Resultant Velocity Calculator

Add two velocity vectors by magnitude and direction to find the resultant velocity's magnitude, direction, and x/y components using vector addition.

Quick Facts

Vector addition
Rx = v₁cosθ₁ + v₂cosθ₂, Ry = v₁sinθ₁ + v₂sinθ₂
Add velocities by their x/y components, never just their magnitudes.
Magnitude
R = √(Rx² + Ry²)
Pythagorean theorem applied to the combined components.
Direction
θ = atan2(Ry, Rx)
Measured counterclockwise from the positive x-axis (east = 0°, north = 90°).
Perpendicular shortcut
R = √(v₁² + v₂²)
Applies when the two vectors are 90° apart, e.g. a boat crossing a current.

Your Results

Calculated
Resultant Velocity
-
R = √(Rx² + Ry²)
Direction Angle
-
θ = atan2(Ry, Rx), from +x axis
X-Component (Rx)
-
v₁cosθ₁ + v₂cosθ₂
Y-Component (Ry)
-
v₁sinθ₁ + v₂sinθ₂

Ready

Enter both velocity vectors (magnitude and direction), then press Calculate.

Formula and Method for Resultant Velocity

When an object is influenced by two or more velocities at the same time — a plane flying through wind, a boat crossing a moving current, or two conveyor belts pushing a package in different directions — the object does not follow either velocity alone. It follows their vector sum, called the resultant velocity. This calculator takes two velocity vectors, each defined by a magnitude and a direction angle, and returns the magnitude, direction, and x/y components of the single equivalent velocity that replaces them.

Breaking velocities into components

Every velocity vector can be split into a horizontal (x) and vertical (y) component using vx = v·cosθ and vy = v·sinθ, where θ is the direction angle measured counterclockwise from the positive x-axis (0° = east, 90° = north). To combine two velocities, add their components separately — never their magnitudes: Rx = v₁cosθ₁ + v₂cosθ₂ and Ry = v₁sinθ₁ + v₂sinθ₂. The magnitude of the resultant then follows from the Pythagorean theorem, R = √(Rx² + Ry²), and its direction is θ = atan2(Ry, Rx), converted to degrees and normalized to the 0-360° range.

Common mistakes and practical notes

  • Mixing angle conventions: this calculator uses the standard math/physics convention (counterclockwise from east). If you have a compass bearing (clockwise from north), convert it first: standard angle = 90° − bearing, adjusted into 0-360°.
  • Mixing units: both velocities must be in the same unit before you add them — m/s and km/h do not combine directly. Pick one unit for both inputs.
  • The river-crossing shortcut: the classic "boat crossing a current" problem is just this formula with the angle between the two vectors equal to 90°, which reduces to R = √(v₁² + v₂²).
  • Resultant ≠ v₁ + v₂: simple addition of magnitudes only works when both vectors point in exactly the same direction (θ₁ = θ₂); at any other angle the resultant is smaller than the arithmetic sum.

Frequently Asked Questions

What is resultant velocity?
Resultant velocity is the single vector that represents the combined effect of two or more velocities acting on an object at the same time — for example, a plane's airspeed combined with wind, or a boat's speed combined with a river current. It is found by adding the velocity vectors together, not by adding their magnitudes.
How do you calculate resultant velocity from two vectors?
Break each velocity into x and y components with v_x = v·cosθ and v_y = v·sinθ, add the components separately (Rx = v1cosθ1 + v2cosθ2, Ry = v1sinθ1 + v2sinθ2), then find the magnitude with the Pythagorean theorem, R = √(Rx² + Ry²), and the direction with θ = atan2(Ry, Rx).
What if the two velocities are perpendicular, like a boat crossing a river?
When the angle between the two vectors is 90°, the general formula simplifies to the Pythagorean shortcut R = √(v1² + v2²), with direction angle θ = atan(v2/v1) measured from v1's direction. This is the classic boat-crossing-a-current problem.
Does the direction angle use compass bearing or standard math convention?
This calculator uses the standard physics/math convention: angles are measured counterclockwise from the positive x-axis, so 0° points east and 90° points north. If you have a compass bearing (measured clockwise from north), convert it first with standard angle = 90° − bearing, adjusted into the 0-360° range.

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