Resultant Velocity Calculator
Add two velocity vectors to find the resultant speed and direction, or work out how a boat crossing a river drifts, how long it takes and where to aim.
➕ What is Resultant Velocity?
Resultant velocity is the single velocity vector you get when two or more velocities act on the same object at once. Velocity has both a size (speed) and a direction, so it is added the way all vectors are added: place the vectors head to tail and draw the arrow from the first tail to the last head, or equivalently add the x components and the y components separately. A boat that points across a river is moving relative to the water, and the water is moving relative to the bank, so the boat's velocity relative to the bank is the vector sum of the two. The same idea describes an aircraft in a crosswind, a swimmer in a current, a walker on a moving walkway and a person running along the deck of a moving train.
This calculator has two modes. The first adds any two velocities that you give as a speed and a direction angle, and reports the resultant components, speed, direction as a math angle and as a compass bearing, and a check on the special cases of parallel, antiparallel and perpendicular vectors. The second mode is the classic river problem: a boat with a given speed relative to the water, a given heading, a river current and a river width. It returns the ground velocity, the time to cross, the downstream drift and the heading that would take the boat straight across, or a clear message when the current is too strong for that.
A few points are easy to get wrong. Speeds never add arithmetically unless the vectors point the same way. The resultant is not the average of the two velocities. Time to cross a river depends only on the part of the boat's velocity that points across the river, so a faster current never makes the crossing slower or faster, it only moves where you land. And the direction you enter must be the direction each velocity points toward: aviation wind is quoted as the direction it comes from, so it must be reversed by 180 degrees before it is entered here. For the aviation wind triangle with true airspeed and a wind correction angle, use the ground speed calculator; for a single speed from distance and time, use the velocity calculator.
The page works in m/s, km/h, mph or knots, using 1 mph = 0.44704 m/s and 1 knot = 1852 / 3600 m/s. Each mode draws the vector triangle to scale and a chart: the resultant speed against the angle between the two vectors, and the downstream drift against the boat heading. The charts mark your current inputs so you can see how sensitive the answer is to the angle.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Two Perpendicular Velocities
3 m/s east (0°) plus 4 m/s north (90°), math angles
Example 2 — Airplane in a Crosswind
240 km/h airspeed heading east (bearing 90°), wind blowing toward bearing 150° at 40 km/h
Example 3 — A Swimmer in a River
Swimmer at 1.2 m/s heading straight across a 50 m river with a 0.5 m/s current
Example 4 — A Boat Across a Current
Boat at 5 m/s heading straight across a 200 m river with a 3 m/s current
Example 5 — Opposite Velocities
A boat making 8 m/s east against a 5 m/s flow toward the west, math angles
Example 6 — Heading Upstream to Land Straight Across
The Example 4 boat (5 m/s, 3 m/s current, 200 m) aimed at 126.87° from the downstream bank
❓ Frequently Asked Questions
🔗 Related Calculators
How do you find the resultant of two velocities?
Resolve each velocity into x and y components, add the components, then rebuild the result. Rx = v1 cos a1 + v2 cos a2 and Ry = v1 sin a1 + v2 sin a2. The resultant speed is the square root of Rx squared plus Ry squared, and the direction is atan2(Ry, Rx).
What is the resultant velocity of two perpendicular velocities?
Use Pythagoras: |R| = sqrt(v1 squared + v2 squared). For 3 m/s and 4 m/s at right angles the resultant is 5 m/s, pointing at atan(4/3) = 53.13 degrees from the 3 m/s direction toward the 4 m/s direction.
What is the resultant of two velocities in the same or opposite direction?
In the same direction the speeds add: 10 m/s plus 10 m/s gives 20 m/s. In opposite directions they subtract: 8 m/s east and 5 m/s west give 3 m/s east. The calculator labels these special cases and checks them against the cosine rule.
What is the cosine rule for adding two velocities?
If theta is the angle between the two velocity vectors, |R| = sqrt(v1 squared + v2 squared + 2 v1 v2 cos theta). It gives 0 degrees as v1 + v2, 90 degrees as Pythagoras and 180 degrees as the difference of the speeds.
How do I convert a compass bearing to a math angle?
Math angle = 90 degrees minus the bearing, wrapped into 0 to 360. North (bearing 0) is 90 degrees, east (bearing 90) is 0 degrees and south (bearing 180) is 270 degrees. The calculator shows the answer both ways.
How long does it take a boat to cross a river?
Only the component of the boat velocity across the river matters. With heading measured from the bank, t = w / (vb sin heading). A 200 m wide river and a 5 m/s boat heading straight across takes 40 s, whatever the current is.
How far does the current carry a boat downstream?
Drift d = (vr + vb cos heading) times t. For a 5 m/s boat heading straight across a 3 m/s current, 200 m wide, d = 3 x 40 = 120 m. The current only changes where you land, never how long the crossing takes.
At what angle should a boat head to cross a river straight across?
Aim upstream of the opposite point by asin(vr / vb) so the upstream part of the boat velocity cancels the current. With vb = 5 m/s and vr = 3 m/s that is 36.87 degrees, and the crossing then takes w / sqrt(vb squared minus vr squared).
What if the river current is faster than the boat?
If vr is at least vb, no heading cancels the current, so you cannot land directly opposite. The calculator reports that straight crossing is not possible. You will always be carried downstream, though you can still reach the far bank by heading across.
How is this different from the ground speed calculator?
The ground speed calculator works the aviation wind triangle with a wind coming FROM a direction and gives a wind correction angle. This one adds any two velocity vectors toward their stated directions, and adds a river crossing mode with drift and crossing time.