Calculate resultant velocity from vector addition of object and medium velocities. Boat crossing river, airplane in wind scenarios.
The **Resultant Velocity Calculator** determines the actual velocity of an object moving through a medium — such as a boat crossing a river, an airplane flying in wind, or a swimmer in ocean current. By adding the object's velocity vector to the medium's velocity vector, you get the true speed and direction of travel.
This is the classic relative motion problem from introductory physics. A boat heading straight across a river at 5 m/s encounters a 3 m/s current flowing downstream. The boat doesn't reach the opposite bank directly across — it drifts downstream. The actual path, speed, and drift angle all depend on the vector sum of the two velocities.
The calculator supports multiple speed units (m/s, km/h, mph, knots) and includes presets for common scenarios. The visual velocity diagram shows all three vectors: object, medium, and resultant. A travel time comparison table shows how the medium velocity affects travel time over various distances. This is indispensable for navigation, river crossing problems, and aviation wind correction.
Anytime an object moves through a flowing or moving medium, the actual path differs from the intended heading. Pilots must calculate wind correction angles to reach their destination; sailors must account for current and tide; even hikers crossing a river need to aim upstream to avoid being swept past their target.
The travel time comparison is particularly useful for planning: it shows how much time you gain or lose due to the medium velocity. A tailwind shortens flight time dramatically; a headwind extends it. Understanding these effects is essential for fuel planning, navigation, and safety.
Vector addition: Vx = V_obj × cos(θ_obj) + V_med × cos(θ_med) Vy = V_obj × sin(θ_obj) + V_med × sin(θ_med) V_resultant = √(Vx² + Vy²) θ_resultant = arctan(Vy / Vx) Drift angle = θ_resultant − θ_obj Variables: V = velocity, θ = direction angle (° CCW from east)
Result: 5.83 m/s at 59.0°
A boat heading north (90°) at 5 m/s in a 3 m/s eastward (0°) current. Vx = 0 + 3 = 3 m/s, Vy = 5 + 0 = 5 m/s. V_result = √(9+25) = 5.83 m/s. Direction = arctan(5/3) = 59°. The boat drifts 31° east of its intended north heading.
The most classic application is a boat crossing a river. The boat has its own velocity relative to the water, and the river has a current velocity relative to the ground. The boat's actual motion relative to the ground is the vector sum. Two key questions arise: what is the minimum time to cross (head straight across, accept drift), and how must the boat aim to cross straight to a point directly opposite (aim upstream, accept longer time)?
For minimum crossing time, the boat heads perpendicular to the bank. The crossing time depends only on the boat speed and river width — the current doesn't affect crossing time, only drift. For a straight crossing, the boat aims upstream at an angle determined by arcsin(V_current/V_boat) — but this only works if the boat is faster than the current.
Pilots call this the "wind triangle" — heading + wind = track. The three sides represent: - **Heading/Airspeed**: Where the plane points and how fast it flies through the air - **Wind**: Direction and speed of the wind aloft - **Track/Ground speed**: The actual path and speed over the ground
Flight planning requires solving this triangle for every leg of a flight to determine fuel consumption, timing, and the heading to fly. Modern avionics do this automatically, but every pilot learns the manual computation.
This calculator uses Galilean (classical) velocity addition where V_result = V1 + V2. At everyday speeds, this is perfectly accurate. At speeds approaching the speed of light, relativistic velocity addition applies: V_result = (V1 + V2)/(1 + V1·V2/c²). For all practical navigation and engineering problems, classical addition is correct.
This IS relative velocity — the resultant velocity is the object velocity relative to the ground (fixed frame). The object moves relative to the medium, and the medium moves relative to ground. Adding these gives ground velocity.
The object can never reach certain directions. A boat in a 5 m/s current that can only go 3 m/s through water will always drift downstream. It cannot travel straight upstream. The drift angle exceeds 90° for certain headings.
Pilots compute the wind correction angle (WCA) to maintain their desired ground track. They set the heading to compensate for crosswind, and compute ground speed to determine fuel requirements and estimated time of arrival.
Small streams: 0.3-0.5 m/s. Medium rivers: 0.5-1.5 m/s. Large rivers like the Mississippi: 1-2 m/s. Tidal estuaries: 1-3 m/s. Flood conditions can exceed 5 m/s.
This calculator handles two velocity vectors (object + medium). For three or more, use the resultant force calculator with the same vector addition principle — the math is identical for velocity vectors.
1 m/s = 3.6 km/h = 2.237 mph = 1.944 knots. Select your preferred unit from the dropdown and all outputs will convert automatically.