To find displacement, subtract the initial position from the final position: Δx = x₂ – x₁. The result is a vector with a magnitude and a direction, measured in meters (m), feet (ft), or miles (mi). Other situations use velocity, acceleration, graphs, vector components, or angles. This guide explains how to find displacement with 8 methods, the units of displacement, and a worked example for each method.
What Is Displacement and What Are Examples of Displacement?
Displacement is the change in position of an object, measured as a straight-line vector from the starting point to the ending point. Displacement has a magnitude and a direction, and its SI unit is the meter (m).
According to OpenStax University Physics Volume 1, displacement is a vector that is positive or negative depending on the chosen positive direction. NASA Glenn Research Center lists displacement, velocity, and acceleration as vector quantities.
Here are 4 examples of displacement:
- A runner finishing 100 m east of the starting line
- A professor pacing 2 m right and 4 m left, ending 2 m left of the start
- A hiker walking 3 km northeast from camp
- An elevator rising 30 ft from the ground floor
How Does Displacement Differ From Distance?
Displacement is the straight-line change in position with direction, and distance is the total length of the path traveled. The table compares displacement and distance across 5 properties.
| Property | Displacement | Distance |
|---|---|---|
| Quantity type | Vector | Scalar |
| Measures | Change in position | Total path length |
| Direction | Included | Not included |
| Possible values | Positive, negative, or zero | Zero or positive |
| Example (professor pacing 2 m right, then 4 m left) | -2 m | 6 m |
A 2018 study by Handhika et al. tested 12 physics education students at PGRI Madiun University and found that 75% of students assumed displacement equals distance.
How Do You Find Displacement With the Basic Formula?
Subtract the initial position from the final position to find displacement: Δx = x₂ – x₁. OpenStax University Physics Volume 1 states that displacement is always solved by subtracting the initial position from the final position. The 3 variables are:
- Δx: displacement in meters (m), positive or negative
- x₂: final position on the chosen axis
- x₁: initial position on the chosen axis
What Are the Steps to Find Displacement?
Follow 5 steps to calculate displacement:
- Choose a positive direction, such as east or right.
- Record the initial position (x₁) on that axis.
- Record the final position (x₂) on that axis.
- Subtract the initial position from the final position.
- State the result with units and direction.
Example 1: A car starts at the 20-mile marker and stops at the 95-mile marker, with north as the positive direction. Δx = 95 – 20 = 75 mi north.
Example 2: The same car starts at the 95-mile marker and stops at the 20-mile marker. Δx = 20 – 95 = -75 mi, which equals 75 mi south.
How Do You Find Total Displacement for Several Legs?
Add the individual displacements to find total displacement: Δx(total) = Δx₁ + Δx₂ + Δx₃. OpenStax defines total displacement as the sum of the individual displacements.
Example: A delivery driver travels 4 km east, 1 km west, and 6 km east, with east as positive. Δx(total) = 4 + (-1) + 6 = 9 km east. The distance traveled is 4 + 1 + 6 = 11 km.
What Units Measure Displacement?
Displacement is measured in units of length. The SI unit is the meter (m), and US customary units include feet (ft), inches (in), and miles (mi).
The table lists 5 length units and the value of 1 unit in meters.
| Unit | Symbol | System | Value of 1 unit in m |
|---|---|---|---|
| Meter | m | SI | 1 |
| Kilometer | km | Metric | 1,000 |
| Mile | mi | US customary | 1,609.344 |
| Foot | ft | US customary | 0.3048 |
| Inch | in | US customary | 0.0254 |
How Do You Convert Displacement Units?
Multiply the displacement by the conversion factor between the two units. Here are 2 conversions:
- ft to m: multiply by 0.3048, so 30 ft × 0.3048 = 9.144 m
- mi to km: multiply by 1.609344, so 75 mi × 1.609344 = 120.70 km
How Do You Find Displacement From Velocity and Time?
Multiply velocity by time to find displacement when velocity is constant: Δx = v × t. For changing velocity, multiply average velocity by time: Δx = v(avg) × t.
In position form, OpenStax University Physics Volume 1 writes x = x₀ + v(avg)t, which rearranges to Δx = v(avg) × t. Velocity carries a direction, so the displacement carries the same direction.
Example 1: A cyclist rides at 12 mph east for 2.5 hours. Δx = 12 × 2.5 = 30 mi east.
Example 2: A train travels at 25 m/s north for 3 minutes (180 s). Δx = 25 × 180 = 4,500 m north, which equals 4.5 km.
How Do You Find Displacement With Constant Acceleration?
Displacement under constant acceleration is found with Δx = v₀t + ½at² when initial velocity, acceleration, and time are known. According to OpenStax University Physics Volume 1, these equations are valid only when acceleration is constant.
