Section 1
Distance and Displacement
THINGS TO KNOW
When you take a walk to your friend’s house, you may start at your front door, walk
1. Distance is the amount of ground covered
when something moves from one place to
the next.
to the sidewalk, turn and walk down the street, and maybe even turn a corner and
2. Displacement is the distance between two
points “as the crow flies” from one point to
the next as well as the direction the crow
must fly.
walk a litle further until you get to your friend’s house. You probably don’t travel in
a straight line from your front door to your friend’s house.
So there’s a couple different ways to specify the distance between your house and
your friend’s. One is “how the crow flies,” the direct distance between your house
and your friend’s with no turns or cares for obstacles. The crow flies right over all
obstacles. The other way is to specify the actual distance that you have to walk as
you turn corners. If you need to know how fast you can get to your friends house,
you need to know the actual ground that you have to cover. If
you want to launch a
rocket to your friends
house, the distance to
the house “as the crow
flies” and the direction is
all you need.
Distance is the amount
of ground covered and
displacement is the
distance between two
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points and the direction from the first point to the second point.
For example, if you walk five meters to your sidewalk, then turn
right and walk twenty meters to the corner of your street, then
turn right again and walk sixty meters to your friend’s street, then
walk 30 meters to the front of your friend’s house, and then five
meters up to her front door. The total distance that you walked is
found by adding each of the parts of the trip, or 5m + 20m + 60m
Northwest. When you need to be more precise, you will need to
specify the actual angle on the compass.
Example 1
You walk 10 meters North and 10 meters East. What distance did
you walk and what is your displacement?
For distance, you add the length of each path,
+ 30m + 5m = 120m.
10m + 10m = 20m
For displacement, you have walked the two sides of a right
triangle with equal sides, or a 45° triangle, and you need to find
the hypotenuse.
The hypotenuse of an 45° right triangle is
Hypotenuse =
•Opposite
or hypotenuse ≃ 1.4•Opposite
Your displacement is then 14 meters 45° NorthEast.
Example 2
However, your displacement is only determined by the distance
between your door and your friend’s door, or 64 meters, and the
direction from your door to your friend’s. In this case we’ll call it
You drive your car 100 meters North and then turn around and
drive 100 meters South. What distance have you traveled and
what is your displacement?
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For distance, you add the length of each path,
100m + 100m = 200m
For a bike ride down the street, if you start at zero (Xi = 0m) and
end your bike ride at ten meters (Xf = 10m), then the distance you
rode is calculated with the equation for distance above,
The distance traveled is 200m.
Displacement only cares about where you start and where you
end and not how you got there. So if you end exactly where you
Example 3
start, your displacement is zero.
If you ride your bike in reverse from 10m to 5 m, what distance
Mathematically, to calculate distances, you add the absolute
have you traveled?
value of each segment of a trip. For one segment of a trip,
For this problem the initial position is 10m, or xi = 10m, and the
final position is 5m, or xf = 5m.
Using the distance formula gives
where the vertical line brackets on the right mean absolute value,
or the value inside made into a positive value.
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