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One trip, two lengths: how much ground was covered, and how far the finish is from the start, with a sign for direction.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to find the distance and the signed displacement of a trip along a line, and read a negative answer as a direction.
You know how to measure lengths and times, and you have found a speed by dividing a distance by a time. You can read a number line with positive and negative numbers. This lesson uses that number line to describe where things go, not just how far.
| Term | What it means |
|---|---|
| Position | Where an object is on a line, measured from a chosen origin. |
| Origin | The zero point that positions are measured from. |
| Distance | The total length of ground covered, never negative. |
| Displacement | The change in position, finish minus start, with a sign for direction. |
| Average velocity | Displacement divided by time, with a direction. |
| Average speed | Distance divided by time, with no direction. |
Any trip along a straight line can be described with two different lengths:
$$\Delta x = x_{\text{end}} - x_{\text{start}}.$$
For the same walk, the displacement is $5$ m east.
To give displacement a direction, choose one way as positive. Movement the other way gets a minus sign. A negative displacement is not a mistake; it tells you which way the object ended up.
Another way: picture
Picture walking your dog around the block. You might cover half a mile, but you end up right back at your front door. Your distance is half a mile; your displacement is zero, because you finished exactly where you started.
Another way: steps
Before you can give a displacement a sign, you must decide which direction counts as positive. Any choice works: east, north, up, or to the right. What matters is that you stick with it for the whole problem.
Once the choice is made, every movement in that direction is positive and every movement the other way is negative. Two people who choose opposite directions will get answers with opposite signs, and both will be right.
A position is a place on a line, measured from a zero point called the origin. A mailbox might be at $+40$ m from your front door, and a stop sign at $-25$ m, on the other side.
The origin can be anywhere that is convenient: the starting line of a race, the ground floor of a building, or your front door. Moving the origin changes every position, but it does not change any displacement, because displacement is a difference.
Displacement is the finishing position minus the starting position. If a runner starts at $10$ m and finishes at $60$ m, the displacement is $60 - 10 = 50$ m. If she starts at $60$ m and finishes at $10$ m, it is $10 - 60 = -50$ m.
The order matters: always finish minus start. Getting it backward flips the sign and points the answer the wrong way.
Distance is the total length of every leg of the trip, added up with no signs. A trip of $40$ m forward and $15$ m back has a distance of $55$ m. Distance never goes down and is never negative.
If the trip never turns around, the distance equals the size of the displacement. As soon as it doubles back, the distance becomes larger, because the backward leg adds to the distance but subtracts from the displacement.
A trip that ends where it started has zero displacement, however long it was. A runner who finishes four laps of a school track has run $1600$ m but has a displacement of zero.
This surprises many people at first, but it is exactly what displacement measures: not how hard you worked, but how far your finishing point is from your starting point.
Average speed is distance divided by time; it has no direction. Average velocity is displacement divided by time; it has the same sign as the displacement.
For a runner who does two laps of a $400$ m track in $200$ s, the average speed is $4$ m/s, but the average velocity is zero, since she ends where she began. Velocity tells you about where you got to; speed tells you about how fast you moved along the way.
Checking an answer. The distance must be at least as large as the size of the displacement. A trip ending at its start has zero displacement. The sign of the displacement must match the side of the start where the trip ended.
Displacement is defined as the change in position, so subtracting the start from the finish is simply what the word means. Signed numbers add like steps on a number line: a step of $-3$ undoes a step of $+3$.
Distance, by contrast, adds up the length of the path. Because every piece of path counts, whatever its direction, the signs are dropped before adding.
An elevator gives a clean example of up-and-down displacement. Riding from the lobby up to the tenth floor and back down to the fifth, the elevator travels up ten floors and down five, fifteen floors of distance.
Its displacement, however, is only five floors up, since it ended five floors above where it started. Choosing up as positive, the ride up is $+10$ floors and the ride down is $-5$ floors.
