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Relativistic Addition of Velocities

The addition of velocities is the kinematic relation used to determine the velocity of an object in one inertial frame given its velocity in another frame. In special relativity, it modifies the classical Galilean transformation to preserve the invariance of the speed of light.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Galilean velocity addition in classical mechanics, relative velocity in one and two dimensions, and frame of reference concepts.

PGT extension

Einstein's relativistic velocity addition formula, length contraction and time dilation implications, and Fizeau's water-tube experiment.

Prerequisites

Inertial and non-inertial frames of reference
Galilean coordinate transformations
Lorentz coordinate transformations

Galilean Velocity Addition

In classical mechanics, velocities simply add vectorially according to $u = v + u'$.
Assumes absolute time where intervals dt are identical in all inertial frames.
Fails at speeds approaching the speed of light c because it allows resultant velocities to exceed c.

Postulates of Special Relativity

The laws of physics are invariant in all inertial reference frames.
The speed of light in vacuum c is isotropic and independent of the motion of the source or observer.
Requires the abandonment of absolute simultaneity and the adoption of Lorentz transformations.

Relativistic Velocity Addition Formula

Derived by taking differentials of the Lorentz coordinate transformations x' = γ(x - vt) and t' = γ(t - vx/c²).
Ensures that the composition of any velocity less than c with another velocity less than c never equals or exceeds c.
Reduces to the classical Galilean expression when the relative velocity v is much smaller than c.

Fizeau's Experiment on Ether Drag

Measured the speed of light in flowing water to test the Fresnel dragging coefficient.
Provided historical empirical support for the relativistic velocity addition formula before Einstein's formulation.
Demonstrated that moving media do not simply add their velocity to light linearly.

Derivations

Galilean Velocity Addition

Conditions: Low speeds where v << c

Consider position coordinate relation $x = x' + vt between two frames S$ and S'.
Take the time derivative of the position relation assuming dt = dt'.
Obtain the velocity relation dx/d$t = dx'/dt' + v$.

Result: $u = u' + v$

Relativistic Velocity Addition

Conditions: Inertial frames S and S' moving with relative velocity v along the x-axis

Start with the differential Lorentz transformations: dx' = gamma(dx - v dt) and dt' = gamma(dt - v dx / c^2).
Express the velocity in frame S' as $u_x' = dx' / dt' = \gamma(dx - v dt) / [\gamma(dt - v dx / c^2)]$.
Divide numerator and denominator by dt, substituting $u_x = dx/dt$ to get $u_x' = (u_x - v) / (1 - v u_x / c^2)$.
Invert to solve for u_x in terms of u_x': $u_x = (u_x' + v) / (1 + v u_x' / c^2)$.

Result: $u_x = \frac{u_x' + v}{1 + \frac{v u_x'}{c^2}}$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Classical Velocity Addition	$$u = u' + v$$	u = velocity in lab frame, u' = velocity in moving frame, v = velocity of moving frame m s^-1
Relativistic Velocity Addition (x-component)	$$u_x = \frac{u_x' + v}{1 + \frac{v u_x'}{c^2}}$$	$u_x = velocity in S$, $u_x' = velocity in S'$, v = relative velocity between frames m s^-1	Valid for motion along a common axis
Relativistic Velocity Addition (inverse x-component)	$$u_x' = \frac{u_x - v}{1 - \frac{v u_x}{c^2}}$$	u_x, $u_x' = respective velocities$, v = relative frame velocity m s^-1	Forward transformation
Relativistic Velocity Addition (y-component)	$$u_y = \frac{u_y' \sqrt{1 - \frac{v^2}{c^2}}}{1 + \frac{v u_x'}{c^2}}$$	$u_y = transverse velocity in S$, $u_y' = transverse velocity in S'$ m s^-1	Perpendicular to relative motion
Relativistic Velocity Addition (z-component)	$$u_z = \frac{u_z' \sqrt{1 - \frac{v^2}{c^2}}}{1 + \frac{v u_x'}{c^2}}$$	$u_z = transverse velocity in S$, $u_z' = transverse velocity in S'$ m s^-1	Perpendicular to relative motion
Composition of two velocities equal to c	$$u = \frac{c + c}{1 + \frac{c \cdot c}{c^2}} = c$$	c = speed of light m s^-1	Light speed invariance check
Fresnel Dragging Coefficient Formula	$$V = v \left(1 - \frac{1}{n^2}\right)$$	V = dragged velocity, v = fluid velocity, n = refractive index m s^-1	Low-speed approximation of relativistic addition in media
Lorentz Factor	$$\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$	v = velocity, c = speed of light in vacuum Dimensionless	v < c

Graphs and experiments

Velocity Addition Asymptotic Plot

Draw: Plot of resultant velocity u_x versus source velocity u_x' for various frame velocities v, bounded strictly between -c and +c.

