Difference Between Distance and Displacement: 10 Key Differences, Formulas, and Examples
The difference between distance and displacement is that distance measures the total length of the path traveled, and displacement measures the straight-line change in position from the start point to the end point. Distance is a scalar, and displacement is a vector. Both use the meter (m) as the SI unit. The 10 differences below cover definition, direction, path dependence, sign, zero values, formulas, and measurement.
For a reliable reference, consult OpenStax section on position and displacement.
What Are Distance and Displacement, and What Are Examples of Each?
Distance and displacement both describe how far an object moves, but they measure different things. OpenStax University Physics Volume 1 defines displacement as the change in position of an object and defines distance traveled as the total length of the path between 2 positions.
What Is Distance?
Distance is the total length of the path an object travels between 2 points. Distance is a scalar quantity, so it has a magnitude and no direction. Scalars are quantities described by magnitude only. Examples include distance, speed, mass, time, and energy.
Distance never decreases during motion. An odometer reading, a 400 m lap, and a 42.195 km marathon course are examples of distance.
What Is Displacement?
Displacement is the change in position of an object, measured as a straight-line vector from the start point to the end point. Displacement is a vector quantity, so it has a magnitude and a direction. Vectors are quantities described by magnitude and direction. Examples include displacement, velocity, acceleration, and force.
Displacement depends only on the start point and the end point. A car that moves 40 km east has a displacement of 40 km east.
The table below defines the symbols used for distance and displacement. All lengths are in meters (m), and time is in seconds (s).
| Symbol | Meaning | Quantity Type |
|---|---|---|
| d | Distance | Scalar |
| Δx | Displacement along 1 axis | Vector (the sign shows direction) |
| Δr⃗ | Displacement vector in 2 or 3 dimensions | Vector |
| x_i | Initial position | Vector (the sign shows direction) |
| x_f | Final position | Vector (the sign shows direction) |
| t | Elapsed time | Scalar |
What Are the 10 Key Differences Between Distance and Displacement?
The 10 key differences are definition, quantity type, direction, path dependence, possible values, zero value, formula, size relationship, related average rate, and measuring instrument. The table below compares distance and displacement by these 10 features.
| No. | Feature | Distance | Displacement |
|---|---|---|---|
| 1 | Definition | Total length of the path traveled | Straight-line change in position from start to end |
| 2 | Quantity type | Scalar | Vector |
| 3 | Direction | None | Points from the start point to the end point |
| 4 | Dependence on path | Depends on the path | Does not depend on the path |
| 5 | Possible values | Zero or positive | Positive, negative, or zero |
| 6 | Zero for a moving object | Never | Yes, if the object returns to the start |
| 7 | Formula | d = sum of the path segment lengths | Δx = x_f − x_i |
| 8 | Size relationship | Greater than or equal to the displacement magnitude | Less than or equal to the distance |
| 9 | Related average rate | Average speed = d / Δt | Average velocity = Δx / Δt |
| 10 | Measuring instrument | Odometer, pedometer, trundle wheel | Tape measure from start to end, GPS coordinate difference, laser rangefinder |
What Are the Formulas for Distance and Displacement?
Distance is calculated by adding the lengths of all path segments, and displacement is calculated by subtracting the initial position from the final position. The table below lists the formulas for 6 situations.
| Situation | Distance Formula | Displacement Formula |
|---|---|---|
| 1 axis, no reversal | d = |x_f − x_i| | Δx = x_f − x_i |
| 1 axis, with reversals | d = |Δx₁| + |Δx₂| + … | Δx = Δx₁ + Δx₂ + … |
| 2 dimensions, several legs | d = sum of the leg lengths | Δr = √((ΣΔx)² + (ΣΔy)²) |
| Constant velocity | d = speed × t | Δx = v × t |
| Velocity function v(t) | d = ∫ |v(t)| dt | Δx = ∫ v(t) dt |
| Arc of a circle with radius r and angle θ in radians | d = rθ | Δr = 2r sin(θ / 2) |
How Do You Calculate Distance and Displacement Step by Step?
Distance and displacement are calculated by listing each movement leg, adding the leg lengths for distance, and adding the leg vectors for displacement.
- Choose a positive direction for each axis.
- List the start point, the end point, and every turning point.
- Add the absolute length of each leg to find the distance.
- Subtract the initial position from the final position to find the displacement.
- Attach the unit, and attach the direction to the displacement.
How Do You Calculate Distance and Displacement in 1 Dimension?
In 1 dimension, distance is the sum of the absolute leg lengths, and displacement is the final position minus the initial position. Example: An object starts at x = 2 m, moves to x = 10 m, and then moves back to x = 6 m.
