3. Relativistic Velocity Addition

Can Two Velocities Add to More Than the Speed of Light?

Imagine a spacecraft travelling away from Earth at:

0.80c

It launches a probe forward at:

0.70c

relative to the spacecraft.

Classical physics would suggest:

0.80c + 0.70c = 1.50c

But Special Relativity tells us that the probe cannot be measured travelling at:

1.50c.

Instead, velocities combine according to a different rule:

relativistic velocity addition.

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Classical Velocity Addition

At ordinary speeds, velocities simply add or subtract.

Suppose a train moves at:

20 m/s

and a passenger walks forward inside the train at:

2 m/s.

An observer standing beside the tracks measures approximately:

20 + 2 = 22 m/s.

So:

u = u′ + v

where:

• u = object's velocity measured in S
• u′ = object's velocity measured in S′
• v = velocity of S′ relative to S

For everyday motion, this works extremely well.

---

Why Does Classical Addition Work?

For speeds much smaller than:

c

relativistic corrections are tiny.

A car travelling at 30 m/s and another object moving at 10 m/s relative to it do not require complicated relativistic calculations.

We can simply use:

30 + 10 = 40 m/s.

But this approximation breaks down when velocities become a significant fraction of:

the speed of light.

---

The Problem with Classical Addition

Suppose a spacecraft travels at:

0.80c

and fires a probe forward at:

0.70c.

Classically:

u = 0.80c + 0.70c

u = 1.50c

This creates a problem.

Special Relativity requires that massive objects cannot be accelerated through the light-speed limit, and all inertial observers measure light in vacuum travelling at:

c.

Therefore, ordinary velocity addition cannot apply at:

relativistic speeds.

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The Relativistic Velocity Addition Equation

For motion along the same straight line:

u = (u′ + v)/(1 + u′v/c²)

This equation gives the velocity u measured in S when:

• S′ moves at velocity v relative to S
• an object moves at velocity u′ relative to S′

The denominator is the key difference from:

classical velocity addition.

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Another Form of the Equation

Sometimes we know the object's velocity in S and want its velocity in S′.

Then:

u′ = (u − v)/(1 − uv/c²)

This is the relativistic equivalent of the classical equation:

u′ = u − v.

Which form you use depends on:

which frame contains the known velocity.

---

Understanding the Variables

Always define your:

reference frames

before substituting numbers.

This prevents many sign and direction errors.

---

Where Does the Equation Come From?

The relativistic velocity equation follows directly from the:

Lorentz transformations.

Recall:

x′ = γ(x − vt)

and:

t′ = γ(t − vx/c²).

Velocity is:

u = dx/dt

and:

u′ = dx′/dt′.

Therefore:

u′ = dx′/dt′

becomes:

u′ = [γ(dx − vdt)] / [γ(dt − vdx/c²)].

The γ factors cancel:

u′ = (dx − vdt)/(dt − vdx/c²).

Divide numerator and denominator by dt:

u′ = (dx/dt − v)/(1 − v(dx/dt)/c²).

Since:

dx/dt = u,

we obtain:

u′ = (u − v)/(1 − uv/c²).

So relativistic velocity addition is not an extra rule added separately to Special Relativity.

It follows from the:

Lorentz transformations.

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Worked Example 1: Two Spacecraft

A spacecraft travels at:

0.80c

relative to Earth.

It launches a probe forward at:

0.70c

relative to the spacecraft.

What velocity does Earth measure?

Use:

u = (u′ + v)/(1 + u′v/c²).

Substitute:

u = (0.70c + 0.80c)/(1 + (0.70c)(0.80c)/c²)

Simplify:

u = 1.50c/(1 + 0.56)

u = 1.50c/1.56

Therefore:

u ≈ 0.962c.

Earth measures the probe travelling at about:

0.962c.

Not:

1.50c.