Interpret: Shows that regardless of how high u_x' and v are, the sum never crosses the horizontal asymptote at $u_x = c$.

Exam use: Interpreting asymptotic behavior and upper speed limits in PGT conceptual questions.

Fizeau Water Tube Experiment

Draw: Interferometer setup with flowing water columns and split light paths.

Interpret: Demonstrates interference shift proportional to water velocity modified by 1 - 1/n^2.

Exam use: Questions linking classical fluid drag and relativistic velocity addition limits.

Galilean vs Relativistic Comparison Graph

Draw: Linear curve for Galilean addition intersecting/violating c versus bounded curve for relativistic addition.

Interpret: Highlights where classical mechanics breaks down at high velocities.

Exam use: Differentiating domain of validity between TGT and PGT levels.

Solved numericals

TGT core

A train is moving at 60 km/h east relative to the ground. A passenger walks inside the train at 5 km/h east relative to the train. Find the velocity of the passenger relative to the ground.

Identify velocities in classical framework: $v = 60 km/h$, u' = 5 km/h.
Apply Galilean velocity addition: $u = u' + v$.
Calculate sum: $u = 5 + 60 = 65 km/h$.

Answer: 65 km/h east

PGT extension

Two spaceships approach each other, each moving at a speed of 0.8c relative to a stationary observer on Earth. What is the speed of one spaceship relative to the other?

Let frame S be Earth. Spaceship 1 moves at $v = -0.8c$. Spaceship 2 moves at $u_x = 0.8c in Earth frame$.
We need the velocity u_x' of spaceship 2 relative to spaceship 1.
Use inverse relativistic formula: $u_x' = (u_x - v) / (1 - v u_x / c^2)$.
Substitute values: $u_x' = (0.8c - (-0.8c)) / (1 - (-0.8c)(0.8c) / c^2)$.
Simplify denominator: 1 - (-0.64) = 1.64.
Calculate numerator: 1.6c.
Result: $u_x' = 1.6c / 1.64 = 0.9756c$.

Answer: 0.9756 c

PGT extension

A particle emits a photon in the direction of its motion. If the particle is moving at speed v = 0.6c relative to the laboratory, what is the speed of the photon in the laboratory frame?

Identify the speed of the photon in the particle's rest frame: $u_x' = c$.
Identify the velocity of the particle frame: v = 0.6c.
Apply the relativistic velocity addition formula: $u_x = (u_x' + v) / (1 + v u_x' / c^2)$.
Substitute c for u_x': $u_x = (c + 0.6c) / (1 + (0.6c * c) / c^2)$.
Simplify: $u_x = 1.6c / (1 + 0.6) = 1.6c / 1.6 = c$.

Answer: c

Assumptions and limitations

Formulas assume motion restricted to collinear directions (standard configuration). Transverse components require separate transformation equations involving the Lorentz factor gamma.

Common misconceptions

Common error: Velocities simply add as $u = u' + v even at speeds close$ to the speed of light.

Correct: Velocities must combine via the relativistic relation $u = (u' + v)/(1 + vu'/c^2)$.

Linear addition permits speeds greater than c, which violates the second postulate of special relativity.

Common error: The speed of light changes when measured from a moving inertial frame.

Correct: The speed of light c is invariant in all inertial frames.

Einstein's postulate states that c is constant regardless of the relative motion of source and observer.

Quick revision

Relativistic velocity addition guarantees that no material object or signal can exceed the speed of light c.
Classical Velocity Addition: $u = u' + v$
Relativistic Velocity Addition (x-component): $u_x = \frac{u_x' + v}{1 + \frac{v u_x'}{c^2}}$

Must remember

In classical mechanics, velocities add vectorially as $u = v + u'$, which fails as speeds approach the speed of light $c$.
The relativistic velocity addition formula along the $x$-axis is given by $u_x = \frac{u_x' + v}{1 + \frac{v u_x'}{c^2}}$.
The composition of any velocity less than $c$ with another velocity less than $c$ never exceeds or equals $c$.
Transverse components of velocity transform under special relativity with a scaling factor of $\sqrt{1 - \frac{v^2}{c^2}}$ due to time dilation.
Fizeau's experiment on ether drag provided historical empirical support for relativistic velocity composition before Einstein's formulation.

Glossary

Galilean Velocity AdditionThe classical rule $u = u' + v$ based on absolute time and space intervals.

Lorentz FactorThe factor $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ appearing in relativistic transformations.

Fizeau's ExperimentAn experiment measuring the speed of light in flowing water to test the Fresnel dragging coefficient.

