Resultant Velocity Calculator — 2D Vector Addition
Calculate the resultant velocity when two velocities act simultaneously at different directions — enter the magnitude and direction of each velocity to get the combined speed and direction.
m/s
°
m/s
°
Magnitude of the combined velocity vector
- 1
x-components (vx)
3 × cos(0°) + 4 × cos(90°) = 3 - 2
y-components (vy)
3 × sin(0°) + 4 × sin(90°) = 4 - 3
Resultant speed
√(3² + 4²) = 5
How does this calculator work?
Add two velocity vectors by summing their x- and y-components (vx = v₁cosθ₁ + v₂cosθ₂, vy = v₁sinθ₁ + v₂sinθ₂), then find the resultant speed = √(vx² + vy²) at θ = atan2(vy, vx). The classic river-crossing example: 3 m/s east + 4 m/s north = 5 m/s at 53.13° N of E.
Formula
How this is calculated
Velocity is a vector quantity — it has both magnitude (speed) and direction. When two velocities are present simultaneously (a boat crossing a river with a current, an aircraft flying in a crosswind, a ball rolling on a moving surface), they combine by vector addition: each velocity is broken into its x- (horizontal) and y- (vertical) components using vx = v·cosθ and vy = v·sinθ, where θ is measured counter-clockwise from east. The components are summed separately, then the resultant speed = √(vx² + vy²) and direction θ = atan2(vy, vx) are found.
The classic example: a boat moves at 3 m/s eastward (θ₁ = 0°) while a river current flows at 4 m/s northward (θ₂ = 90°). The resultant speed is √(3² + 4²) = 5 m/s at atan2(4, 3) ≈ 53.13° north of east — the default values in this calculator. This is identical mathematically to the resultant force problem; the only difference is the physical quantity being combined.
This calculator covers two concurrent velocities. For relative velocity problems (e.g., velocity of A relative to B), set v₂ to the negative of the reference-frame velocity by adjusting the angle by 180°. The method extends to three or more velocities by summing all x- and y-components first.
Frequently asked questions
A plane heading north at 200 km/h in a 50 km/h eastward crosswind has a resultant velocity of √(200² + 50²) ≈ 206 km/h at an angle slightly east of north. The pilot must aim a few degrees west to compensate and maintain the desired track.
To find the velocity of object A as seen from moving frame B, subtract B's velocity vector from A's: v_A_rel_B = v_A − v_B. In this calculator, enter v₁ as A's speed/direction and v₂ as B's speed with its direction rotated 180° to subtract it.
Speed is the magnitude (how fast — a scalar), while velocity includes direction (a vector). The resultant of this calculator is a velocity: it has both a speed (the magnitude output) and a direction (the angle output).
Also known as
TG we-Calculate Editorial Team. (2026). Resultant Velocity Calculator — 2D Vector Addition [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/resultant-velocity-calculator
TG we-Calculate Editorial Team. "Resultant Velocity Calculator — 2D Vector Addition." TG we-Calculate. 2026. https://we-calculate.com/calculator/resultant-velocity-calculator.
TG we-Calculate Editorial Team, "Resultant Velocity Calculator — 2D Vector Addition," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/resultant-velocity-calculator
@misc{wecalculate_resultant_velocity_calculator, title = {Resultant Velocity Calculator — 2D Vector Addition}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/resultant-velocity-calculator}}, year = {2026}, note = {TG we-Calculate} }
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