Momentum at Everyday Speeds

In classical mechanics, momentum describes the quantity of motion of an object.

For ordinary speeds, momentum depends on two quantities:

• mass
• velocity

p = mv

where:

• p = momentum, measured in kg·m/s
• m = mass, measured in kg
• v = velocity, measured in m/s

For cars, balls, aircraft, and most everyday objects, this classical relationship works extremely well.

But when an object's speed becomes a significant fraction of the speed of light:

p = mv is no longer sufficient.

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Why Classical Momentum Must Be Modified

Special Relativity requires the laws of physics to work consistently for all:

inertial observers.

If we continued to use only:

p = mv

at speeds approaching c, momentum would not transform correctly between reference frames, and conservation of momentum would not remain consistent with:

Special Relativity.

The solution is relativistic momentum.

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Relativistic Momentum

For a particle with rest mass m moving at velocity v:

p = γmv

where:

γ = 1 / √(1 − v²/c²)

and:

• p = relativistic momentum
• m = rest mass
• v = velocity
• c = speed of light
• γ = Lorentz factor

The Lorentz factor is the same factor encountered in:

time dilation, length contraction, and relativistic energy.

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The Lorentz Factor

The Lorentz factor is:

γ = 1 / √(1 − v²/c²)

At low speeds:

v ≪ c

so:

v²/c² ≈ 0.

Therefore:

γ ≈ 1.

Relativistic momentum becomes:

p ≈ mv.

This explains why classical momentum works so well in:

everyday situations.

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How γ Changes with Speed

At relatively low speeds, γ is close to:

1.

Near the speed of light, however, γ increases dramatically.

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Classical vs Relativistic Momentum

Classical momentum:

p = mv

Relativistic momentum:

p = γmv

The relativistic equation contains an additional factor:

γ.

Therefore:

p_rel = γp_classical

for the same m and v.

Because:

γ ≥ 1,

relativistic momentum is always at least as large in magnitude as the corresponding classical value.

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At Low Speeds

Suppose:

v = 0.01c.

Then:

γ ≈ 1.00005.

Therefore:

p = γmv ≈ mv.

The difference is extremely small.

Classical mechanics provides an excellent:

approximation.

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At High Speeds

Suppose:

v = 0.90c.

Then:

γ ≈ 2.294.

Therefore:

p = 2.294mv.

The relativistic momentum is more than twice the value predicted by:

classical momentum.

At this speed, the classical approximation is no longer appropriate.

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Worked Example 1: Relativistic Momentum

A particle has rest mass:

m = 2.0 × 10⁻²⁷ kg

and travels at:

v = 0.80c.

Calculate its relativistic momentum.

First find γ:

γ = 1 / √(1 − 0.80²)

γ = 1 / √(0.36)

γ = 1.667

Now calculate the velocity:

v = 0.80(3.00 × 10⁸)

v = 2.40 × 10⁸ m/s.

Use:

p = γmv

p = (1.667)(2.0 × 10⁻²⁷)(2.40 × 10⁸)

Therefore:

p ≈ 8.0 × 10⁻¹⁹ kg·m/s.

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Compare with Classical Momentum

For the same particle, classical mechanics predicts:

p = mv

p = (2.0 × 10⁻²⁷)(2.40 × 10⁸)

p = 4.8 × 10⁻¹⁹ kg·m/s.

So:

classical prediction = 4.8 × 10⁻¹⁹ kg·m/s

while:

relativistic prediction = 8.0 × 10⁻¹⁹ kg·m/s.

That is a substantial difference.

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A Useful Calculation Method

For most relativistic momentum problems:

Step 1: Identify the speed

If given as a fraction of c:

v = βc

where:

β = v/c.

Step 2: Calculate γ

γ = 1/√(1 − β²)

Step 3: Convert v into m/s if necessary

v = β(3.00 × 10⁸)

Step 4: Calculate momentum

p = γmv

Step 5: Give appropriate units

kg·m/s

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Worked Example 2: A Proton at 0.60c

A proton has rest mass approximately:

1.67 × 10⁻²⁷ kg.

Suppose it travels at:

0.60c.

First:

γ = 1/√(1 − 0.60²)

γ = 1/√0.64

γ = 1.25.

The velocity is:

v = 0.60(3.00 × 10⁸)

v = 1.80 × 10⁸ m/s.

Now:

p = γmv

p = (1.25)(1.67 × 10⁻²⁷)(1.80 × 10⁸)

p ≈ 3.76 × 10⁻¹⁹ kg·m/s.

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What Would Classical Physics Predict?