- Distance: |10 − 2| + |6 − 10| = 8 + 4 = 12 m
- Displacement: 6 − 2 = +4 m, which is 4 m in the positive direction
How Do You Calculate Distance and Displacement in 2 Dimensions?
In 2 dimensions, distance is the sum of the leg lengths, and displacement is the straight-line separation found with the Pythagorean theorem. Example: A person walks 6 km east and then 8 km north.
- Distance: 6 + 8 = 14 km
- Displacement magnitude: √(6² + 8²) = 10 km
- Displacement direction: arctan(8 / 6) ≈ 53.1° north of east
How Do You Calculate Distance and Displacement for Several Legs?
For several legs, the east-west components and the north-south components are added separately. Example: A taxi drives 3 km east, 4 km north, and 3 km west.
- East-west component: +3 − 3 = 0 km
- North-south component: +4 km
- Displacement: 4 km north
- Distance: 3 + 4 + 3 = 10 km
What Are Examples of Distance and Displacement?
The 7 examples below show that distance equals or exceeds the displacement magnitude in every case. The table lists the situation, the distance, and the displacement.
| No. | Situation | Distance | Displacement |
|---|---|---|---|
| 1 | A car drives straight for 40 km east | 40 km | 40 km east |
| 2 | A person walks 3 m east and then 4 m north | 7 m | 5 m at 53.1° north of east |
| 3 | A person walks 6 m east and then 6 m west | 12 m | 0 m |
| 4 | A taxi drives 3 km east, 4 km north, and 3 km west | 10 km | 4 km north |
| 5 | A runner completes 1 lap of a circular track with a circumference of 400 m | 400 m | 0 m |
| 6 | A ball drops from 5 m, hits the ground, and rebounds to 3 m | 8 m | 2 m downward |
| 7 | A ball thrown upward at 20 m/s is caught at the release point | 40.82 m | 0 m |
Example 7 uses g = 9.8 m/s². The ball rises v² / (2g) = 400 / 19.6 = 20.41 m and falls the same height.
When Are Distance and Displacement Equal?
Distance equals the displacement magnitude when an object moves along a straight line without reversing direction. The table below lists 4 path types and the relationship between distance and displacement magnitude.
| Path Type | Example | Relationship |
|---|---|---|
| Straight line, no reversal | A car drives 40 km east | d = |Δr| |
| Straight line, with reversal | A person walks 6 m east and 6 m west | d > |Δr| |
| Curved path | A runner follows a semicircle | d > |Δr| |
| Closed loop | A runner completes 1 lap | d > 0 and Δr = 0 |
Can Displacement Be Larger Than Distance, Zero, or Negative?
Displacement magnitude is never larger than distance, displacement can be zero or negative, and distance is never negative. The 3 answers below explain each case.
Can Displacement Be Greater Than Distance?
Displacement magnitude cannot exceed distance, because a straight line is the shortest path between 2 points. Any other path between the same points is longer or equal. The 3-4-5 triangle shows the relationship: the displacement is 5 m, and the distance along the 2 legs is 7 m.
Can Displacement Be Zero While Distance Is Not Zero?
Displacement is zero when the final position equals the initial position, even if the object traveled a long path. A full lap of a 400 m track gives a distance of 400 m and a displacement of 0 m. The reverse case does not occur, because zero distance means the object never moved.
Can Distance Be Negative?
Distance cannot be negative, because it is a sum of absolute path lengths. Displacement carries a sign along a chosen axis. A displacement of −7 m means 7 m in the negative direction.
What Do the Sign and Direction of Displacement Show?
The sign of displacement shows the direction of the net movement along the chosen positive axis. The table below lists 3 cases with the positive direction set as east.
| Initial Position | Final Position | Displacement | Meaning |
|---|---|---|---|
| 2 m | 9 m | +7 m | 7 m east |
| 9 m | 2 m | −7 m | 7 m west |
| 5 m | 5 m | 0 m | No net change in position |
Displacement in 2 dimensions uses a compass bearing or an angle from the positive x-axis, such as 53.1° north of east.
How Do You Find Distance and Displacement From a Velocity-Time Graph?
Displacement equals the signed area between the velocity line and the time axis, and distance equals the total area with every region counted as positive. Area above the axis is positive displacement, and area below the axis is negative displacement.
Example: Velocity is +4 m/s for 5 s and then −2 m/s for 5 s.
- Displacement: 4 × 5 + (−2 × 5) = 20 − 10 = +10 m
- Distance: 20 + 10 = 30 m
Beichner published a study of kinematics graph interpretation in the American Journal of Physics in 1994. The study found that students have difficulty interpreting the area under a velocity-time graph.
How Do You Find Distance and Displacement From a Position-Time Graph?
