Distance vs Displacement

One lap on a track: displacement? (preview)

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Distance is the length of the actual path. A scalar: magnitude = length of the path. Let me give an example. Say I'm on a 400 m track. I run one lap and come back. Now, my displacement. Usually written x. What is it? It's 0, right? Let's continue our Basics lessons. Today: displacement and distance. Both are core to straight-line motion. Displacement is especially important.

In this lesson: four key points

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After this lesson, if a problem gives you a motion and asks for the displacement and the distance, or asks you to compare distance and displacement in size, can they be equal? you'll have no trouble with these. Your quizzes, midterms and finals test this a lot. Students new to this often miss these. You should fully master four points: First, the definitions: what are displacement and distance, and how do they differ? Second, how to write 1D displacement. Third, calculating displacement from coordinates, quickly. Finally, crucial pitfalls in comparing distance and displacement.

Why we need displacement

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Let's start with the concepts of displacement and distance. Build a habit in physics: with every new quantity, ask what it is used for and why learn it. We're studying straight-line motion. Unless it's purely spinning in place, its motion must change its position. Living room to bathroom, bedroom to kitchen: position changes. So we need displacement to describe it. It describes the change in position. For example, Qi Xiaoyu and I once walked in a small neighborhood park. Beijing has many such parks. Suddenly she got a phone call. She left to help a friend. Alone, I wandered aimlessly to the right. I walked 2 km and found a new milk tea shop by the road. I don't like milk tea. It's unhealthy. Drink less, especially high schoolers: Sugar makes you sleepy at night, bad for your brain.

2:23
Remember: go easy on sugar. Sugar harms the brain, so eat less. But Qi Xiaoyu loves milk tea, so I got dragged into it. But after a while: hey, it's good! I saw the shop, went in, bought one. Great taste. Excited, I told Qi Xiaoyu about the new shop. She asked how to get there. What do I say? How to describe the shop? Where is it, relative to the park? If I just say '2 km from the park', is that enough? Of course not. Why? 2 km around the park is a whole circle. 2 km north? 2 km west? Which way? You can't tell. So? Just add a direction, and it's exact, right? I tell her: the shop is 2 km to the right of the park. Now she can pinpoint the milk tea shop at once. Think: I said '2 km right of the park.' Besides the shop's location, I also told her my position change, right?

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I started in the park: initial position. I ended at the shop: final position. From initial to final position, I moved 2 km to the right. That 2 km to the right is the change in position, right? This is displacement. You can see: A position change needs two factors. One is how far apart they are, 2 km here; the other is the direction. So, for displacement: magnitude and direction, both essential.

Definitions: displacement and distance

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Now let's look at displacement properly. It's a vector for changes in position. Note: it's a vector. One strong recommendation: if you haven't seen Lesson 4, on vectors and scalars, the one on signs. Go watch! Whatever your physics level, you don't understand this deeply, and that may affect later study. Fix problems early, enjoy later. Displacement is a vector, so it has magnitude and direction. Its magnitude is the length of the straight line from initial to final position. Displacement relates two positions, not how the person moved. Whether you came straight or curved, we take the straight line between them. Direction: from initial to final position. As a vector, displacement is an arrow from initial position to final position. The arrow's length is the straight line between the positions. That's the definition.

5:03
It's called displacement. 'Displace' means to shift: a verb. Add -ment: a noun for shifting. So displacement is a shift of position: the quantity for position change. That covers displacement. Now distance. Distance is a quantity you know very well from middle school. Distance is the length of the actual path. A scalar: magnitude = length of the path. Let me give an example. Say I'm on a 400 m track. I run one lap and come back. Now, my displacement. Usually written x. What is it? It should be 0, right? Because it describes the position change. I started here, and I ended right here: no position change, so displacement is 0. And distance? Distance, written s, is the actual path. I ran 400 m around, so distance is 400 m. From this example, know how they differ: displacement is path-independent, only describing the change in position.

6:12
Distance depends on the path, it's the length of the actual path.

1D displacement: a sign plus a number

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Next: writing 1D displacement simply. In this series, displacements are all 1D. What is 1D? Straight-line displacement. On a line there are two directions: the two ends. That's 1D. Lesson 4 covered this in detail, on vectors and scalars. That's the sign convention. Quick review. Use a sign plus a number: sign: direction; number: magnitude. Say you have a line, right is positive. Displacement from O to A, x_OA, can be written as O to A points right: positive. Along the positive direction, 3 m: +3 m. The + sign may be dropped: 3 m is fine. Now O to B: it points left, So it's negative. Its magnitude is 2 units: −2 meters. A to B also points left, so it's negative too: 5 units, −5 meters. A simple way to write 1D displacement: a sign plus a number.