Classically:

p = mv

p = (1.67 × 10⁻²⁷)(1.80 × 10⁸)

p ≈ 3.01 × 10⁻¹⁹ kg·m/s.

Compare:

Classical: 3.01 × 10⁻¹⁹ kg·m/s

Relativistic: 3.76 × 10⁻¹⁹ kg·m/s

The relativistic result is:

25% larger.

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Worked Example 3: A Particle at 0.90c

Suppose:

m = 5.0 × 10⁻²⁷ kg

and:

v = 0.90c.

First:

γ ≈ 2.294.

Velocity:

v = 2.70 × 10⁸ m/s.

Then:

p = γmv

p = (2.294)(5.0 × 10⁻²⁷)(2.70 × 10⁸)

p ≈ 3.10 × 10⁻¹⁸ kg·m/s.

Classically:

p = mv

p = 1.35 × 10⁻¹⁸ kg·m/s.

The classical value severely underestimates the particle's:

momentum.

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What Happens as v Approaches c?

Consider:

γ = 1/√(1 − v²/c²).

As:

v → c,

then:

v²/c² → 1.

Therefore:

1 − v²/c² → 0.

The denominator approaches zero, so:

γ → ∞.

Since:

p = γmv,

the momentum of a massive particle grows without bound as its speed approaches:

c.

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Momentum Does Not Level Off at mc

A common mistake is to think that because velocity cannot exceed c, momentum must have some maximum value near:

mc.

It does not.

Velocity approaches c, but:

γ increases without bound.

Therefore, the momentum of a massive particle can continue increasing even while its speed changes by only a very small amount.

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Why Massive Objects Cannot Reach c

Suppose we keep accelerating a massive particle.

At first, additional energy produces noticeable increases in:

speed.

As the particle approaches c, additional energy still increases its:

energy and momentum,

but the speed increases by progressively smaller amounts.

Reaching exactly c would require:

unbounded energy and momentum.

Therefore, an object with nonzero rest mass cannot be accelerated to:

the speed of light.

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Momentum Is Still a Vector

Relativistic momentum is still a:

vector quantity.

Its direction is the same as the direction of the particle's velocity.

In one dimension:

right → positive momentum

left → negative momentum.

This becomes especially important when solving:

conservation problems.

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Conservation of Relativistic Momentum

Momentum remains conserved in Special Relativity.

For an isolated system:

total momentum before = total momentum after.

However, at relativistic speeds we must calculate the momentum of massive particles using:

p = γmv.

So:

Σp_before = Σp_after.

The conservation principle remains.

The expression used to calculate momentum is:

relativistic.

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Why Conservation Matters

Consider two high-speed particles colliding.

We cannot simply use classical momentum if their speeds are close to:

c.

Instead, calculate each particle's relativistic momentum and then apply:

momentum conservation.

Relativistic collision problems generally also require conservation of:

total energy.

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Worked Example 4: Opposite Momenta

Two identical particles have the same rest mass.

Particle A moves right at:

0.80c.

Particle B moves left at:

0.80c.

Their Lorentz factors are identical.

Particle A has momentum:

+γmv.

Particle B has momentum:

−γmv.

Therefore:

p_total = γmv − γmv

p_total = 0.

The total momentum of the system is:

zero.

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Zero Total Momentum Does Not Mean Zero Energy

Although the two particles have:

zero total momentum,

both particles are moving.

They therefore possess:

energy.

This distinction becomes extremely important in:

particle physics.

Two particles travelling in opposite directions can have enormous energies while the system's total momentum remains:

zero.

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Why Particle Colliders Use Opposing Beams

Modern particle colliders often accelerate particles in:

opposite directions.

If the two beams have equal and opposite momenta:

total momentum ≈ 0.

A large fraction of the available energy can then contribute to the energy of:

new particles and their motion.

This makes colliding-beam experiments particularly useful for studying:

high-energy physics.

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Relativistic Collision Example

Suppose two identical particles approach each other with equal speeds:

0.60c.

For each particle:

γ = 1.25.

If the rest mass of each particle is m, their momenta are:

p₁ = +1.25m(0.60c)

and:

p₂ = −1.25m(0.60c).

Therefore:

p_total = 0.

Any products produced in the collision must also have a combined momentum of:

zero

in this reference frame.

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Conservation Does Not Mean Individual Momentum Is Constant

During a collision, an individual particle's momentum can:

• increase
• decrease
• reverse direction
• be redistributed among new particles

What remains conserved is:

the total momentum of the isolated system.

This principle works in both classical and:

relativistic physics.

