Resultant Velocity Calculator

Related Calculators

Velocity (vector): A quantity with both speed and direction, so you add it using vector addition, not regular number addition.[1]

Component (x, y): The horizontal part (x) and vertical part (y) of a vector. You add x parts together and y parts together.[1]

Resultant velocity: The single velocity vector that equals the sum of all the velocity vectors you entered.[2]

Magnitude (speed): The size of the velocity vector (always 0 or positive here).[2]

Direction convention: The rule for what 0 degrees means and which way angles increase (Math vs Compass). The calculator still uses x = right and y = up for components.

Unit direction components (u_x, u_y): The direction-only version of the resultant split into x and y parts. It is unitless and only exists when the magnitude is not 0.

Mistake: Entering compass bearings (0 = North) while Direction convention is set to Math (0 = +x, counterclockwise).
Fix: Match the setting to your problem statement: choose Compass for N/E/S/W style bearings, or choose Math for standard x-y diagrams.

Mistake: Leaving a required angle blank in Magnitude and angle mode for a vector you are using.
Fix: Check Input type and then recalculate. Enter a direction in degrees for every enabled vector (0 is allowed), or switch to Components mode if you only know x and y.

Mistake: Typing a negative value for Velocity 1 magnitude (or any magnitude) to try to flip the direction.
Fix: Check Input type and then recalculate. Keep magnitudes non-negative and change the direction angle, or use Components mode and enter negative x/y values to represent left/down.

Mistake: Setting Number of velocity vectors (2 to 5) to 3 to 5 but not entering values for the added vectors, so you accidentally add zeros (or blanks) and get an unexpected result.
Fix: Either fill in each extra enabled vector, or set the vector count back to the number you actually have.

Mistake: Expecting a direction when the resultant is the zero vector (both resultant x and y are 0).
Fix: Check Input type and then recalculate. Report magnitude 0 and no direction. The calculator should show direction and unit direction components as N/A in this case.

Mistake: Mixing speed units across vectors (for example, one in m/s and another in km/h) and trusting the combined magnitude.
Fix: Check Input type and then recalculate. Convert all inputs to the same unit first, then add them.

2D only: This calculator adds vectors in an x-y plane (x is right, y is up). It does not include any z-component.

No unit conversion: You can use any speed unit, but all vectors must use the same unit. If you mix units, the outputs will not be meaningful.

Components keep the same axes even in Compass mode: Choosing Compass only changes how angles are interpreted and displayed. The x/y components are always based on x = right and y = up.

Zero resultant has undefined direction: If resultant x = 0 and resultant y = 0, the magnitude is 0 and the direction cannot be defined, so direction and unit direction components should show N/A. Very small rounding can make a near-zero result look like a tiny nonzero value.

Magnitude-and-angle uses non-negative magnitudes: Negative direction effects should be represented by changing the angle or by using negative x/y components in Components mode.

Rounding and display choices can change what you see: Fixed decimals can round small components to 0. If you need to see tiny values, increase decimals or switch the display format to Scientific.

The calculator adds velocity vectors by converting each one to x and y components (if needed), summing components, then converting the sum back into magnitude and direction.[1][2]

1) Convert each vector to components (only in Magnitude and angle mode)

First, the direction is treated as degrees and converted to radians for trig. Then cosine gives the x-part and sine gives the y-part (using the selected direction convention, internally aligned to the standard math axes).[1]

θ (degrees) to radians: θrad = θ * (π/180)

vx,i = vi * cos(θrad)

vy,i = vi * sin(θrad)

If you enter components, the calculator uses your inputs directly as v_x,i and v_y,i.

2) Add components

vx,res = sum of vx,i for i = 1..N

vy,res = sum of vy,i for i = 1..N

Only vectors 1 through N are included, where N is your Number of velocity vectors (2 to 5).

3) Convert the summed components to magnitude (speed)

|vres| = sqrt( vx,res2 + vy,res2 )

4) Find direction with correct quadrant (atan2)

Direction uses atan2 so the quadrant is correct (for example, v_x,res negative and v_y,res positive points up-left, not up-right).[1]

θmath (radians) = atan2(vy,res, vx,res)

θmath (degrees) = θmath * (180/π)

If you selected Compass convention, the calculator converts between math angle and bearing like this:

θcompass = mod(90 - θmath, 360)

Finally, it maps the displayed angle into your chosen Output angle range (0 to 360 or -180 to +180).

5) Unit direction components (only if magnitude is not 0)

ux = vx,res / |vres|

uy = vy,res / |vres|

Mini example (check by hand)

Using Math convention with 2 vectors: v1 = 10 at 0 degrees and v2 = 10 at 90 degrees. Components add to v_x,res = 10 and v_y,res = 10, so |v_res| = sqrt(10^2 + 10^2) = 14.142... and direction = 45 degrees.[2]