Here are 3 equations for displacement with constant acceleration:
- Initial velocity, acceleration, and time known: Δx = v₀t + ½at²
- Initial velocity, final velocity, and time known: Δx = ½(v₀ + v)t
- Initial velocity, final velocity, and acceleration known: Δx = (v² – v₀²) / (2a)
Example 1: A car starts from rest (v₀ = 0) and accelerates at 3 m/s² for 4 s. Δx = 0 + ½ × 3 × 4² = 24 m.
Example 2: A car speeds up from 10 m/s to 22 m/s in 4 s. Δx = ½ × (10 + 22) × 4 = 64 m.
Example 3: A car brakes from 30 m/s (67.1 mph) to 0 m/s at -7.00 m/s², an example from OpenStax. Δx = (0² – 30²) / (2 × -7.00) = 900 / 14 = 64.3 m.
How Do You Find Displacement From a Graph or Calculus?
Displacement equals the signed area between a velocity-time graph and the time axis. Calculus finds the same area with the integral Δx = ∫ v(t) dt, taken from the starting time to the ending time. OpenStax University Physics Volume 1 recovers position from velocity with x(t) = ∫ v(t) dt + C.
Here are 3 rules for finding displacement from graphs and functions:
- A position-time graph gives displacement as the final position minus the initial position on the vertical axis.
- A velocity-time graph gives displacement as the signed area between the line and the time axis.
- A velocity function gives displacement as the integral of velocity over the time interval.
Area above the time axis counts as positive displacement, and area below the axis counts as negative displacement.
Example 1: Velocity is +4 m/s for 3 s, then -2 m/s for 3 s. Δx = (4 × 3) + (-2 × 3) = 12 – 6 = 6 m forward. The distance traveled is 12 + 6 = 18 m.
Example 2: Velocity is v(t) = 6t + 2 in m/s, from t = 0 to t = 4 s. The integral is x(t) = 3t² + 2t, so Δx = 3(4)² + 2(4) = 48 + 8 = 56 m.
How Do You Find Displacement in Two Dimensions?
Subtract the initial position vector from the final position vector, then find the magnitude with the Pythagorean theorem: |Δr| = √(Δx² + Δy²). OpenStax University Physics Volume 1 defines the displacement vector as Δr = r(t₂) – r(t₁).
Follow 4 steps to find two-dimensional displacement:
- Subtract the initial x-coordinate from the final x-coordinate to get Δx.
- Subtract the initial y-coordinate from the final y-coordinate to get Δy.
- Calculate the magnitude with √(Δx² + Δy²).
- Calculate the direction with θ = tan⁻¹(Δy / Δx).
Example 1: A drone moves from (2 m, 1 m) to (8 m, 9 m). Δx = 8 – 2 = 6 m and Δy = 9 – 1 = 8 m. |Δr| = √(6² + 8²) = √100 = 10 m, at θ = tan⁻¹(8 / 6) = 53.1° above the +x axis.
Example 2: A polar-orbiting satellite at 400 km altitude moves from (0 km, 6,770 km) at the North Pole to (4,787 km, -4,787 km). Δx = 4,787 km and Δy = -4,787 – 6,770 = -11,557 km. |Δr| = √(4,787² + 11,557²) = 12,509 km, at θ = -67.5°. This matches OpenStax Example 4.1.
How Do You Add Several Displacements in Two Dimensions?
Add the x-components and the y-components separately, then find the magnitude and direction of the totals.
Example: A hiker walks 3 km east, then 4 km north. The total components are Δx = 3 km and Δy = 4 km. |Δr| = √(3² + 4²) = 5 km, at θ = tan⁻¹(4 / 3) = 53.1° north of east. The distance traveled is 3 + 4 = 7 km.
How Do You Find Displacement in Free Fall and Projectile Motion?
Vertical displacement in free fall equals Δy = v₀t – ½gt², with up as the positive direction and g = 9.8 m/s² (32.2 ft/s²). An object dropped from rest has v₀ = 0, so Δy = -½gt². Both examples ignore air resistance.
Example: A ball falls from rest for 3 s. Δy = -½ × 9.8 × 3² = -44.1 m, which equals 44.1 m downward (144.7 ft).
How Do You Find the Displacement of a Projectile?
Find horizontal displacement with Δx = v₀ cos θ × t and vertical displacement with Δy = v₀ sin θ × t – ½gt², then combine both with the Pythagorean theorem. In both equations, θ is the launch angle above the horizontal.
Example: A ball is launched at 20 m/s and 30°. After 1.5 s, Δx = 20 × cos 30° × 1.5 = 25.98 m, and Δy = 20 × sin 30° × 1.5 – ½ × 9.8 × 1.5² = 15 – 11.025 = 3.975 m. The total displacement is √(25.98² + 3.975²) = 26.28 m, at 8.7° above the horizontal.