Hiking trails zigzag up mountainsides in switchbacks, so the path is much longer than the straight line from trailhead to summit. The trail's length is the distance; the change in elevation is the vertical displacement.
Hiking guides in national parks list both. The trail up Pikes Peak in Colorado is about twenty-one kilometers long, but the vertical displacement is only about two kilometers, the gain in elevation from the start to the summit.
On a map, displacement is the straight arrow from the starting point to the finishing point. A road trip from Chicago to Milwaukee along a winding route may cover more than a hundred and forty kilometers, but the displacement is about one hundred and thirty kilometers, roughly north.
In this course, trips stay on one straight line, so directions are just plus and minus. Later physics courses give displacement a direction on a map, using arrows called vectors.
The most common error is to add distances when the question asks for displacement. Another is to subtract in the wrong order, giving start minus finish, which flips the sign.
A third is to treat a negative answer as wrong and drop the minus sign. The sign is part of the answer. A displacement of $-20$ m means twenty meters in the negative direction.
Distance and displacement are both lengths, measured in meters, centimeters or kilometers. A classroom might be ten meters long; a school track, four hundred meters around; a highway trip, hundreds of kilometers.
Always write the unit, and when a problem mixes units, such as kilometers and meters, convert them to the same unit before subtracting or adding.
A quick sketch makes most displacement problems easy. Draw a straight line, mark the origin with a zero, and put an arrow on the positive end. Then draw each leg of the trip as an arrow along the line: arrows pointing the positive way are positive, arrows pointing back are negative.
The displacement is a single arrow from the tail of the first leg to the head of the last. The distance is the total length of all the leg arrows laid end to end. Seeing both on one sketch shows at a glance why they differ whenever the trip turns around.
Up and down work the same way as east and west. Choosing up as positive, climbing stairs gives a positive displacement and going down to a basement gives a negative one. Elevators, hikes and thrown balls all use this vertical number line.
A ball thrown straight up and caught at the same height has a vertical displacement of zero, even though it traveled up and back down. That simple fact becomes important later, when you study how gravity changes a ball's motion.
Some questions need distance: how much gas a car used, or how tired a runner is. Others need displacement: how far a package moved from the warehouse, or where a lost hiker ended up. Physics keeps both words so each question gets the right answer.
The Barr Trail climbs Pikes Peak, the Colorado mountain that inspired the song America the Beautiful. The trail winds for about twenty-one kilometers through forest and switchbacks from the town of Manitou Springs to the summit.
The distance hikers cover is that whole winding path. Their vertical displacement is much smaller: the trailhead sits at about two thousand meters and the summit at about four thousand three hundred, so they rise about two thousand three hundred meters. Rangers use the vertical displacement to warn hikers about thin air and tired legs, and the trail distance to estimate how many hours the climb will take. Coming back down, the vertical displacement is the same size but negative.
School districts across the country track their buses with GPS, which records the bus's position every few seconds. From those positions, software computes both the distance the bus drives, used to plan fuel and maintenance, and its displacement from the depot at any moment.
At the end of the day the bus is back at the depot, so its displacement is zero, even though it may have driven over a hundred kilometers. Parents watching an app see the bus's position, which is its displacement from the school, and can tell which way it is heading from whether that displacement is growing or shrinking.
When a calculation gives a negative displacement, it is tempting to think something went wrong and drop the minus sign. But the sign is part of the answer. It tells you the object ended up on the negative side of where it started.
A related error is to treat distance and displacement as the same thing. They are equal only when the trip never turns back. The moment it does, the distance grows while the displacement shrinks.
A student walks $60$ m east to a corner store, then $25$ m back west to a friend's house. Choose east as positive.
$+60\ \text{m}, \ -25\ \text{m}$
Signs for direction.
Find the displacement.
$60 + (-25) = 35\ \text{m}$
Add with signs.
Find the distance.
$60 + 25 = 85\ \text{m}$
Add without signs.
State the displacement with a direction.
$35\ \text{m east}$
Positive means east.
Check the sizes.
$85 \ge 35$
Distance is never smaller.