---

Classical vs Relativistic Result

For the same problem:

Classical

u = 0.70c + 0.80c

u = 1.50c

Relativistic

u = (0.70c + 0.80c)/(1 + 0.70 × 0.80)

u ≈ 0.962c

The difference is enormous because the speeds are:

relativistic.

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Why the Denominator Matters

Consider:

u = (u′ + v)/(1 + u′v/c²).

The denominator:

1 + u′v/c²

reduces the result compared with simple addition.

At low speeds:

u′v ≪ c²

so:

u′v/c² ≈ 0.

Then:

u ≈ (u′ + v)/1

and therefore:

u ≈ u′ + v.

So classical velocity addition appears naturally as the:

low-speed approximation.

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Worked Example 2: Moderate Speeds

Suppose:

v = 0.30c

and:

u′ = 0.40c.

Classically:

u = 0.30c + 0.40c

u = 0.70c.

Relativistically:

u = (0.40c + 0.30c)/(1 + 0.40 × 0.30)

u = 0.70c/1.12

u = 0.625c.

Even at these speeds, the difference is already noticeable.

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Worked Example 3: Lower Speeds

Suppose:

v = 0.01c

and:

u′ = 0.02c.

Classically:

u = 0.03c.

Relativistically:

u = 0.03c/(1 + 0.0002)

u ≈ 0.029994c.

The difference is extremely small.

This explains why we normally use:

classical velocity addition

in everyday life.

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A Visual Comparison

Classical velocity addition increases without a built-in limit.

Relativistic velocity addition approaches:

c

without allowing massive objects to cross it.

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What Happens If the Object Is Light?

This is one of the most important tests of the equation.

Suppose S′ moves at velocity:

v

relative to S.

A light pulse moves forward in S′ at:

u′ = c.

Use:

u = (u′ + v)/(1 + u′v/c²).

Substitute:

u = (c + v)/(1 + cv/c²).

Simplify the denominator:

1 + v/c.

Therefore:

u = (c + v)/(1 + v/c).

Factor c from the numerator:

u = c(1 + v/c)/(1 + v/c).

So:

u = c.

Every inertial observer still measures the light travelling at:

c.

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Example: Chasing a Beam of Light

Suppose a spacecraft travels at:

0.90c.

A light beam travels forward past the spacecraft.

Classical physics might suggest that the spacecraft measures the light travelling at:

c − 0.90c = 0.10c.

But use the relativistic transformation:

u′ = (u − v)/(1 − uv/c²).

Set:

u = c

and:

v = 0.90c.

Then:

u′ = (c − 0.90c)/(1 − 0.90)

u′ = 0.10c/0.10

Therefore:

u′ = c.

The spacecraft still measures:

the full speed of light.

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Why This Is So Important

The invariance of the speed of light is one of the foundations of:

Special Relativity.

Relativistic velocity addition ensures that different inertial observers do not obtain:

different vacuum light speeds.

This is one reason classical velocity addition must be replaced at:

relativistic speeds.

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Multiple Moving Observers

Now consider three observers:

Earth

Spacecraft A

Spacecraft B

Suppose A travels at:

0.70c

relative to Earth.

B travels forward at:

0.60c

relative to A.

What velocity does Earth measure for B?

Use:

u = (u′ + v)/(1 + u′v/c²).

Substitute:

u = (0.60c + 0.70c)/(1 + 0.60 × 0.70)

u = 1.30c/1.42

Therefore:

u ≈ 0.915c.

Earth measures B travelling at:

0.915c.

---

Three Frames, Three Measurements

This example demonstrates an important idea.

The velocity of an object is always measured:

relative to a reference frame.

Spacecraft B can have:

0.60c relative to A

while simultaneously having:

0.915c relative to Earth.

There is no contradiction.

Velocity is:

frame-dependent.

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Opposite Directions

Signs become particularly important when objects move in:

opposite directions.

Define rightward as:

positive.

Then leftward velocities are:

negative.

The same equation still works:

u′ = (u − v)/(1 − uv/c²).

