Resultant Velocity Calculator

Resultant velocity is the vector sum of all velocity vectors acting on an object: v_res = v1 + v2 + ... + vn. In components: vx_res = v1*cos(theta1) + v2*cos(theta2) + ... , vy_res = v1*sin(theta1) + v2*sin(theta2) + ... . The magnitude is |v_res| = sqrt(vx_res^2 + vy_res^2) and the direction is theta = atan2(vy_res, vx_res), measured counter-clockwise from the +x axis. This calculator adds up to 10 velocity vectors, in components or magnitude-plus-angle form.

Take v1 = 3 m/s at 30 degrees and v2 = 6 m/s at 60 degrees. Step 1, resolve into components: vx1 = 3*cos(30 deg) = 2.59808 m/s, vy1 = 3*sin(30 deg) = 1.50000 m/s; vx2 = 6*cos(60 deg) = 3.00000 m/s, vy2 = 6*sin(60 deg) = 5.19615 m/s. Step 2, sum the components: vx_res = 2.59808 + 3.00000 = 5.59808 m/s, vy_res = 1.50000 + 5.19615 = 6.69615 m/s. Step 3, magnitude: |v_res| = sqrt(5.59808^2 + 6.69615^2) = 8.7279 m/s. Step 4, direction: theta = atan2(6.69615, 5.59808) = 50.10 degrees. Entering these two vectors in this calculator gives exactly 8.7279 m/s at 50.10 degrees.

Angles here use standard position: measured counter-clockwise from the positive x-axis, in the range (-180 deg, 180 deg]. A plain arctan(vy/vx) loses the quadrant: (vx,vy) = (1,1) and (-1,-1) both give arctan(1) = 45 deg, but the second vector points at -135 deg (225 deg). The two-argument atan2(vy, vx) keeps the signs and always returns the correct quadrant, which is why this calculator (and every physics text) uses atan2. For navigation, the compass bearing (clockwise from north) is also shown.

Yes. Each vector row has its own input-type switch: enter (vx, vy) components directly, or a magnitude with an angle in standard position. The two forms are equivalent: vx = v*cos(theta), vy = v*sin(theta), and conversely v = sqrt(vx^2 + vy^2), theta = atan2(vy, vx). A 10 m/s vector at 30 deg is exactly the same as (8.6603, 5.0000) m/s, and this calculator gives identical results either way.

The velocity of A relative to B is v_A/B = v_A - v_B: subtract the observer's velocity vector from the object's velocity vector, component by component. Example: a swimmer crosses a river heading north at 2 m/s while the current flows east at 1.5 m/s. Relative to the bank, v = (1.5, 2.0) m/s, so the ground speed is sqrt(1.5^2 + 2^2) = 2.5000 m/s at atan2(2, 1.5) = 53.13 degrees. If A and B move identically, the relative velocity is zero.

Yes. When the vectors cancel, the resultant is the zero vector. Two 10 m/s velocities at 0 deg and 180 deg sum to (0, 0) m/s with no defined direction. Physically this means the contributions balance: e.g. a boat motoring upstream at exactly the current's speed stays still relative to the bank. The calculator shows a magnitude of 0 and reports the direction as undefined instead of inventing an angle.

Speed is a scalar: just how fast, e.g. 50 m/s. Velocity is a vector: how fast plus in which direction, e.g. 50 m/s at 53.13 deg north of east. Resultant velocity is the single vector equivalent to several simultaneous velocities: 30 m/s east plus 40 m/s north gives a resultant of sqrt(30^2 + 40^2) = 50 m/s at atan2(40, 30) = 53.13 deg - the 3-4-5 triangle. Average velocity is different again: total displacement divided by total time. Adding speeds as scalars (30 + 40 = 70) is wrong whenever directions differ.

The vector mathematics is identical - both add vectors tip-to-tail - but the physical quantities differ: velocity is a kinematic quantity in m/s, force is a dynamic quantity in newtons. This tool computes resultants of velocities only. For adding forces, use the companion net-force calculator, which sums force vectors and reports the result in N, kN or lbf.

Metres per second (m/s), kilometres per hour (km/h), feet per second (ft/s), miles per hour (mph), and knots. Conversions are exact definitions: 1 m/s = 3.6 km/h = 2.23694 mph = 3.28084 ft/s = 1.94384 kn. For example 10 mph = 4.4704 m/s exactly (10 x 0.44704). All vectors in one calculation share the selected unit; results are shown in that unit.

This calculator uses classical (Galilean) vector addition, which is exact for all everyday speeds where v is far below the speed of light c; relativistic addition (Einstein's formula) only matters near c and is not included. For aviation wind triangles - true airspeed plus wind giving ground speed and wind-correction angle - use the dedicated ground-speed calculator, which solves the full E6B wind triangle in both directions.