A car starts at $x = 200$ m and ends at $x = -50$ m. Write the rule.
$\Delta x = x_{\text{end}} - x_{\text{start}}$
Finish minus start.
Substitute the positions.
$\Delta x = -50 - 200$
Keep the signs.
Evaluate the displacement.
$\Delta x = -250\ \text{m}$
Negative direction.
The car never turned around. Find the distance.
$250\ \text{m}$
Same size as the displacement.
Move the origin $100$ m and recompute.
$-150 - 100 = -250\ \text{m}$
Displacement does not change.
Explain why it stays the same.
$\text{both positions shift equally}$
The difference stays the same.
A runner does three laps of a $400$ m track in $300$ s, finishing at the start. Find the distance.
$3 \times 400 = 1200\ \text{m}$
Every lap counts.
Find the displacement.
$0\ \text{m}$
Back where she began.
Find the average speed.
$\dfrac{1200}{300} = 4\ \text{m/s}$
Distance over time.
Find the average velocity.
$\dfrac{0}{300} = 0\ \text{m/s}$
Displacement over time.
She stops halfway around the next lap, $100$ m straight across the field from the start. Find the displacement.
$100\ \text{m}$
Straight line from start.
Find the distance now.
$1200 + 200 = 1400\ \text{m}$
Half a lap more.
Compare the two numbers.
$1400 \gg 100$
Very different questions.
Write the rule.
$\Delta x = x_{\text{end}} - x_{\text{start}}$
Finish minus start.
Substitute the positions.
$\Delta x = 17 - 5$
In meters.
Evaluate the displacement.
A student walks $20$ m east along a hallway, then $9$ m back west. With east as positive, what is the student's displacement, in m?
Complete the worked solution: an elevator starts $12$ m above the lobby, rises to $39$ m, then comes down to $3$ m. With up as positive, find the displacement going up in m, the displacement coming down in m, and the total distance traveled in m.
Find the displacement going up.
$\Delta y_1 = y_{\text{top}} - y_{\text{start}} =$ a
Positive: upward.
Find the displacement coming down.
$\Delta y_2 = y_{\text{end}} - y_{\text{top}} =$ b
Negative: downward.
Find the total distance.
$|\Delta y_1| + |\Delta y_2| =$ d
Sizes added.
Note the overall displacement.
$\Delta y_1 + \Delta y_2$
End minus start.
Match each word to its meaning.
| the total length of ground covered | how far the finish is from the start, with a direction | where something is, measured from an origin | which way the object ended up | |
|---|---|---|---|---|
| distance | ||||
| displacement | ||||
| position | ||||
| the sign of a displacement |
A bus moves along a straight street. It starts at position $0$ m, stops at $250$ m, then ends at $-100$ m. Fill in the first leg's displacement in m, the second leg's displacement in m, and the total distance in m.
| value | |
|---|---|
| first leg's displacement (m) | |
| second leg's displacement (m) | |
| total distance (m) |
A robot on a straight track starts at position $3$ m and moves in the positive direction at a steady $4$ m/s. Write its position, in m, as a function of the time $t$ in seconds.
Answer:
A jogger runs $500$ m east along a straight path, then turns and runs $200$ m west. The whole run takes $100$ s. With east as positive, what is the jogger's average velocity, in m/s?
Answer: m/s
Hikers on the South Kaibab Trail into the Grand Canyon start at an elevation of $2200$ m and finish at $740$ m. With up as positive, what is their vertical displacement, in m?
Answer: m
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A robot on a straight track starts at position $5$ m and moves in the positive direction at a steady $2$ m/s. Write its position, in m, as a function of the time $t$ in seconds.
Answer:
You can give a trip both a distance and a displacement. Explain to someone why a runner who finishes four laps where she started has run a long way but has no displacement.
27. Your turn: a toy car rolls from $x = 5$ m to $x = 17$ m. What is its displacement?, step 3
$\Delta x = 12\ \text{m}$
Positive direction.