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### Path Length or Displacement?

```
Date: 10/17/2001 at 21:52:41
From: C0rrie McDowell
Subject: Linear velocity

A body moves from A due east 5m to B, then from B due north 6m to C,
and from C due west 5m to D. Calculate total distance covered from A
to D.

I tried adding all the units together and got an answer of 16.
```

```
Date: 10/18/2001 at 10:33:19
From: Doctor Ian
Subject: Re: Linear velocity

Hi Corrie,

It's important to distinguish between path length and displacement.
Suppose I start somewhere, move 10 units north, then turn around and
move 10 units south. I end up back where I started, which means that
my displacement is zero. But I did move 20 units, so that's my path
length.

Or suppose I run around a 440 meter track 10 times. My path length is
4400 meters, but my displacement is zero, since I end up back where I
started.

Or suppose my friend and I decide to drive from Chicago to New York
along different routes. (I'm going to take the most direct route,
while he's going to drive through Tennessee so he can visit relatives
there.) If we start from the same place, and meet at some other place,
then our displacements will be the same. But our path lengths will be
very different.

The path in your problem looks like this:

5 m
D<------------ C
|
|
|  6 m
|
|
A------------> B
5 m

I honestly don't know whether 'distance covered' refers to
clarify that.

If you really want to find the path length, then you did the right
thing, adding the lengths of the individual paths.

On the other hand, if you want to find displacement (and it sounds
like that's what you're being asked for, even though 'distance
covered' isn't a very good way to refer to displacement), then you
need to figure out how far you would have to move from A to go
directly to D.

Does this help?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry

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