Displacement from coordinates

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Now, an interesting point: a quick way to find displacement from coordinates. Another example. Here's an axis. From 0 to 1: my displacement? Using the sign convention we just saw. An arrow pointing right is positive, so 0 to 1 is positive: +1 meter. From −2 to 2: count one, two, three, four: +4 meters. From 3 to −2 it's −5 meters. Put them together. Spot the pattern? Look: +1 is exactly 1 minus 0. +4 is exactly −2 minus 2? And −5 is exactly −2 minus 3. See the pattern? Oops, I wrote the middle one wrong. The +4 comes from 2 minus −2. Seen the pattern? Each time: final position minus initial. So given position coordinates, take the final coordinate minus the initial coordinate to find displacement quickly.

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One thing to stress. Note: Numbers for coordinates, like −1 or 3: their sign isn't a math sign (bigger/smaller). Lesson 4: signs have three meanings. Here the sign shows if the position is on the positive or negative side of the origin. Subtracting doesn't make it a math sign. Here I treated them as math signs, and the result matches the displacement. It just happens to match. Coordinate signs are never math signs. Two different things. Take care. That's displacement from coordinates. Always final minus initial. Never subtract the other way around. I just did it backwards myself. Final coordinate minus initial coordinate. Once more: a coordinate's sign isn't a math sign. I only treated it as a math sign in the calculation, and the result looks just like the displacement.

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It's just that. Don't misunderstand it. Last: comparing distance and displacement.

Comparing distance and displacement

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First: distance and displacement can't be compared. Key point! Many still miss it in final year. Why not? Displacement is a vector, an arrow; distance is a number. Just a plain number. Compare an arrow with a number? No way. Eating or sleeping: which matters more? No way to compare. But distance can be compared directly with the magnitude of displacement. Magnitude of displacement drops direction, leaving only the gap between positions. Magnitude of displacement and distance are both lengths, so comparable. For example, take Sichuan. Everyone knows it. Great food, if you like spicy: hotpot, ice jelly, skewer hotpot, braised pork belly. I love Sichuan food. Beijing has Sichuan restaurants, but it's unsatisfying, especially hotpot isn't authentic. On holidays, I want to eat in Sichuan.

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Beijing to Sichuan has three routes: fly, or go by car, or go by boat. Look at these three ways. By definition, is displacement the same? Its magnitude is always the length of the straight line, Beijing to Sichuan, the same as the flight path. By car, distance is this curve's length. By boat, this curve's length. Compare them: is distance always at least the magnitude of displacement? Two points: the straight line is shortest, so the shortest distance between them is the magnitude of displacement. If you go straight, with no detour, distance equals that magnitude. Detour: distance is definitely larger. It's like taking a taxi. When the driver drops you off, you try to haggle: "Hey pal, let's not charge by distance, by displacement, OK? He'd be furious.

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If he drives a full loop of Beijing's ring road, displacement is 0: a free lap. Absurd. So distance is always at least the magnitude of displacement. Now a very important point:

When does distance equal displacement?

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when does equality hold? Many students say: it holds in straight-line motion? Is that right? No. Why? Here's an example. I set out from Beijing, walk to here, then turn back, walk here, and turn back again. Still a straight line, with turns. Still straight-line motion: anything on one line counts. But after turning back, does distance still exceed the magnitude? Equality fails here too. Only in one-way straight-line motion. "One-way" is easy to overlook. Take extra care: it's often tested. Only in one-way straight-line motion is distance equal to the magnitude, because one-way rules out turning back. Only then are the two equal. This page is crucial for exams. Again: Can't compare distance and displacement, only with its magnitude: distance is at least that magnitude, equal only in one-way straight-line motion.

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That covers this lesson's points. Now, let's practice.

Example 1: ball thrown straight up

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Example 1: a ball moves as shown. The ball is at A, 3 meters above ground, thrown straight up, it rises to B, then falls, lands at C. Find the displacement A to C. Note: displacement is just the change in position, not the ball's motion. Just the position change from A to C: A and C are 3 meters apart in a line, so the magnitude is 3 meters. The answer is A. Note: for displacement, we only look at the change in position. I don't care if it went via B first. If I calculate the distance from A to C, I need 5 plus 5 plus 3. 5 plus 5 is 10, plus 3 is 13 meters. That's my distance. Tell distance from displacement.

