PhysicsSolver360 logoPhysicsSolver360°
Powered by Solver360°

Relative Velocity Calculator

Find relative speed between two objects: same direction, opposite direction, or at an angle

Parameters

m/sⓘ
m/sⓘ
°ⓘ
Angle between the two velocity directions (0° = same direction)
Show Trail

Controls

xⓘ

Calculated Values

1D Relative Speed (|v₁−v₂|):
5.00;m/s5.00;m/s
2D Relative Speed:
5.00;m/s5.00;m/s
Head-on (θ=180°):
35.00;m/s35.00;m/s

Examples

Same direction

20 m/s and 15 m/s same way.

  • 1D Relative Speed: 5.005.00

Head-on

20 m/s and 15 m/s opposite.

  • 2D Relative Speed: 35.0035.00

Visualization

Relative Velocity and Reference Frames

Relative velocity answers: how fast does object A move as seen by object B? Vector definition: v_AB = v_A − v_B (velocity of A relative to B).

On a straight line with the same direction convention: speed of A relative to B is |v_A − v_B|. Same speed same way → relative speed 0.

Head-on motion (opposite directions): relative speed = v_A + v_B. Classic train and river-boat problems use 1D relative velocity.

In 2D, subtract velocity vectors and take magnitude: |v_12| = √(v₁² + v₂² − 2v₁v₂ cos θ) where θ is the angle between velocity directions (law of cosines).

Galilean relativity (non-relativistic): add the observer’s velocity to convert between frames. Not valid near the speed of light.

Key Concepts

  • v_AB = v_A − v_B (vector subtraction)
  • 1D same direction: |v₁ − v₂|
  • 1D opposite: v₁ + v₂
  • 2D magnitude: law of cosines on velocities
  • Frame-dependent: not an absolute quantity
  • Galilean transformation (v << c)

Real-World Applications

  • River boat and swimmer problems
  • Trains passing each other
  • Aircraft wind correction angles
  • Collision approach speeds
  • Competitive exam relative motion numericals

Explore Further

More mechanics tools

Physics Equations

1D Relative Speed:
vrel=∣v1−v2∣ (same line)v_{rel} = |v_1 - v_2| \text{ (same line)}
2D Magnitude:
vrel=v12+v22−2v1v2cos⁡θv_{rel} = \sqrt{v_1^2 + v_2^2 - 2v_1 v_2 \cos\theta}

Step-by-Step Solution

See how the main results are calculated.

1

One-Dimensional Relative Speed

When both velocities lie on the same line (θ = 0° or 180°):

Equation:

vrel,1D=∣v1−v2∣ (same direction, same sign convention)v_{rel,1D} = |v_1 - v_2| \text{ (same direction, same sign convention)}

Calculation:

∣v1−v2∣=∣20−15∣=5.00 m/s|v_1 - v_2| = |20 - 15| = 5.00 \text{ m/s}

Result:

vrel,1D=5.00 m/sv_{rel,1D} = 5.00 \text{ m/s}

Explanation:

Same direction & same speed → relative speed 0. Opposite directions → add speeds: v₁ + v₂.

2

Relative Velocity as Vector Difference

Velocity of object 1 relative to object 2:

Equation:

v⃗12=v⃗1−v⃗2\vec{v}_{12} = \vec{v}_1 - \vec{v}_2

Calculation:

∣v⃗12∣=v12+v22−2v1v2cos⁡θ|\vec{v}_{12}| = \sqrt{v_1^2 + v_2^2 - 2v_1 v_2 \cos\theta}

Result:

Use magnitude below

Explanation:

Relative velocity depends on the observer’s frame. Galilean transformation: add the observer’s velocity to convert frames.

3

Calculate Magnitude (Law of Cosines)

With θ = 0° between velocity directions:

Equation:

∣v⃗12∣=v12+v22−2v1v2cos⁡θ|\vec{v}_{12}| = \sqrt{v_1^2 + v_2^2 - 2v_1 v_2 \cos\theta}

Calculation:

=202+152−2(20)(15)cos⁡(0°)= \sqrt{20^2 + 15^2 - 2(20)(15)\cos(0°)}
=5.00 m/s= 5.00 \text{ m/s}

Result:

∣v⃗12∣=5.00 m/s|\vec{v}_{12}| = 5.00 \text{ m/s}

Explanation:

θ = 0° (same direction): |v₁ − v₂|. θ = 180° (head-on): v₁ + v₂. θ = 90°: √(v₁² + v₂²).

Frequently Asked Questions (FAQ)

Who moves faster in relative terms?

It depends on the observer’s frame—relative velocity is not a single absolute number.

River boat: boat speed relative to water vs ground?

Ground speed is vector sum of boat velocity relative to water and water velocity relative to ground.

When do we add speeds in 1D?

When objects move in opposite directions along the same line (head-on).

Is relative velocity symmetric?

v_AB = −v_BA; magnitudes are equal but directions oppose.

Practice MCQs

  1. Two cars at 60 km/h same direction: relative speed is:
  2. 20 m/s east and 15 m/s west: relative speed is:
  3. Relative velocity formula uses:

Related Calculators

These tools connect to the same physics concepts used in this calculator.

Projectile MotionCalculate and visualize the path of objects under the influence of gravity.Newton's LawsApply the three laws of motion to solve force and acceleration problems.Circular MotionAnalyze objects moving in circular paths with centripetal acceleration.Conservation of MomentumSolve problems involving collisions and momentum conservation.Error PropagationCombine measurement uncertainties for sums, products, quotients, and powers in experimental formulas.Related Measurement & Uncertainty topic.Ideal Gas LawCalculate pressure, volume, temperature, and moles for ideal gases.Related Thermodynamics topic.Bernoulli's PrincipleCalculate fluid velocities, pressures, and flow rates using Bernoulli's equation.Related Fluid Mechanics topic.