Inertial Reference FrameA frame of reference in which Newton's laws of motion hold true and the laws of physics are invariant.

Classical vs Relativistic Velocity Addition

Parameter	Galilean (Classical)	Einsteinian (Relativistic)
Velocity limit	Can exceed the speed of light $c$	Never exceeds the speed of light $c$
Formula ($x$-component)	$u = u' + v$	$u_x = \frac{u_x' + v}{1 + \frac{v u_x'}{c^2}}$
Time assumption	Absolute time ($dt = dt'$)	Relative time intervals via Lorentz transformation
Low-speed limit ($v \ll c$)	Exact match	Reduces to classical expression

Practice Quiz

Choose an option to reveal the explanation.

1. What is the classical expression for velocity addition in a moving frame?

Explanation: In classical mechanics, velocities simply add vectorially according to $u = u' + v$ under the assumption of absolute time.

2. The relativistic velocity addition formula for the $x$-component is given by:

Explanation: The correct forward relativistic velocity addition formula for motion along the $x$-axis is $u_x = \frac{u_x' + v}{1 + \frac{v u_x'}{c^2}}$.

3. Under what condition does the relativistic velocity addition formula reduce to the classical Galilean expression?

Explanation: When the relative velocity $v$ is much smaller than $c$ ($v \ll c$), the denominator $\frac{v u_x'}{c^2}$ approaches zero, reducing the formula to $u = u' + v$.

4. What is the result of combining two velocities equal to $c$ using the relativistic velocity addition formula?

Explanation: Substituting $u_x' = c$ and $v = c$ into the formula yields $\frac{c + c}{1 + \frac{c \cdot c}{c^2}} = \frac{2c}{2} = c$, preserving the invariance of the speed of light.

5. Fizeau's experiment on ether drag was designed to test:

Explanation: Fizeau measured the speed of light in flowing water to test the Fresnel dragging coefficient, giving historical support to relativistic addition.

6. The transformation for the transverse velocity component $u_y$ in special relativity includes which scaling factor due to time dilation?

Explanation: The transverse velocity component contains $\sqrt{1 - \frac{v^2}{c^2}}$ in the numerator because time intervals transform via the Lorentz factor.

7. Which of the following is a postulate of Special Relativity?

Explanation: Einstein's second postulate states that the speed of light in vacuum $c$ is constant and independent of the source or observer motion.

8. What does the Lorentz factor $\gamma$ equal when $v \ll c$?

Explanation: For $v \ll c$, $\frac{v^2}{c^2} \approx 0$, so $\gamma = \frac{1}{\sqrt{1 - 0}} = 1$.

9. Why does Galilean velocity addition fail at speeds approaching $c$?

Explanation: Linear addition $u = u' + v$ can result in velocities greater than $c$, which violates the second postulate of special relativity.

10. In the Fresnel dragging coefficient formula $V = v \left(1 - \frac{1}{n^2}\right)$, what does $n$ represent?

Explanation: In Fresnel's formula, $n$ is the refractive index of the moving fluid medium.

11. If an object moves with velocity $u_x'$ in frame $S'$ and frame $S'$ moves with velocity $v$ relative to $S$, the inverse transformation for $u_x'$ is:

Explanation: The inverse relativistic velocity addition formula is $u_x' = \frac{u_x - v}{1 - \frac{v u_x}{c^2}}$.

12. Which branch of physics primarily deals with Galilean relativity and absolute time intervals?

Explanation: Classical mechanics assumes absolute time intervals and uses Galilean transformations.

13. What is the primary implication of abandoning absolute simultaneity in relativity?

Explanation: Abandoning absolute simultaneity leads directly to Lorentz transformations, causing time dilation and length contraction.

14. If $u_x' = 0.5c$ and $v = 0.5c$, what is the relativistic velocity $u_x$?

Explanation: Using $u_x = \frac{0.5c + 0.5c}{1 + \frac{(0.5c)(0.5c)}{c^2}} = \frac{1.0c}{1 + 0.25} = \frac{1.0}{1.25}c = 0.8c$.

15. What fundamental mathematical tool is used to derive the relativistic velocity addition formula?

Explanation: Taking differentials of the Lorentz coordinate transformations $x' = \gamma(x - vt)$ and $t' = \gamma(t - \frac{vx}{c^2})$ yields the velocity addition formulas.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. Two spaceships approach each other, each moving at speed $0.6c$ relative to a stationary observer on Earth. What is the velocity of one spaceship relative to the other?

Explanation: Using relativistic addition with $u' = 0.6c$ and $v = 0.6c$: $u = \frac{0.6c + 0.6c}{1 + \frac{(0.6c)(0.6c)}{c^2}} = \frac{1.2c}{1 + 0.36} = \frac{1.2}{1.36}c \approx 0.882c$.

