Relative Velocity Calculator

Calculate relative velocity in 1D and 2D, solve riverboat and airplane wind problems, and visualize motion from the ground frame or moving observer frame.

Background

Relative velocity describes how fast one object appears to move from another object's point of view. The key idea is frame of reference: the same motion can look different from the ground, from a moving car, from a boat in a river, or from an airplane moving through wind.

Calculate relative velocity

Choose problem type

Start with 1D for cars, trains, or moving walkways. Use riverboat or airplane mode for classic 2D vector problems.

Frame and unit settings

View frame

The animation changes to show what motion looks like from each frame.

Speed unit

Angle convention

Solve mode

Use normal mode to calculate v_A/B. In 1D and 2D modes, you can also solve for a missing A velocity, B velocity, or relative velocity.

Solve for

For riverboat and airplane modes, normal vector-sum mode is used.

1D relative motion

Positive direction is to the right. Use the solve mode above to calculate a missing value when needed.

Velocity of object A

Velocity of observer B

Known relative velocity v_A/B

Used only when solving for vA or vB.

Optional starting separation

Used to estimate meet/pass/overtake time.

Distance unit

Optional time for animation

Used to show displacement over time.

Riverboat problem

Boat velocity is usually relative to the water. River current is relative to the ground. Their vector sum gives ground velocity.

Boat speed relative to water

Boat heading angle

Use 90° for straight across the river in math angle mode.

River current speed

River current direction angle

Optional river width

Used for time to cross and downstream drift.

Width unit

Airplane + wind problem

Airspeed is the plane's velocity relative to the air. Wind is the air's velocity relative to the ground. Their vector sum gives ground velocity.

Airspeed

Plane heading angle

Wind speed

Wind direction angle

Optional desired course angle

Used to estimate required heading to stay on course.

2D vector relative velocity

Enter components for any two known vectors. The solve mode can find v_A/B, v_A, or v_B.

A x-component

A y-component

B x-component

B y-component

Relative x-component

Relative y-component

Animation controls

Supported formats

30-122.51D signsx/y componentsangles in degreesriver currentwind vector

Use negative values in 1D when an object moves opposite the positive direction.
For vector modes, angles are measured from the +x axis unless compass bearing mode is selected.
Relative velocity is about viewpoint: v_A/B means “velocity of A as seen by B.”
The animation is scaled for learning, so it shows direction and relative motion rather than exact screen distance.

Quick examples (click to see result)

Result

No result yet. Enter motion values or choose a quick example, then click Calculate Relative Velocity.

How to use this calculator

Choose a problem mode: 1D motion, riverboat, airplane and wind, or 2D vector relative velocity.
Enter the velocities, directions, or vector components requested by that mode.
Use solve mode to calculate a missing velocity in 1D or 2D vector problems.
Choose the frame view: ground frame or moving observer frame.
Click Calculate Relative Velocity.
Use the animation and vector breakdown to understand what changes when the observer is moving.

How this calculator works

The calculator converts all speeds to a common internal unit.
For 1D motion, it subtracts the observer velocity from the object velocity.
For 2D motion, it subtracts x-components and y-components separately.
For riverboat problems, it adds boat velocity relative to water and river current relative to ground.
For airplane problems, it adds airspeed and wind velocity to get ground velocity.
It reports magnitude, direction angle, components, special outputs, and a student-friendly interpretation.

Formula & Equations Used

Relative velocity: v_A/B = v_A − v_B

Solve for A: v_A = v_A/B + v_B

Solve for B: v_B = v_A − v_A/B

2D components: v_A/B,x = v_A,x − v_B,x, v_A/B,y = v_A,y − v_B,y

Magnitude: |v| = √(v_x² + v_y²)

Direction angle: θ = atan2(v_y, v_x)

Riverboat ground velocity: v_boat/ground = v_boat/water + v_water/ground

Airplane ground velocity: v_plane/ground = v_plane/air + v_air/ground

Example Problems & Step-by-Step Solutions

Example 1: Two cars moving in the same direction

Car A moves at 30 m/s and car B moves at 20 m/s in the same direction.

The relative velocity of A as seen by B is v_A/B = 30 − 20 = 10 m/s.

So from car B, car A appears to move forward at 10 m/s.

Example 2: Boat crossing a river

A boat moves across the river while the river current flows downstream.

The boat's velocity relative to water and the river current are added as vectors.

If river width is given, the calculator also estimates crossing time and downstream drift.

Example 3: Plane flying through wind

An airplane's airspeed tells how fast it moves relative to the air, not the ground.

To find ground velocity, add the wind vector to the plane's airspeed vector.

The calculator can also estimate the heading needed to maintain a desired course.

Common mistakes to avoid

Do not forget the order: v_A/B means velocity of A relative to B, so subtract B from A.
Do not add magnitudes when the velocities point in different directions. Add or subtract vector components.
In riverboat problems, do not confuse velocity relative to water with velocity relative to ground.
In airplane problems, do not confuse heading, airspeed, wind, and ground velocity.
Watch signs in 1D motion. Opposite directions should use opposite signs.

Frequently Asked Questions

What is relative velocity?

Relative velocity is the velocity of one object as measured from another object's frame of reference.

Why do we subtract velocities?

Changing to a moving observer's frame means removing the observer's own velocity. That is why v_A/B = v_A − v_B.

How do riverboat problems use relative velocity?

The boat's velocity relative to water combines with the water's velocity relative to ground. The vector sum gives the boat's velocity relative to ground.

How does wind affect airplane velocity?

The plane's airspeed is relative to the air. Wind is the air's motion relative to the ground, so the ground velocity is the vector sum of airspeed and wind.