Horizontal displacement at landing on level ground is the range: R = v₀² sin(2θ) / g = 20² × sin 60° / 9.8 = 35.35 m.
How Do You Find Displacement and Angular Displacement in Circular Motion?
Angular displacement equals the final angle minus the initial angle: Δθ = θ₂ – θ₁, measured in radians (rad). OpenStax College Physics 2e defines the rotation angle as the ratio of arc length to radius (Δθ = Δs / r) and states that 2π rad equals 1 revolution, or 360°.
Here are 3 formulas for circular motion:
- Arc length (distance along the circle): s = rΔθ
- Straight-line displacement (chord): |Δr| = 2r sin(Δθ / 2)
- Degrees to radians: radians = degrees × π / 180
Example 1: A wheel turns 3 revolutions. Δθ = 3 × 2π = 18.85 rad.
Example 2: A point at r = 0.5 m rotates through 90° (1.5708 rad). The arc length is s = 0.5 × 1.5708 = 0.785 m, and the straight-line displacement is 2 × 0.5 × sin 45° = 0.707 m.
Example 3: NASA states that the International Space Station travels at 5 miles per second and orbits Earth about every 90 minutes. One orbit covers about 5 × 90 × 60 = 27,000 mi of distance, but the position returns to its start, so the displacement after one orbit is 0 mi.
What Are Common Mistakes When Finding Displacement?
Finding displacement involves 5 common mistakes:
- Using distance in place of displacement
- Subtracting the final position from the initial position
- Omitting direction or sign
- Applying constant-acceleration equations to changing acceleration
- Mixing units, such as feet with meters
What Are People Also Asking About How to Find Displacement?
How do you find displacement with distance and time?
Multiply velocity with direction by time, because distance and time alone give average speed.
How do you find displacement without time?
Use Δx = (v² – v₀²) / (2a) with initial velocity, final velocity, and acceleration.
How do you find displacement from force and mass?
Divide force by mass to get acceleration, then apply Δx = v₀t + ½at².
How do you find displacement from work and force?
Divide work by the force component along the motion: d = W / (F cos θ).
How do you find the displacement of a spring?
Divide the force by the spring constant: x = F / k, with force in newtons and k in N/m.
How do you find displacement in simple harmonic motion?
Use x(t) = A cos(ωt), where A is amplitude, ω is angular frequency, and t is time.
How is displacement measured in real life?
GPS receivers, laser rangefinders, ultrasonic motion sensors, and inertial measurement units measure displacement from position changes, distance changes, echo times, and integrated acceleration.
Can displacement be zero?
Yes, displacement is zero when the final position equals the initial position, such as after one full lap of a track.
Can displacement be negative?
Yes, negative displacement indicates movement in the negative direction of the chosen axis.
How Do You Find Displacement: Which Method Fits Each Situation?
Displacement is found by matching the known quantities to the right formula. The table summarizes 8 methods by situation, formula, and known quantities.
| Method | Situation | Formula | Known quantities |
|---|---|---|---|
| 1 | Position change | Δx = x₂ – x₁ | Initial position, final position |
| 2 | Constant velocity | Δx = v × t | Velocity, time |
| 3 | Constant acceleration, time known | Δx = v₀t + ½at² or Δx = ½(v₀ + v)t | Initial velocity, acceleration or final velocity, time |
| 4 | Constant acceleration, time unknown | Δx = (v² – v₀²) / (2a) | Initial velocity, final velocity, acceleration |
| 5 | Graph or calculus | Δx = ∫ v(t) dt | Velocity function or velocity-time graph |
| 6 | Two dimensions | Magnitude = √(Δx² + Δy²) | Coordinates or components |
| 7 | Free fall and projectile | Δy = v₀t – ½gt² | Initial velocity, angle, time |
| 8 | Angular displacement | Δθ = θ₂ – θ₁ | Initial angle, final angle |
The basic formula to find displacement is Δx = x₂ – x₁ when the initial and final positions are known. The other 7 methods apply when velocity, acceleration, graphs, coordinates, or angles are known instead.
Sources:
- 3.1 Position, Displacement, and Average Velocity – University Physics Volume 1
- 3.4 Motion with Constant Acceleration – University Physics Volume 1
- 3.6 Finding Velocity and Displacement from Acceleration – University Physics Volume 1
- 4.1 Displacement and Velocity Vectors – University Physics Volume 1
- 6.1 Rotation Angle and Angular Velocity – College Physics 2e
- Scalars and Vectors – NASA Glenn Research Center
- International Space Station Facts and Figures – NASA
- The Students Conception About Kinematics – Displacement and Distance Concept (Handhika et al., 2018)
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