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Relativistic Velocity Addition

Calculates the relative velocity of objects moving at a significant fraction of the speed of light.

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u = \frac{v + u ^{'}}{1 + \frac{v u ^{'}}{c ^{2}}}

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Core idea

Overview

The Relativistic Velocity Addition formula determines the velocity of an object as perceived by an observer in a different inertial frame moving at a constant speed. It is a core component of Special Relativity, ensuring that the combined speed of two objects never exceeds the universal speed limit of light.

When to use: Apply this formula when objects are moving at relativistic speeds, typically exceeding 10% of the speed of light, relative to an observer. It is used when a simple addition of velocities in classical mechanics would inaccurately suggest a speed close to or beyond the speed of light.

Why it matters: This equation preserves the second postulate of special relativity: that the speed of light is constant for all observers regardless of their motion. It is practically applied in high-energy physics to track subatomic particles and in the timing corrections required for global satellite navigation systems.

Symbols

Variables

u = Resultant Velocity, v = Frame Velocity, u' = Relative Velocity, c = Speed of Light

u

Resultant Velocity

m/s

v

Frame Velocity

m/s

Relative Velocity

m/s

c

Speed of Light

m/s

Walkthrough

Derivation

Derivation: Relativistic Velocity Addition

Special relativity requires a modified velocity addition rule so that no object exceeds the speed of light, even when velocities are combined.

Frame S' moves at velocity v relative to S along the x-axis.
An object moves at u' in S'; we seek its speed u in S.
c is the speed of light (invariant in all inertial frames).

Apply the Lorentz transformation differentials:

Differentiate the Lorentz transformations for position and time.

d x = γ (d x^{'} + v d t^{'}), d t = γ (d t^{'} + \frac{v d x ^{'}}{c ^{2}})

Divide to get velocity u = dx/dt:

The γ factors cancel. When u' = c, u = (v+c)/(1+v/c) = c — the speed of light is the same in all frames.

u = \frac{d x}{d t} = \frac{v + u ^{'}}{1 + \frac{v u ^{'}}{c ^{2}}}

Result

u = \frac{d x}{d t} = \frac{v + u ^{'}}{1 + \frac{v u ^{'}}{c ^{2}}}

Source: University Physics — Special Relativity (Griffiths / Helliwell)

Free formulas

Rearrangements

Solve for $u$

Relativistic Velocity Addition

\frac{v + u ^{'}}{1 + \frac{v \cdot u ^{'}}{c ^{2}}}

The Relativistic Velocity Addition formula directly defines the resultant velocity $u$ . This problem presents the formula and clarifies its notation.

Difficulty: 4/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Why it behaves this way

Intuition

The combination of velocities in special relativity is not a simple linear sum but rather a transformation that 'bends' the resultant velocity to always stay below the speed of light, much like adding vectors on a curved

u

Velocity of the object as measured by the first (stationary) observer.

The final combined velocity we are trying to calculate, relative to the initial reference frame.

v

Velocity of the second inertial frame relative to the first (stationary) frame.

How fast the 'moving' observer's frame is traveling relative to the 'stationary' observer.

Velocity of the object as measured by an observer in the second inertial frame.

How fast the object is moving relative to the 'moving' observer's frame.

c

Speed of light in a vacuum.

The universal speed limit that no object with mass can reach or exceed.

Signs and relationships

1 + vu'/c^2: The denominator term `1 + vu'/ $c^{2}$ ` acts as a relativistic correction factor. As `v` and `u'` approach `c`, the term `vu'/ $c^{2}$ ` approaches 1, making the denominator approach 2.

Free study cues

Insight

Canonical usage

All velocity terms (u, v, u', c) must be expressed in consistent units, typically meters per second (m/s) in the International System of Units (SI).

Common confusion

A common mistake is to use inconsistent units for the velocities (v, u', c) or to forget that 'c' must be the speed of light in vacuum, not just any speed.

Dimension note

The term $\frac{v u ^{'}}{c ^{2}}$ in the denominator is dimensionless, ensuring the overall expression correctly yields a velocity.

Unit systems

u, v, u'm/s · All velocities in the equation must be expressed in the same consistent units.

$c$ m/s · The speed of light in vacuum, a fundamental physical constant.

One free problem

Practice Problem

A spacecraft moves at 0.6c relative to a planet. It launches a probe forward at 0.5c relative to the spacecraft's own frame. What is the velocity of the probe as measured by an observer on the planet?

Frame Velocity0.6 m/s

Relative Velocity0.5 m/s

Speed of Light1 m/s

Solve for: $u$

Hint: Express velocities as decimals where c = 1 to simplify the denominator calculation.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

A particle accelerator launches an electron at 0.9c. If that electron emits a photon forward, the photon travels at exactly 1.0c relative to the laboratory, not 1.9c.

Study smarter

Tips

Use consistent units; treating the speed of light 'c' as 1 is a common convention for simplifying calculations.
Pay close attention to direction; assign a negative sign to any velocity moving in the opposite direction of the reference frame.
The result 'u' will always be less than or equal to 'c' if the component velocities are less than or equal to 'c'.

Avoid these traps

Common Mistakes

Applying simple Newtonian addition (v + u') for particles moving at high fractions of c.
Incorrectly squaring the entire denominator instead of just the speed of light term.
Mismatched units between the velocities and the speed of light constant.

Common questions

Frequently Asked Questions

Special relativity requires a modified velocity addition rule so that no object exceeds the speed of light, even when velocities are combined.

Apply this formula when objects are moving at relativistic speeds, typically exceeding 10% of the speed of light, relative to an observer. It is used when a simple addition of velocities in classical mechanics would inaccurately suggest a speed close to or beyond the speed of light.

This equation preserves the second postulate of special relativity: that the speed of light is constant for all observers regardless of their motion. It is practically applied in high-energy physics to track subatomic particles and in the timing corrections required for global satellite navigation systems.

Applying simple Newtonian addition (v + u') for particles moving at high fractions of c. Incorrectly squaring the entire denominator instead of just the speed of light term. Mismatched units between the velocities and the speed of light constant.