Resultant Velocity Calculator adds up to five velocity vectors and gives the size and direction of the resultant, with its x and y parts. It also handles a boat crossing a current.
Angles run counterclockwise from east. More vectors are under the section below.
More Vectors and Units
Step-by-Step Working
- Known values in SI units: 2 velocities
- Vector 1: 12 m/s at 25° gives x = 10.88, y = 5.07
- Vector 2: 8 m/s at 115° gives x = −3.38, y = 7.25
- Add the parts: Σvx = 7.49 m/s, Σvy = 12.32 m/s
- Size: √(7.49² + 12.32²) = 14.42 m/s
- Direction: tan⁻¹(Σvy ÷ Σvx), placed in the right quadrant = 58.69°
- Result: 14.42 m/s at 58.69°
Parts of Each Velocity
| Velocity | x Part | y Part |
|---|---|---|
| Vector 1 | 10.88 m/s | 5.07 m/s |
| Vector 2 | −3.38 m/s | 7.25 m/s |
| Resultant | 7.49 m/s | 12.32 m/s |
Vector Diagram
What the Resultant Velocity Calculator Adds
In physics, the resultant velocity is the single velocity that has the same effect as two or more velocities acting together. A boat pushed by its engine and by a current has one resultant velocity over the ground. Velocities are vectors, so they must be added with their directions, not as plain numbers.
The Resultant Velocity Calculator has four settings under Combine. Velocities With Directions adds up to five vectors, each with a speed and an angle.
Two Speeds and Their Angle handles the common two-vector problem. The x and y Parts setting builds a velocity from its components, and Boat Crossing a Current solves river problems.
It opens with 12 m/s at 25° and 8 m/s at 115°. Their resultant is 14.42 m/s at 58.69°, which is a compass bearing of 31.31°. The cards also show the x and y parts, how much speed the directions cancel, and the result in km/h, mph and knots.
Resultant Velocity Formula
To calculate a resultant velocity, split every velocity into an x part and a y part. A velocity of size v at angle θ, measured counterclockwise from the x-axis, has these two parts. East is 0° and north is 90°.
$$v_x = v \cos\theta$$
$$v_y = v \sin\theta$$
Add all the x parts together and all the y parts together. The size of the resultant then comes from the Pythagorean theorem, and its direction from the inverse tangent. OpenStax College Physics uses this same method for adding velocities.
$$v = \sqrt{\left(\sum v_x\right)^2 + \left(\sum v_y\right)^2}$$
$$\theta = \tan^{-1}\left(\frac{\sum v_y}{\sum v_x}\right)$$
The table works the opening example through, as the parts table in the Resultant Velocity Calculator does. The second velocity points partly west, so its x part is negative and takes away from the first.
| Velocity | x part | y part |
|---|---|---|
| 12 m/s at 25° | 10.88 m/s | 5.07 m/s |
| 8 m/s at 115° | −3.38 m/s | 7.25 m/s |
| Resultant | 7.49 m/s | 12.32 m/s |
The square root of 7.49² plus 12.32² is 14.42 m/s. The inverse tangent of 12.32 ÷ 7.49 is 58.69°. The Resultant Velocity Calculator places the angle in the correct quadrant, which a plain inverse tangent on a hand calculator does not always do. The two speeds add up to 20 m/s, so 5.58 m/s is lost to their directions.
Resultant Velocity of Two Vectors
When only two velocities are involved and the angle between them is known, there is a direct resultant velocity formula with the angle. It is the law of cosines, with θ as the angle between the two arrows drawn from one point.
$$v = \sqrt{v_1^2 + v_2^2 + 2 v_1 v_2 \cos\theta}$$
The table runs 15 m/s and 25 m/s through the Resultant Velocity Calculator at different angles. The resultant runs from the sum of the two speeds, when they point the same way, down to their difference, when they point opposite ways.
| Angle between them | Resultant velocity | Angle from the first |
|---|---|---|
| 0° | 40 m/s | 0° |
| 30° | 38.72 m/s | 18.83° |
| 60° | 35 m/s | 38.21° |
| 90° | 29.15 m/s | 59.04° |
| 120° | 21.79 m/s | 83.41° |
| 150° | 14.16 m/s | 118.02° |
| 180° | 10 m/s | 180° |
A well-known example is a pair of 100 km/h velocity vectors at right angles to each other. Their resultant is 141.42 km/h, at 45° to each. At right angles the cosine term is zero, so the formula becomes the Pythagorean theorem.
