Relative Velocity Calculator With Angle

Calculation History (for CSV and PDF export)

Each calculation is added to the history table. Export the table as CSV or PDF for documentation and reporting.

Example Data Table

Use these sample values to understand how the relative velocity and direction behave for different magnitudes and angles.

Formula Used for Relative Velocity With Angle

Relative velocity in two dimensions is treated as a vector difference between the velocity vectors of two objects. Each vector is defined by a magnitude and an angle.

For Object A and Object B, we define magnitudes vA, vB and angles θA, θB measured from the positive x-axis.

• Components of Object A: vAx = vA · cos(θA), vAy = vA · sin(θA)
• Components of Object B: vBx = vB · cos(θB), vBy = vB · sin(θB)

Velocity of A relative to B is:

→vAB = →vA − →vB

• vABx = vAx − vBx
• vABy = vAy − vBy

The magnitude of the relative velocity is:

|vAB| = √(vABx2 + vABy2)

The direction angle of the relative velocity measured from the positive x-axis is given by:

φ = atan2(vABy, vABx) converted to degrees. The calculator automatically adjusts the angle to the correct quadrant using the two-argument arctangent function.

Velocity of B relative to A is obtained by reversing the order of subtraction: →vBA = →vB − →vA, which simply flips the relative vector direction.

How to Use This Relative Velocity Calculator

1. Choose a consistent velocity unit from the dropdown list, for example metres per second or kilometres per hour.
2. Enter the magnitude of Object A and its direction angle measured from the positive x-axis (usually the horizontal direction in your diagram).
3. Enter the magnitude of Object B and its direction angle using the same sign convention and reference axis as Object A.
4. Select whether you want the velocity of A relative to B or the velocity of B relative to A from the relative type menu.
5. Click the Calculate Relative Velocity button to compute components, angle difference, and the magnitude and direction of the relative velocity vector.
6. Review the detailed results panel and the history table to compare multiple scenarios or parameter changes for your problem.
7. When your table is ready, use the CSV or PDF download buttons to export the history for lab reports, homework solutions, or engineering documentation.

Relative Velocity With Angle: Detailed Explanation

Understanding relative velocity in two dimensions

Relative velocity describes how fast and in what direction one object appears to move when observed from another moving object. In two dimensions, both magnitude and direction matter, so we treat velocities as vectors instead of simple signed numbers along a line. This is essential whenever observers or instruments are themselves moving, which happens in many real-world experiments and applications.

Role of angles and vector components

Every velocity is defined by a magnitude and an angle from a reference axis. The calculator converts each vector into horizontal and vertical components using cosine and sine. This lets you combine, subtract, and compare motions that are not aligned along the same straight line.

Comparing motion between different reference frames

Choosing whether you want A relative to B or B relative to A changes the subtraction order. This represents changing the reference frame. The result shows how motion looks for an observer riding with one object and watching the other move across the plane.

Using the calculator in physics education

Students often struggle with converting between magnitudes, angles, and components. By entering trial values and observing the results panel, learners can see exactly how changing an angle or speed reshapes the relative vector, reinforcing classroom diagrams and textbook examples through interactive exploration.

Applications in navigation and transport problems

Relative velocity with angles appears in aircraft crosswind problems, river boat navigation, ship collision avoidance, and car overtaking scenarios at road junctions. The calculator helps visualise how headings, turning angles, and speed adjustments influence closing speed, lateral drift, and time to meet or separate. Changing just a few degrees of heading can significantly alter the resulting ground track, especially when environmental flows are comparable to vehicle speed.

Analysing experimental and simulation data

When you collect motion data from tracking software or sensors, the measurements often include direction information. You can feed those values into this calculator to compute relative motion between particles, vehicles, or experimental carts and quickly compare theoretical predictions with measured vector results.

Practical tips for reliable calculations

Keep angles measured from the same axis, typically the positive x-direction, and use a consistent unit for all speeds. Verify values by sketching a quick vector diagram. If the relative direction seems counterintuitive, adjust angles or reference frames until the numerical output matches your physical understanding.

Frequently Asked Questions

What is relative velocity with angle?

Relative velocity with angle compares two motions in a plane, using both speed and direction. Each velocity is treated as a vector, and the calculator subtracts them to show how one object appears to move when viewed from the other.

Which angle convention should I use here?

Measure all angles from the same reference direction, typically the positive x-axis pointing right, and increase them counterclockwise. Using a consistent convention for both objects ensures that component calculations and relative directions match your diagrams and standard trigonometric definitions.

Why do some relative velocity components appear negative?

Negative components simply mean the relative motion points opposite to the chosen positive axis. A negative x-component indicates leftward relative motion, while a negative y-component indicates downward motion, according to the reference axes you used for defining the original velocity vectors.

What is the difference between A relative to B and B relative to A?

Velocity of A relative to B shows how A moves as observed from B. Velocity of B relative to A reverses the subtraction order, giving a vector of the same magnitude but opposite direction, representing B’s motion from A’s moving reference frame.

Can I mix different units for the two velocities?

No. Both velocities must use the same unit, such as metres per second, kilometres per hour, or miles per hour. The unit selector only labels results, so always convert speeds beforehand if your original data comes in mixed or incompatible units.

Does this calculator include acceleration or changing speeds?

This tool assumes constant velocities during the interval considered, so it does not directly model acceleration, turning dynamics, or curved paths. However, you can analyse snapshots at different times by entering updated velocities taken from simulation outputs or experimental measurements.