2. According to the theory of special relativity, if a beam of light is emitted by a source moving towards an observer at speed $v$, the measured speed of light by the observer is:

Explanation: By Einstein's second postulate, the speed of light in vacuum $c$ is invariant and independent of the motion of the source.

3. In Fizeau's experiment, the measured velocity of light in water moving with velocity $v$ and refractive index $n$ is approximately given by $V = \frac{c}{n} + v \left(1 - \frac{1}{n^2}\term_dummy\right)$ (ignoring term formatting, the drag factor is):

Explanation: The Fresnel dragging coefficient is $1 - \frac{1}{n^2}$, which represents the low-speed approximation of relativistic velocity addition in a refractive medium.

4. A particle moves in the $x'y'$ plane of frame $S'$ with velocity components $u_x'$ and $u_y'$. If frame $S'$ moves along the $x$-axis with velocity $v$ relative to $S$, the transverse velocity $u_y$ in frame $S$ is proportional to:

Explanation: The exact transverse velocity formula is $u_y = \frac{u_y' \sqrt{1 - v^2/c^2}}{1 + \frac{v u_x'}{c^2}}$, which matches option B when expressed using $\gamma^{-1} = \sqrt{1 - v^2/c^2}$.

5. What is the limiting value of the resultant velocity when two velocities, each arbitrarily close to $c$, are combined relativistically?

Explanation: No matter how large the component velocities are (provided they are $\le c$), the relativistic velocity addition formula strictly ensures the resultant velocity never exceeds $c$.

6. In the low-velocity limit where $v \ll c$ and $u' \ll c$, the denominator of the relativistic velocity addition formula $1 + \frac{vu'}{c^2}$ approaches:

Explanation: As $v \ll c$ and $u' \ll c$, the term $\frac{vu'}{c^2}$ approaches 0, making the denominator $1 + 0 = 1$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. Which transformation equations are fundamentally required to derive the relativistic velocity addition formula?

Explanation: The relativistic velocity addition formula is derived by taking differentials of the Lorentz coordinate transformations.

2. If a rocket moves at $0.8c$ relative to Earth and fires a projectile forward at $0.6c$ relative to the rocket, what is the projectile's speed relative to Earth?

Explanation: Using $u_x' = 0.6c$ and $v = 0.8c$: $u_x = \frac{0.6c + 0.8c}{1 + \frac{(0.6c)(0.8c)}{c^2}} = \frac{1.4c}{1 + 0.48} = \frac{1.4}{1.48}c \approx 0.9459c$.

3. What is the main physical consequence of the relativistic velocity addition denominator $1 + \frac{v u_x'}{c^2}$?

Explanation: The denominator ensures that the composition of sub-luminal velocities always results in a sub-luminal speed.

4. In classical mechanics, if frame $S'$ moves at velocity $v$ relative to $S$, and a particle is stationary in $S'$, its velocity in $S$ is:

Explanation: In classical mechanics, if $u' = 0$, then $u = u' + v = 0 + v = v$.

5. Which of the following velocity addition equations is correct for the inverse transformation along the $x$-axis?

Explanation: The inverse transformation switches the signs of $v$ and $u_x$, yielding $u_x' = \frac{u_x - v}{1 - \frac{v u_x}{c^2}}$.

6. Why did Fizeau's water-tube experiment present a challenge to classical ether theories?

Explanation: Fizeau's result supported Fresnel's dragging coefficient, showing light speed in moving media does not obey simple linear addition $c/n + v$.

7. If $u_x' = c$ and $v = 0.5c$, what is $u_x$?

Explanation: Evaluating $u_x = \frac{c + 0.5c}{1 + \frac{c(0.5c)}{c^2}} = \frac{1.5c}{1 + 0.5} = \frac{1.5c}{1.5} = c$, verifying light speed invariance.

8. The transverse components of velocity ($u_y$ and $u_z$) differ from their classical counterparts primarily due to:

Explanation: Transverse velocity components transform with a factor of $\sqrt{1 - v^2/c^2}$ because time intervals $dt$ are dilated relative to $dt'$.

9. In special relativity, the addition of two velocities in opposite directions (e.g., $u' = -0.5c$ and $v = 0.5c$) yields a resultant velocity of:

Explanation: Using the formula: $\frac{-0.5c + 0.5c}{1 + \frac{(-0.5c)(0.5c)}{c^2}} = \frac{0}{1 - 0.25} = 0$.

10. Which UP TGT/PGT concept core distinguishes classical velocity addition from relativistic velocity addition?

Explanation: The fundamental distinction lies in abandoning absolute time for Lorentz transformations and recognizing the universal invariant speed $c$.