marked location, like a phone 
booth. You start walking away from the phone booth. At any given 
instant, you are some distance from the phone booth, right? 

But at different instants, your distance is different, which means 
that your distance from the phone booth is _changing_ over time. 

Suppose you and a friend started walking away from the booth at the 
same time, but after 10 seconds, you've walked 20 feet, while your 
friend has walked 30 feet. Both of your distances from the phone booth 
are changing with time, but the changes aren't the same, and we'd like 
to have some way to compare them.  

That's where _rates_ come in. We would look at your progress, and say 
that since you've moved 20 feet in 10 seconds, on average your 
distance from the phone booth is changing by 2 feet every second. That 
is, if we know your distance at time T, then if we check it again at 
time T plus 1 second, we should expect the distance to have increased 
by 2 feet.  

But this is a pretty wordy description, and so we abbreviate it by 
saying that the _rate_ of change of your distance from the phone booth 
is 2 feet per second. And the rate of change of your friend's distance 
is 3 feet per second. 

Where this gets tricky is that things don't always move in a straight 
line. For example, suppose you walk in a square path, 

             20 feet
    start ----------->
          ^          |
          |          | 20 feet
          |          |
          |          v
          <----------

and the trip takes you 40 seconds. At any given time, if we measure 
distance from where you were a moment ago, we would find that you were 
moving at 2 feet per second the whole time. That is, the rate of 
change of your distance FROM YOUR CURRENT POSITION was 2 feet per 
second. 

However, note that 40 seconds later, you're right back where you 
started. So the rate of change of your distance FROM YOUR STARTING 
POSITION was sometimes positive (when you were getting farther away 
from the starting position), and sometimes negative (when you were 
getting closer to the starting position); and on average, it was zero.  

In a situation like this, it becomes important to distinguish between 
velocity (which has a direction) and speed (which does not).  We would 
say that your average SPEED was 2 feet per second, while your average 
VELOCITY was zero feet per second.  

Both of these are measures of the rate of change of your distance from 
somewhere; but they differ in where distance is being measured from.  

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/