Do not automatically add magnitudes.

Use:

signed velocities.

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Worked Example 4: Opposite Directions

Earth observes:

Spacecraft A moving right at:

+0.80c

Spacecraft B moving left at:

−0.70c.

What velocity does A measure for B?

Let:

u = −0.70c

and:

v = +0.80c.

Use:

u′ = (u − v)/(1 − uv/c²).

Substitute:

u′ = (−0.70c − 0.80c)/(1 − (−0.70)(0.80))

u′ = −1.50c/(1 + 0.56)

u′ = −1.50c/1.56

u′ ≈ −0.962c.

Therefore A measures B travelling at approximately:

0.962c in the opposite direction.

Not:

1.50c.

---

Relative Speed Between Two Spacecraft

This result is often surprising.

Earth can observe:

• A moving right at 0.80c
• B moving left at 0.70c

Classically their relative speed would be:

1.50c.

But the speed of B measured in A's inertial frame is:

0.962c.

This is an important distinction between:

coordinate-frame relative velocity

and simply adding two speed magnitudes measured by a third observer.

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Worked Example 5: Finding Velocity in the Moving Frame

Earth observes a probe travelling at:

0.90c.

A spacecraft travels in the same direction at:

0.60c.

What velocity does the spacecraft measure for the probe?

Use:

u′ = (u − v)/(1 − uv/c²).

Substitute:

u′ = (0.90c − 0.60c)/(1 − 0.90 × 0.60)

u′ = 0.30c/(1 − 0.54)

u′ = 0.30c/0.46

Therefore:

u′ ≈ 0.652c.

Classically we would obtain:

0.30c.

At relativistic speeds, that classical answer is substantially incorrect.

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Worked Example 6: Finding an Unknown Velocity

Earth measures a probe travelling at:

0.90c.

The probe moves at:

0.50c

relative to a spacecraft travelling in the same direction.

Find the spacecraft's speed relative to Earth.

Start with:

u = (u′ + v)/(1 + u′v/c²).

Substitute:

0.90c = (0.50c + v)/(1 + 0.50v/c).

Let:

β = v/c.

Then:

0.90 = (0.50 + β)/(1 + 0.50β).

Multiply:

0.90(1 + 0.50β) = 0.50 + β

0.90 + 0.45β = 0.50 + β

0.40 = 0.55β

β ≈ 0.727.

Therefore:

v ≈ 0.727c.

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A Useful Dimensionless Form

When all speeds are given as fractions of c, calculations become easier.

Define:

βu = u/c

βu′ = u′/c

βv = v/c.

Then:

βu = (βu′ + βv)/(1 + βu′βv).

For example:

βu′ = 0.70

βv = 0.80

Then:

βu = (0.70 + 0.80)/(1 + 0.70 × 0.80)

βu = 0.962.

Therefore:

u = 0.962c.

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Why Massive Objects Cannot Reach c

The velocity-addition equation ensures that combining sub-light velocities produces another velocity below:

c.

But there is a deeper reason massive objects cannot be accelerated to the speed of light.

Recall the Lorentz factor:

γ = 1/√(1 − v²/c²).

As:

v → c

then:

1 − v²/c² → 0

and therefore:

γ → ∞.

---

Relativistic Energy

The total energy of a particle with rest mass m is:

E = γmc².

As:

v → c

then:

γ → ∞.

Therefore, the energy required to continue accelerating a massive object toward c grows without bound.

Reaching exactly:

v = c

would require unbounded energy in this framework.

A massive object therefore cannot be accelerated from below c to:

c.

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What About Light?

Light is different.

Photons have:

zero rest mass.

They travel in vacuum at:

c.

The equation:

E = γmc²

should not be applied to photons by simply setting m = 0 and v = c, because that produces an undefined limiting expression.

For photons, the appropriate energy relationship is:

E = pc.

---

Can Anything Travel Faster Than Light?