Displacement equals the final position reading minus the initial position reading, and distance equals the sum of the position changes between turning points. A turning point is a moment when the line changes direction.
Example: A graph shows 0 m at 0 s, 8 m at 4 s, and 2 m at 8 s.
- Displacement: 2 − 0 = +2 m
- Distance: |8 − 0| + |2 − 8| = 8 + 6 = 14 m
How Do You Find Distance and Displacement With Calculus?
Displacement is the integral of velocity, and distance is the integral of the absolute value of velocity. The formulas are Δx = ∫ v(t) dt and d = ∫ |v(t)| dt.
Example: v(t) = 4 − 2t, in m/s, from t = 0 to t = 4 s. The velocity changes sign at t = 2 s.
- Displacement: [4t − t²] from 0 to 4 = 16 − 16 = 0 m
- Distance from 0 to 2 s: [4t − t²] from 0 to 2 = 4 m
- Distance from 2 to 4 s: |[4t − t²] from 2 to 4| = |0 − 4| = 4 m
- Total distance: 4 + 4 = 8 m
What Is the Difference Between Speed and Velocity in Relation to Distance and Displacement?
Speed is distance divided by time, and velocity is displacement divided by time. The table below compares average speed and average velocity for the position-time example (distance 14 m, displacement +2 m, time 8 s).
| Quantity | Formula | Quantity Type | SI Unit | Result for the Example |
|---|---|---|---|---|
| Average speed | d / Δt | Scalar | m/s | 14 / 8 = 1.75 m/s |
| Average velocity | Δx / Δt | Vector | m/s | 2 / 8 = +0.25 m/s |
Average speed is greater than or equal to the magnitude of average velocity. A ball thrown upward at 20 m/s and caught at the release point has an average speed of 40.82 / 4.08 = 10.0 m/s and an average velocity of 0 m/s.
How Do Distance and Displacement Differ in Circular Motion?
In circular motion, distance is the arc length (d = rθ), and displacement is the chord length (Δr = 2r sin(θ / 2)). The table below lists 4 fractions of a circular track with a circumference of 400 m, which gives a radius of 63.66 m.
| Fraction of Track | Angle θ | Distance (m) | Displacement Magnitude (m) |
|---|---|---|---|
| Quarter | 90° | 100 | 90.03 |
| Half | 180° | 200 | 127.32 |
| Three quarters | 270° | 300 | 90.03 |
| Full | 360° | 400 | 0 |
The half-turn displacement equals the diameter of the circle. A satellite in orbit has a displacement of 0 after each complete revolution.
How Do Distance and Displacement Differ in Projectile Motion and Free Fall?
In projectile motion, the curved flight path is longer than the straight-line displacement from the launch point to the landing point. Example: A ball launches at 20 m/s and 30° above the horizontal and lands on level ground.
- Time of flight: 2 × 20 × sin 30° / 9.8 = 2.04 s
- Horizontal displacement: 20² × sin 60° / 9.8 = 35.35 m
- Vertical displacement: 0 m
- Path length (distance): about 37.2 m
In free fall, a ball thrown upward at 20 m/s reaches a height of 20.41 m. At t = 3 s, the displacement is 20 × 3 − ½ × 9.8 × 3² = +15.9 m. The distance is 20.41 + (20.41 − 15.9) = 24.92 m.
What Are the Units of Distance and Displacement?
Distance and displacement share the SI unit of length, which is the meter (m). NIST defines the meter as the distance light travels in a vacuum in 1/299,792,458 of a second. The table below lists the value of 6 length units in meters.
| Unit Name | Symbol | Value in Meters |
|---|---|---|
| Kilometer | km | 1,000 |
| Centimeter | cm | 0.01 |
| Millimeter | mm | 0.001 |
| Foot | ft | 0.3048 |
| Mile | mi | 1,609.344 |
| Nautical mile | nmi | 1,852 |
Multiply the value by its meter equivalent to convert to meters. Example: 5 mi × 1,609.344 = 8,046.72 m.
How Do You Measure Distance and Displacement?
Distance is measured by recording the path length with a counting instrument, and displacement is measured by recording the start and end positions. The table below lists 5 instruments with the quantity each measures and the measuring principle.
| Instrument | Quantity Measured | Measuring Principle |
|---|---|---|
| Odometer | Distance | Counts wheel rotations and multiplies by the wheel circumference |
| Pedometer | Distance | Counts steps and multiplies by the step length |
| Trundle wheel | Distance | Counts wheel rotations along the path |
| GPS receiver | Both | Distance sums the segments between recorded points, and displacement is the straight-line separation between the first and last points |
| Laser rangefinder | Displacement | Measures the straight-line separation from the light travel time (d = c × t / 2) |
Example for the odometer: A wheel with a circumference of 2.0 m completes 5,000 rotations. The distance is 10,000 m, which equals 10 km.
