Why These Two Quantities Are Different
When a particle moves along a line, its displacement tells you where it ended up relative to where it started. Its total distance traveled tells you how much ground it covered along the way.
The difference becomes important whenever the particle changes direction. Suppose a particle moves 10 units to the right and then 6 units back to the left. Its displacement is only 4 units to the right, but it traveled a total distance of 16 units.
| Quantity | What it measures | Can it be negative? |
|---|---|---|
| Displacement | Final position minus initial position | Yes |
| Total distance | How far the particle actually traveled | No |
If the particle never changes direction, total distance and the absolute value of displacement are the same. If the particle changes direction, they can be very different.
Displacement from Velocity
If is the velocity of a particle, then the displacement from time to time is the integral of velocity over that interval.
There is an even simpler way to think about this: displacement is simply the particle's final position minus its initial position.
Because velocity can be negative, areas where the velocity is below the time axis contribute negatively to displacement.
Total Distance Traveled from Velocity
Total distance is different because distance cannot be negative. If velocity is negative, the particle is moving in the opposite direction, but it is still traveling a positive amount of distance.
Therefore, to find total distance directly from velocity, integrate the absolute value of velocity.
The absolute value is the critical difference. For displacement, use . For total distance, use .
The AP Calculus Method
When a problem gives you a velocity function and asks for total distance, don't immediately integrate from the starting time to the ending time. First determine whether the particle changes direction.
- Find where velocity equals zero. Solve on the interval.
- Use those times to split the interval. These are the possible direction-change points.
- Determine the sign of velocity on each resulting interval.
- Calculate displacement by integrating velocity normally.
- Calculate total distance by making every piece positive, either with absolute value or by changing the sign of negative pieces.
If the question says "displacement", integrate velocity with its signs intact. If it says "total distance traveled," make sure every contribution to distance is positive.
Worked Examples
A particle moves along the -axis with velocity . Find the displacement and total distance traveled from to .
The position of a particle is . Find the displacement and total distance traveled on .
A particle has velocity on . Find the displacement and total distance traveled.
A particle's velocity is shown at selected times. Suppose the velocity is positive from , negative from , and positive again from .
Geometric Interpretation
Velocity is the rate of change of position, so the integral of velocity measures signed area between the velocity graph and the time axis.
For displacement, areas above the axis count positively and areas below the axis count negatively.
For total distance, both types of areas count positively. In other words, total distance is the sum of the absolute values of the relevant areas.
This is one of the most useful visual interpretations on the AP Calculus exam: signed area gives displacement; total area gives distance.
Common Mistakes
This is only true when the particle does not change direction. If velocity changes sign, distance and displacement must be handled separately.
An integral of velocity gives displacement, not automatically total distance. Before calculating distance, find where and check whether the sign changes.
Negative velocity means the particle is moving in the negative direction. It does not mean the particle travels a negative distance.
A zero of velocity is a possible turning point. Always check the sign of velocity on both sides before deciding whether the particle actually changes direction.
A Quick Decision Rule
Then ask one question: Does velocity change sign?
- No: total distance is the absolute value of displacement.
- Yes: split the interval at the direction changes and add the positive distances from each piece.
Practice Problems
A particle has velocity . Find its displacement from .
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A particle has velocity . Find the total distance traveled from .
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The position of a particle is . Find its displacement on .
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A particle has velocity on . Find both its displacement and total distance traveled.
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