Velocities add as vectors: a boat crossing a river, a plane in a crosswind or a swimmer in a current all move with the vector sum of their own velocity and that of the medium.
How to use the Resultant Velocity Calculator
- Number of velocities
- Velocity 1
- Angle 1
- Velocity 2
- Angle 2
- Velocity 3
- Angle 3
- Velocity 4
- Angle 4
- Velocity 5
- Angle 5
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Choose how many velocities to add and give each a magnitude and an angle measured counter-clockwise from the positive x-axis (east).
Inputs explained
| Field | What to enter |
|---|---|
| Number of velocities | Choose one: 2, 3, 4, 5 |
| Velocity 1 | Units: m/s, km/h, mph, ft/s, knots, km/s |
| Angle 1 | Units: °, rad, arcmin, arcsec, mrad |
| Velocity 2 | Units: m/s, km/h, mph, ft/s, knots, km/s |
| Angle 2 | Units: °, rad, arcmin, arcsec, mrad |
| Velocity 3 | Units: m/s, km/h, mph, ft/s, knots, km/s |
| Angle 3 | Units: °, rad, arcmin, arcsec, mrad |
| Velocity 4 | Units: m/s, km/h, mph, ft/s, knots, km/s |
| Angle 4 | Units: °, rad, arcmin, arcsec, mrad |
| Velocity 5 | Units: m/s, km/h, mph, ft/s, knots, km/s |
| Angle 5 | Units: °, rad, arcmin, arcsec, mrad |
What each result means
- Resultant velocity = sqrt([x component Vₓ]^2 + [y component V_y]^2)
- Direction (from +x, counter-clockwise) = mod(deg(atan2([y component V_y], [x component Vₓ])), 360)
Example values
How the main result changes when Velocity 1 changes and everything else stays at the example values above:
| Velocity 1 (m/s) | Resultant velocity (m/s) |
|---|---|
| 2.5 | 3.9051 |
| 3.75 | 4.8023 |
| 5 | 5.831 |
| 6.25 | 6.9327 |
| 7.5 | 8.0777 |
| 10 | 10.4403 |
Formula
Vₓ = Σ vᵢ cos θᵢ, V_y = Σ vᵢ sin θᵢ, |V| = √(Vₓ² + V_y²), θ = atan2(V_y, Vₓ)
Worked example
Inputs — Number of velocities: 2; Velocity 1: 5 m/s; Angle 1: 0 °; Velocity 2: 3 m/s; Angle 2: 90 °.
Results — x component Vₓ = 5 m/s; y component V_y = 3 m/s; Resultant velocity = 5.831 m/s; Direction (from +x, counter-clockwise) = 30.96 °.
Key terms
- Resultant
- The single vector equal to the sum of several vectors.
Frequently asked questions
How do I add velocities at right angles?
Use Pythagoras: √(v₁² + v₂²).
What angle convention is used?
Counter-clockwise from the positive x-axis (east = 0°, north = 90°).
Can the resultant be smaller than each velocity?
Yes, when they point in roughly opposite directions.
Does this apply near the speed of light?
No – use relativistic velocity addition.