How this instrument works
Resultant velocity is what an object actually does when two velocities act on it at once — a swimmer's stroke plus a river's current, an aircraft's airspeed plus a crosswind. You cannot add the two speeds directly, because direction matters: 10 m/s east and 10 m/s north do not make 20 m/s of anything. The fix is to resolve each velocity into its eastward and northward pieces, add those pieces separately, then recombine them with the Pythagorean theorem.
The formula works because motion along one axis has no effect on motion along a perpendicular axis, sometimes called the independence of perpendicular components. vₓ = v₁cosθ₁ + v₂cosθ₂ sums the eastward contribution of both vectors; v_y = v₁sinθ₁ + v₂sinθ₂ does the same northward. Only after that addition does √(vₓ² + v_y²) turn the pair back into one speed, and atan2(v_y, vₓ) recovers the single heading. The instrument exposes vₓ and v_y as their own output fields so the intermediate arithmetic can be checked before the final number is trusted.
This is Galilean vector addition, the everyday rule that holds for boats, aircraft, and thrown objects, where velocities simply stack. It stops applying once speeds approach a meaningful fraction of the speed of light, where Einstein's relativistic velocity-addition formula takes over and plain component summation overstates the result. A more common trap at ordinary speeds: the resultant direction sits at the angular midpoint of the two inputs only when their magnitudes are equal; make one vector faster than the other and the resultant swings toward it, not toward the average angle.
- Set First velocity, magnitude and First velocity, direction — direction is measured counter-clockwise from east, so 0° is due east and 90° is due north.
- Set Second velocity, magnitude and Second velocity, direction the same way, using whichever heading the second motion actually has.
- Check Eastward (x) component and Northward (y) component to see the two velocities summed axis by axis before they are recombined.
- Read Resultant speed and Resultant direction for the single combined velocity, switching either field's unit to km/h or radians as needed.
Worked example — a swimmer and a current, both at 10 m/s
A swimmer strikes out due east at 10 m/s through still water, while the current carries everything past her due north at 10 m/s — the textbook mover-plus-medium setup. Enter First velocity, magnitude = 10 m/s, First velocity, direction = 0°, Second velocity, magnitude = 10 m/s, Second velocity, direction = 90°.
Eastward (x) component works out to 10·cos0° + 10·cos90° = 10 m/s, and Northward (y) component works out to 10·sin0° + 10·sin90° = 10 m/s — the current adds nothing eastward and the stroke adds nothing northward, so each axis keeps exactly one live term. Resultant speed is √(10² + 10²) = 14.1421356237 m/s, which is 10√2, and Resultant direction is atan2(10, 10) = 0.785398163397 rad, exactly 45° — the angular midpoint, but only because the two speeds happen to match.
Questions
Why can't I just add the two speeds together?
Because speed alone throws away direction, and direction is exactly what changes the answer. Two 10 m/s velocities at a right angle add to 14.14 m/s, not 20 m/s — simple addition only gives the right answer when both vectors point the same way. The component method, splitting each vector into east and north pieces before using Pythagoras, is what handles every other case correctly.
What does an angle of 0 degrees mean here?
Zero degrees points due east on this instrument's convention, which measures angles counter-clockwise from the positive x-axis — the same axis the Eastward (x) component field sums along. Ninety degrees is due north, 180° is due west, and 270° (or −90°) is due south; negative angles and angles past 360° are accepted, since sine and cosine handle them without any adjustment.
Is the resultant direction always halfway between the two input angles?
Only when the two speeds are equal, as in the 10-and-10 m/s example above, where 0° and 90° average to exactly 45°. Make one vector faster — say 15 m/s east and 5 m/s north — and the resultant leans toward the faster one, landing near 18.4°, well short of the 45° midpoint. Resultant direction is a weighted combination, not a simple average, which is why the atan2 step matters rather than eyeballing an angle.
Does this formula work for velocities close to the speed of light?
No — this is Galilean, classical vector addition, valid for the everyday speeds of swimmers, boats, cars, and aircraft. As speeds approach a meaningful fraction of light speed, the plain sum vₓ = v₁cosθ₁ + v₂cosθ₂ overstates the true resultant, and Einstein's relativistic velocity-addition formula is required instead. Below that regime, the classical formula used here is accurate to far more digits than any real measurement could confirm.
What are the Eastward (x) component and Northward (y) component fields for?
They show the intermediate step the formula depends on: each input velocity resolved into its east-west and north-south pieces before the two are summed. Checking them catches entry mistakes — if a velocity meant to point due north shows a nonzero eastward component, the direction field was likely entered in the wrong unit or off by 90°.
Can this instrument combine three or more velocities at once?
Not directly — it sums exactly two velocity vectors. For a third contributor, such as a diagonal wind added to an aircraft's airspeed and a crosswind correction, resolve that third vector into its own eastward and northward pieces by hand, add them to the Eastward (x) component and Northward (y) component results, then take the magnitude and angle of that combined total.