Example for GPS: A commute records a path length of 12 km, and the start and end points are 8 km apart. The distance is 12 km, and the displacement magnitude is 8 km.
A track experiment follows 5 steps:
- Mark the start point on the track.
- Move the object along the full path and record each turning point.
- Measure each leg and add the lengths to find the distance.
- Measure the straight-line separation from the start to the end.
- Repeat 3 trials and average the results to reduce random error.
What Are Real-Life Applications of Distance and Displacement?
Distance and displacement apply to 6 fields. Examples include:
- Vehicle odometers and taxi meters, which record distance traveled.
- GPS navigation, which reports route distance and straight-line separation.
- Aviation, which compares ground track distance with the direct line between airports.
- Sports timing, such as a marathon with a distance of 42.195 km, set by World Athletics.
- Robotics, which estimates position changes from wheel encoders.
- Surveying, which measures the displacement between benchmark points.
Delivery and ride services price trips by route distance, and the straight-line displacement gives the shortest possible separation.
How Did the Concepts of Distance and Displacement Develop?
Distance has been used in motion studies since Galileo Galilei published Two New Sciences in 1638, and displacement became a vector concept through vector algebra. William Rowan Hamilton introduced the term vector in 1846. Josiah Willard Gibbs and Oliver Heaviside developed modern vector analysis in the 1880s, which allowed displacement to be written as Δr⃗ = r⃗_f − r⃗_i.
What Are the Common Mistakes When Comparing Distance and Displacement?
7 common mistakes cause incorrect results. Research by David Trowbridge and Lillian McDermott, published in the American Journal of Physics in 1980, found that introductory physics students commonly confuse position with velocity.
The mistakes include:
- Reporting the path length as the displacement.
- Dropping the direction from the displacement.
- Treating a displacement of 0 as proof that the object did not move.
- Subtracting in the wrong order (x_i − x_f), which reverses the sign.
- Adding vector legs as scalar lengths, instead of adding components.
- Counting area below the time axis as positive when finding displacement.
- Using average velocity in place of average speed.
Example of the first mistake: A person walks 6 m east and 6 m west. The displacement is 0 m, and the distance is 12 m. Reporting 12 m as the displacement is incorrect.
What Do People Also Ask About the Difference Between Distance and Displacement?
What is the main difference between distance and displacement?
Distance is the total path length and has no direction, and displacement is the straight-line change in position and has a direction.
Is distance a scalar or a vector?
Distance is a scalar, because it has a magnitude and no direction.
Is displacement a scalar or a vector?
Displacement is a vector, because it has a magnitude and a direction.
What is the formula for distance?
Distance is the sum of the path segment lengths, written as d = |Δx₁| + |Δx₂| + …, or d = ∫ |v(t)| dt.
What is the formula for displacement?
Displacement is Δx = x_f − x_i, which subtracts the initial position from the final position.
What is the SI unit of distance and displacement?
The SI unit of both quantities is the meter (m).
Can displacement be negative?
Displacement is negative when the net movement points in the negative direction of the chosen axis.
Can displacement be zero when distance is not zero?
Displacement is zero when the object returns to the start point, as in 1 full lap of a track.
Can displacement be greater than distance?
Displacement magnitude is never greater than distance, because a straight line is the shortest path between 2 points.
When are distance and displacement equal?
Distance equals the displacement magnitude for straight-line motion without a reversal.
What is the difference between speed and velocity?
Speed is distance divided by time and has no direction, and velocity is displacement divided by time and has a direction.
What is the distance and displacement for a round trip?
A round trip has a distance equal to the total path length and a displacement of 0.
How do you find displacement from a velocity-time graph?
Calculate the signed area between the line and the time axis, with area below the axis counted as negative.
How do you find distance from a velocity-time graph?
Add the areas between the line and the time axis, with every area counted as positive.
What Is the Difference Between Distance and Displacement and How Are They Applied?
The difference between distance and displacement is that distance is the total path length (a scalar), and displacement is the straight-line change in position with a direction (a vector). Distance is calculated with d = Σ|Δx| or d = ∫ |v| dt, and displacement is calculated with Δx = x_f − x_i or Δx = ∫ v dt. Distance is zero or positive, and displacement is positive, negative, or zero. Displacement magnitude is less than or equal to distance, and the 2 values are equal for straight-line motion without a reversal. Distance gives average speed, and displacement gives average velocity. Applications include odometers, GPS navigation, aviation, sports timing, robotics, and surveying.
Title: Constant Acceleration Formula: 5 Equations, Derivations, and Worked Examples
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