Distance is the total length of the path actually covered while going from one place to another. Displacement is the shortest distance between the starting point and the finishing point, measured in a definite direction. Both are measured in metres, and both describe the same journey, but they are answers to two different questions.
Ramesh lives at A. His school is at B, to the east of his house, and there are two roads that go there. Road kha is the straight one, but the bridge on it has collapsed, so he takes road ka, which winds about. Road ka shows the distance between A and B, because it has no fixed direction. Road kha shows the displacement, which is towards the east.
Two questions, not two names for one thing
Students often treat these two words as posh synonyms, and then lose marks because they are not synonyms at all. Each one answers its own question, and once you can say the two questions out loud you will never confuse them again.
| Distance | Displacement | |
|---|---|---|
| The question it answers | How much ground did you cover? | Where did you end up, compared with where you began? |
| Definition | The total length of the path actually covered | The shortest distance from the start to the finish |
| Direction | No fixed direction is needed | Always has a definite direction |
| Scalar or vector | Scalar, because a magnitude alone describes it fully | Vector, because it needs a magnitude and a direction |
| Symbol and relation | Shown by d, and distance is speed multiplied by time | Shown by S, and displacement is velocity multiplied by time |
| Can it be zero after moving? | Never. Once you move, it only grows | Yes, if you come back to where you started |
| Can it be negative? | No. You cannot cover less than no ground | Yes, when the movement is opposite to the direction taken as positive |
Ask whether a plain number is enough to answer. If somebody asks how far you walked today and you say four kilometres, that is a complete answer, so distance is a scalar. If somebody asks where you are now compared with this morning and you say four kilometres, they will immediately ask which way. That question is not being fussy. It means the answer was genuinely incomplete, and that is exactly what makes displacement a vector.
Walk round your house and prove it in a minute
Go out of your front door, walk all the way round the house, and come back in through the same door. Suppose the walk was seventy metres. Your distance is seventy metres, because your legs really did cover seventy metres of ground. Your displacement is zero, because you finished exactly where you started.
Now notice how strange that sounds. You certainly moved. You may be out of breath. And one of the two measurements says you did not go anywhere at all. Both numbers are correct, because they were never measuring the same thing. Distance is what your legs felt. Displacement is what a map would show.
Displacement throws the journey away on purpose. That sounds like a weakness and it is the entire point of it. By forgetting how you got there and keeping only where you ended up, displacement becomes something you can add, compare and calculate with. It is why the whole of the next topic works.
Three situations the examiner asks about
| Situation | When it happens | Reason |
|---|---|---|
| Distance is not zero but displacement is zero | Roshani walks from A right round and back to A | She covered 16 m of ground, so the distance is 16 m, but she finished where she began, so the displacement is zero |
| Distance and displacement are equal | She goes from A straight to B and stops | The path she took is already the shortest path, so there is nothing extra for the distance to count |
| Displacement is negative | She turns at C and moves back the way she came | Her movement is now opposite to the direction she started in, and the minus sign is how we record a direction, not a size |
A negative displacement does not mean a small displacement, and it certainly does not mean less than nothing. It means the opposite direction. Before you can call anything negative, somebody has to choose which direction counts as positive, and that choice is yours to make as long as you say what it was. If east is positive then eight metres west is written as minus eight metres, and if you had chosen west as positive the very same walk would be plus eight. The size never changed. Only the label did.
Which one is bigger, and can they ever cross?
Here is a rule worth carrying into every problem you will ever meet. For any journey at all, the distance is either greater than the displacement or exactly equal to it, and it is never smaller. Displacement is the shortest possible route between two points, so no real path can beat it. The two are equal only when you travel in a straight line without turning back.
There is a second difference that is easy to miss. As you keep moving, the distance can only ever grow, because every extra step adds to the ground you have covered. Displacement can grow, shrink, sit still or fall back to zero, depending on which way you turn. Walk away from home and it rises. Turn round and walk back and it falls, even though your distance is still climbing all the while.
The S.I. unit of both is the metre. Distance can be found from speed multiplied by time, written as \( d = v \times t \), and displacement from velocity multiplied by time, written as \( S = v \times t \). Those two formulas look almost identical, and the next topic is about why the difference between them matters.
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Distance is the total length of the path actually covered while going from one place to another.
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