Velocity From vx and vy
To find a velocity given vx and vy, square each part, add them and take the square root. Parts of 3 m/s and 4 m/s give 5 m/s at 53.13° from the x-axis. The x and y Parts setting of the Resultant Velocity Calculator also takes a z part, so 2, 3 and 6 m/s give 7 m/s.
$$v = \sqrt{v_x^2 + v_y^2 + v_z^2}$$
The resultant velocity formula in projectile motion is the same line. The horizontal part stays fixed and the vertical part changes with time. For a launch at 25 m/s and 40°, the parts after 1.2 s are 19.15 m/s and 4.3 m/s. The resultant is 19.63 m/s, at 12.66° above level, which matches the Projectile Motion Calculator.
Adding the parts as plain numbers always overstates the speed. For 3 and 4 m/s, simple addition gives 7 m/s against the true 5 m/s. The same vector rule applies to accelerations, which the Magnitude of Acceleration Calculator handles.
Resultant of a Boat and a Current
The Boat Crossing a Current setting opens with the OpenStax example. A boat aims straight across a river at 0.75 m/s while the current runs at 1.20 m/s. The textbook finds a total velocity of 1.42 m/s at 32.0° to the bank, and the Resultant Velocity Calculator returns the same figures.
It then adds what the crossing looks like. On a river 60 m wide, the boat takes 80 s to cross and lands 96 m downstream. The crossing time depends only on the speed straight across. The current does not slow the crossing. It only moves the landing point.
| Heading upstream | Time to cross | Drift downstream |
|---|---|---|
| 0° | 80 s | 96 m |
| 10° | 81.23 s | 86.9 m |
| 20° | 85.13 s | 80.32 m |
| 30° | 92.38 s | 76.21 m |
| 45° | 113.14 s | 75.76 m |
| 60° | 160 s | 88.08 m |
This boat is slower than the current, so no heading takes it straight across. The least drift comes at 38.68° upstream. A faster boat can cancel the current. At 2 m/s it must point 36.87° upstream, and it then crosses at 1.6 m/s in 37.5 s and lands directly opposite.
An aircraft in a crosswind is the same problem with different names. A plane flying north at 200 knots through a 40 knot wind blowing toward the east moves over the ground at 203.96 knots. Wind correction angles for a planned track are worked in the Ground Speed Calculator.
Resultant Velocity Questions
What is the unit of resultant velocity?
The SI unit of velocity is the meter per second, and a resultant velocity has the same unit as the velocities added. All of them must be in one unit before adding. The Resultant Velocity Calculator lets each entry use its own unit and converts first. Its opening result of 14.42 m/s is 51.92 km/h, 32.26 mph or 28.03 knots.
What is the difference between speed and velocity?
Speed is how fast something moves. Velocity is speed with a direction, which makes it a vector. Two cars at 60 mph have the same speed, but if one heads north and one heads east they have different velocities. The resultant speed is simply the size of the resultant velocity.
What are the basic formulas for velocity?
Three cover most problems. Average velocity is displacement divided by time. With steady acceleration, final velocity is initial velocity plus acceleration times time. When time is not known, final velocity squared equals initial velocity squared plus twice the acceleration times the distance.
Two others come up often. A point on a rotating object moves at its angular velocity times its radius, and the flow velocity in a pipe is the flow rate divided by the pipe’s cross-section area. Displacement over several legs is covered by the Displacement Calculator.
Is a resultant force calculated the same way?
A resultant force is calculated the same way. It is the vector sum of the forces, found by the same component method. Enter the forces in place of the velocities and read the answer in newtons. The method works for any vectors of one kind, including displacements and momenta.