Within Special Relativity, ordinary matter and information cannot be accelerated through the invariant speed:

c.

The causal structure of spacetime prevents ordinary signals from being transmitted locally faster than:

light in vacuum.

This does not mean every speed-like quantity encountered in physics must be below c.

For example, apparent motion, certain wave velocities, and the increasing distance between sufficiently distant galaxies in cosmology require more careful interpretation.

They do not represent an ordinary local object overtaking a nearby beam of light.

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The Speed Limit Is Local

This distinction is important.

Special Relativity states that locally, in an inertial frame:

c is the invariant speed of light in vacuum.

No massive object can locally accelerate through:

c.

In cosmology, however, General Relativity allows the distance between very distant objects to increase in ways that can correspond to recession rates greater than c because:

space itself is dynamically evolving.

That is not ordinary velocity addition.

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What Happens as Speeds Approach c?

Suppose a spacecraft travels at:

0.99c.

It launches a probe forward at:

0.99c

relative to itself.

Classically:

u = 1.98c.

Relativistically:

u = (0.99c + 0.99c)/(1 + 0.99²)

u = 1.98c/1.9801

u ≈ 0.99995c.

Even two velocities extremely close to c combine to produce:

less than c.

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What If One Velocity Equals c?

Suppose:

u′ = c.

Then:

u = (c + v)/(1 + v/c).

This always simplifies to:

u = c.

No matter how quickly the observer moves, the transformed speed of light remains:

c.

This is built directly into the mathematics of:

Lorentz transformations.

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Mathematical Proof for Two Sub-Light Speeds

Suppose:

|u′| < c

and:

|v| < c.

Relativistic addition gives:

u = (u′ + v)/(1 + u′v/c²).

For same-direction positive velocities, define:

a = u′/c

and:

b = v/c

where:

0 ≤ a < 1

and:

0 ≤ b < 1.

Then:

u/c = (a + b)/(1 + ab).

To ask whether this is below 1, compare:

a + b

with:

1 + ab.

Their difference is:

1 + ab − a − b

which factors as:

(1 − a)(1 − b).

Because both factors are positive:

1 + ab > a + b.

Therefore:

u/c < 1

and hence:

u < c.

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Classical and Relativistic Addition Compared

The two models agree closely at:

low speeds.

They diverge dramatically as velocities approach:

c.

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Relativistic Addition Is Symmetric in One Dimension

For two velocities in the same direction:

u = (u′ + v)/(1 + u′v/c²).

Notice that swapping u′ and v gives:

u = (v + u′)/(1 + vu′/c²).

The result is unchanged.

This symmetry reflects the structure of:

one-dimensional relativistic velocity composition.

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Direction Still Matters

Velocity is a:

vector quantity.

In one-dimensional problems, direction can be represented using:

positive and negative signs.

For example:

right:

+

left:

−

A negative final answer does not mean the speed is negative.

It means the velocity points in the:

negative direction.

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Beyond One Dimension

So far, we have considered motion along one:

straight line.

If an object also has velocity components perpendicular to the motion of the reference frame, the transformation becomes more complicated.

For a frame moving along x:

u′x = (ux − v)/(1 − uxv/c²)

while the perpendicular components also involve:

γ.

This means relativistic velocity transformation is fundamentally:

three-dimensional.

For introductory problems, however, one-dimensional motion is usually the most important case.

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Connection to Lorentz Transformations

Relativistic velocity addition is a direct consequence of transforming both:

space

and:

time.

Classically, only the position transformation matters because:

t′ = t.

Relativity instead gives:

t′ = γ(t − vx/c²).

Because time itself transforms, the ratio:

distance/time

also transforms differently.

That is why velocities cannot simply:

add and subtract classically.

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Worldlines and Velocity

On a spacetime diagram, an object's motion is represented by a:

worldline.

The slope of that worldline is related to:

velocity.

Light forms the boundary:

x = ±ct.