Example 2: back-and-forth on the x-axis

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Example 2: displacement from coordinates. A particle starts at A, heading in the +x direction to B, then turns back to the negative side. Which are correct? A: from A via B to C, displacement 2 m, and distance 4 m. Displacement first. Two ways. One: just draw an arrow. From A to C, ignoring the middle, displacement needs only start and end. Start A, end C: an arrow like this. Opposite to positive, so negative; magnitude 2 meters, so it's −2 meters. So the option is wrong, right? Never drop the sign of a displacement. It always has a sign. Minus: opposite to positive. For magnitude, it's only positive. The second way: subtract coordinates. A is at +1 m, C at −1 m, so −1 minus 1 is also −2. End minus start coordinate works too. So displacement is −2 m.

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Distance is 4 m, obviously. So A is wrong. B: displacement and distance from B to D. Use D's coordinate minus B's. D is −2, B is 2, so it's −4. Distance is 4, so B is correct. C: when the particle is at D, its position is D's coordinate, −2 m. So C is right. In Lesson 2, on reference frames, on a 1D axis, one number gives a position. The sign shows which side of the origin; the number is the distance to the origin. So −2 meters is fine. D: when the particle reaches D, its displacement relative to A is −3. Start A, end D: D's coordinate minus A's. D is −2, A is 1, so it's −3 meters. No problem, so D is correct too. Practice both ways to find displacement. You can draw an arrow: judge direction, then magnitude. Or: final coordinate minus initial coordinate.

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I prefer final minus initial, so you won't lose the minus sign.

Example 3: 1 3/4 laps around a circle

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Example 3: someone runs 1 and 3/4 laps on a circle of radius R. Distance? Displacement? Draw a circle; he starts from this point. He runs around the circle: a full lap, then 3/4 of a lap. So he ends here. First a full lap, like this, then three quarters of a lap. So he ends right here. This is the end. Now, how do we find his distance? Distance is the length of his actual path. He ran 1 and 3/4 laps in all, each lap is 2πR, so it comes to 3.5πR. Calculate carefully. 1 and 3/4 laps is 7/4 × 2πR, which is 7/2 πR, that is 3.5πR. So distance is 3.5πR. Displacement? Magnitude: the straight line between the positions. So connect them with a line, and add an auxiliary line. He ran 1 and 3/4 laps, so exactly a quarter arc is left.

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A right angle, 90 degrees. Look: this is R, this is R, and this is the triangle's hypotenuse. So it's √2 R. His distance is 3.5πR, and displacement is √2 R. The answer is B.

Example 4: five true-or-false statements

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Last example: distance vs. displacement. We'll go one by one: true or false. A: in straight-line motion, magnitude of displacement equals distance. What's wrong? "One-way" is missing. We stressed it: Equal only in one-way straight-line motion With turnarounds, distance exceeds the magnitude of displacement. So A is wrong. B: in one-way straight-line motion, distance equals displacement. Looks right, so B is right? No. Why? Read it: distance equals displacement. Different things: distance is a number, displacement is an arrow. Can they be equal? No. Here's the mistake. It should say: distance equals the magnitude of displacement. See? B is wrong on this point: Compare distance only with that magnitude. So B is wrong. C: zero displacement, distance maybe not. True. A lap and back: zero displacement, distance may not be.

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D: more distance means more displacement? Of course not. Same: I run one lap, or a big loop: distance grows, but displacement is still zero. E: the 25 km on a highway sign means displacement to next exit is 25 km. Wrong. It means distance. For example, say there's a curve here, and the exit is at the end of the curve. So the 25 km is the curve's length, not the straight line, which would mean nothing. Because you drive the actual distance. It tells you how far to drive, so it must be distance, not displacement. So E is wrong. So only C is right; the rest are wrong.

Summary

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Distance and displacement: All four points nailed? First, definitions: displacement is an arrow, with magnitude and direction; distance is just a number. Second, writing 1D displacement: a sign plus a number will do. Third, coordinates: final coordinate minus initial coordinate. Again, the sign of a coordinate isn't a math sign; it shows which side of the origin. Only when I treat it as a pure number does the result happen to be displacement. Its sign isn't a math sign. Watch out here. Fourth: distance vs. displacement Distance can only be compared with the magnitude of displacement, and only in one-way straight-line motion are the two equal. If it's not one-way, distance is definitely greater.