Massive objects have worldlines that remain:

inside the light cone.

They cannot be continuously accelerated so that their worldlines cross the light-cone boundary.

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A Practical Problem-Solving Method

When solving relativistic velocity problems:

Step 1: Identify the reference frames.

Step 2: Choose a positive direction.

Step 3: Assign signs to every velocity.

Step 4: Identify u, u′, and v.

Step 5: Choose the appropriate transformation.

Step 6: Substitute carefully.

Step 7: Check that the result makes physical sense.

For ordinary massive objects, you should expect:

|u| < c.

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Example Problem-Solving Setup

Write:

S = Earth

S′ = spacecraft

v = velocity of spacecraft relative to Earth

u′ = velocity of probe relative to spacecraft

u = velocity of probe relative to Earth

Then use:

u = (u′ + v)/(1 + u′v/c²).

Writing the frames explicitly prevents one of the most common errors:

mixing velocities measured in different frames.

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Common Misconception: 0.8c + 0.8c = 1.6c

Only under classical velocity addition.

Relativistically:

u = (0.8c + 0.8c)/(1 + 0.8²)

u = 1.6c/1.64

u ≈ 0.976c.

The combined velocity remains below:

c.

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Common Misconception: A Fast Spacecraft Almost Catches Light

Suppose a spacecraft moves at:

0.999c.

It does not measure a forward-moving light beam travelling at:

0.001c.

It still measures:

c.

This is one of the most important departures from:

classical intuition.

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Common Misconception: Light Gets an Extra c from a Moving Source

Suppose a spacecraft travels at:

0.70c

and switches on a laser pointing forward.

Earth does not measure the light at:

1.70c.

Both Earth and the spacecraft measure the light travelling locally at:

c.

The motion of the source does not add to the vacuum speed of:

light.

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Common Misconception: Opposite Spacecraft Can Measure Each Other Above c

Suppose Earth sees:

A = +0.90c

and:

B = −0.90c.

Earth may note that their coordinate separation is increasing at:

1.80c.

But when A measures B's velocity in A's inertial frame:

u′ = (−0.90c − 0.90c)/(1 + 0.81)

u′ = −1.80c/1.81

u′ ≈ −0.9945c.

So neither spacecraft measures the other locally moving faster than:

c.

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Common Misconception: Relativistic Addition Is Needed for Cars

Technically, relativity applies to:

all velocities.

But at everyday speeds:

u′v/c²

is extraordinarily small.

Therefore:

u ≈ u′ + v.

Classical velocity addition is usually more than accurate enough.

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When Should You Use Relativistic Velocity Addition?

Use it when:

• velocities are significant fractions of c
• high-energy particles are involved
• spacecraft move at relativistic speeds
• comparing measurements between rapidly moving frames
• light or other relativistic signals are involved

For ordinary everyday speeds, classical addition remains an excellent:

approximation.

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Real-World Applications

Relativistic velocity transformations are important in:

• particle accelerator physics
• cosmic-ray studies
• astrophysics
• relativistic jets
• high-energy particle collisions
• theoretical spacecraft problems
• fundamental tests of Special Relativity

Particles in accelerators routinely travel at velocities extremely close to:

c.

Their motion cannot be accurately analyzed using ordinary:

Galilean velocity addition.

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A Final Example

Spacecraft A travels at:

0.95c

relative to Earth.

It launches a probe forward at:

0.80c

relative to itself.

Classically:

u = 1.75c.

Relativistically:

u = (0.80c + 0.95c)/(1 + 0.80 × 0.95)

u = 1.75c/1.76

u ≈ 0.9943c.

Despite both velocities being very large, the final result remains:

below c.

This is exactly what Special Relativity requires.

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Check Your Understanding

1. State the classical velocity addition equation.

2. Why does classical velocity addition work well at everyday speeds?

3. Write the relativistic velocity addition equation for motion in the same direction.

4. Explain the meanings of u, u′, and v.

5. A spacecraft moves at 0.60c and launches a probe forward at 0.50c relative to itself. Calculate the probe's velocity relative to Earth.

6. Compare the classical and relativistic answers to Question 5.

7. A spacecraft travels at 0.80c and fires a probe forward at 0.80c. Calculate the probe's velocity relative to Earth.

8. Explain why the answer to Question 7 is not 1.60c.

9. Show mathematically that if u′ = c, then u = c.

10. A spacecraft travels at 0.90c while a light pulse travels in the same direction. What speed does the spacecraft measure for the light?

11. Earth sees spacecraft A travelling at +0.70c and spacecraft B at −0.60c. Calculate B's velocity in A's frame.

12. Why are signs important in velocity transformation problems?

13. Explain why relativistic velocity addition approaches classical addition at low speeds.

14. What happens to γ as a massive object's speed approaches c?

15. Explain why a massive object cannot be accelerated to c.

16. Two velocities of 0.99c are combined in the same direction. Calculate the resulting velocity.

17. Explain why the motion of a light source does not increase the measured vacuum speed of its light.

18. How does relativistic velocity addition follow from the Lorentz transformations?

19. Why can two observers measure different velocities for the same object without either being wrong?

20. Explain how relativistic velocity addition preserves the invariant speed c.

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Key Terms

• Velocity addition: Rule used to determine an object's velocity when measurements are made from different moving reference frames.
• Classical velocity addition: Low-speed approximation u = u′ + v.
• Relativistic velocity addition: Special Relativity equation u = (u′ + v)/(1 + u′v/c²).
• Velocity transformation: Conversion of a measured velocity from one inertial frame to another.
• Reference frame: Coordinate system relative to which position, time and motion are measured.
• Inertial frame: Non-accelerating reference frame.
• Relative velocity: Velocity of one object measured from another reference frame.
• Speed of light (c): Invariant vacuum speed approximately 3.00 × 10⁸ m/s.
• Lorentz transformation: Transformation connecting space and time coordinates between inertial frames.
• Lorentz factor (γ): 1/√(1 − v²/c²).
• Beta (β): Dimensionless velocity ratio v/c.
• Light cone: Spacetime boundary formed by light travelling from an event.
• Worldline: Path of an object through spacetime.
• Invariant: Quantity that remains the same under the relevant transformation.
• Rest mass: Invariant mass of an object measured in its rest frame.

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Key Takeaways

• Velocities do not simply add at relativistic speeds.
• Classical velocity addition is u = u′ + v.
• Relativistic velocity addition is u = (u′ + v)/(1 + u′v/c²).
• To transform the other way, use u′ = (u − v)/(1 − uv/c²).
• Relativistic velocity addition follows directly from the Lorentz transformations.
• At low speeds, u′v/c² ≈ 0, so the relativistic equation reduces to classical velocity addition.
• At speeds approaching c, the difference between classical and relativistic predictions becomes large.
• Two sub-light velocities do not combine to produce a massive object's speed greater than c.
• If one of the velocities is exactly c, the transformed velocity remains c.
• Every inertial observer measures light in vacuum travelling locally at c.
• A spacecraft cannot reduce the measured speed of a light beam simply by chasing it.
• Motion of a light source does not add its speed to the vacuum speed of the emitted light.
• Velocities are frame-dependent, so the same object can have different velocities in different inertial frames.
• Direction matters, so relativistic velocity problems should use signed velocities.
• Multiple moving-observer problems become easier when each reference frame is identified explicitly.
• The Lorentz factor grows without bound as a massive object's speed approaches c.
• Accelerating an object with nonzero rest mass to exactly c would require unbounded energy.
• Light follows a different energy-momentum relationship because photons have zero rest mass.
• Relativistic velocity addition is essential in particle physics, accelerator physics and astrophysics.
• The equation preserves the invariant speed c, making it a fundamental consequence of Special Relativity.