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Relativistic Momentum and Energy

Momentum and energy are closely connected in Special Relativity.

The fundamental relationship is:

E² = p²c² + m²c⁴.

This connects:

• total energy
• momentum
• rest mass

into a single relativistic relationship.

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A Particle at Rest

If a particle is at rest:

p = 0.

Then:

E² = m²c⁴.

Therefore:

E = mc².

So Einstein's famous mass-energy equation is a special case of the more general:

energy-momentum relationship.

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A Massless Particle

For a photon:

m = 0.

Therefore:

E² = p²c².

So:

E = pc.

This means photons have:

momentum

even though they have:

zero rest mass.

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How Can Light Have Momentum?

Classical momentum:

p = mv

would suggest that something with zero mass must have:

zero momentum.

But this classical equation does not apply to photons.

For photons:

p = E/c.

Since photons carry energy, they also carry:

momentum.

This has measurable physical effects.

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Radiation Pressure

When light strikes or reflects from a surface, it transfers:

momentum.

This produces a small pressure called:

radiation pressure.

One proposed application is the:

solar sail.

A solar sail uses momentum transferred by sunlight to gradually accelerate a spacecraft.

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Momentum in Particle Physics

Relativistic momentum is essential in:

particle accelerators.

Particles such as:

• electrons
• protons
• muons
• ions

can move extremely close to:

the speed of light.

At these speeds:

p = mv

would give seriously incorrect results.

Scientists must use:

relativistic momentum.

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Momentum at Nearly c

Imagine a proton travelling at:

0.999c.

Its Lorentz factor is approximately:

22.37.

Therefore:

p ≈ 22.37mv.

Classical mechanics would predict only:

mv.

So the relativistic momentum is more than:

22 times

the classical prediction for the same m and v.

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Why Speed Barely Changes Near c

At extremely high energies, adding more energy to a particle produces a large increase in:

momentum and energy,

but only a tiny increase in:

speed.

The speed approaches c asymptotically.

It never reaches or exceeds:

c.

This is why accelerator physicists often describe high-energy particles in terms of their:

energy and momentum

rather than simply their speed.

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Units Used in Particle Physics

The SI unit of momentum is:

kg·m/s.

However, particle physicists commonly use units such as:

eV/c

MeV/c

and:

GeV/c.

These units are convenient because particle energies are commonly measured in:

electronvolts.

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The Electronvolt

One electronvolt is:

1 eV ≈ 1.602 × 10⁻¹⁹ J.

Common multiples include:

1 keV = 10³ eV

1 MeV = 10⁶ eV

1 GeV = 10⁹ eV

1 TeV = 10¹² eV.

High-energy particle physics often involves energies in the:

GeV and TeV ranges.

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Worked Example 5: Momentum from Energy

Suppose a photon has energy:

6.0 × 10⁻¹⁹ J.

For a photon:

p = E/c.

Therefore:

p = (6.0 × 10⁻¹⁹)/(3.00 × 10⁸)

p = 2.0 × 10⁻²⁷ kg·m/s.

The photon has momentum even though its rest mass is:

zero.

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Worked Example 6: Finding Velocity from Momentum

Sometimes we know a particle's momentum and want to determine its:

velocity.

Starting from:

p = γmv,

direct rearrangement is inconvenient because γ also depends on v.

A useful form is:

v = pc²/E.

Together with:

E² = p²c² + m²c⁴,

we can determine the particle's:

velocity.

This is particularly useful in advanced particle physics problems.

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Classical Limit

A good relativistic equation should reproduce classical physics when:

v ≪ c.

For low velocities:

γ ≈ 1.

Therefore:

p = γmv

becomes:

p ≈ mv.

This is known as the:

classical limit.

Relativity does not say Newtonian mechanics is useless.

Instead, Newtonian mechanics is an excellent approximation under:

appropriate conditions.

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How Large Is the Relativistic Correction?

Because:

p_rel/p_classical = γ,

the percentage increase is:

(γ − 1) × 100%.

For example, at:

0.60c

γ = 1.25.

Therefore:

percentage increase = 25%.

At:

0.80c

γ ≈ 1.667.

Therefore:

percentage increase ≈ 66.7%.

At:

0.90c

γ ≈ 2.294.

Therefore:

percentage increase ≈ 129%.

The relativistic correction grows rapidly as v approaches:

c.

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Worked Comparison

Consider a particle of rest mass:

1.0 × 10⁻²⁷ kg

travelling at different speeds.

Notice that the two predictions are similar at low speeds but diverge dramatically near:

c.

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Momentum and Reference Frames

Momentum depends on the observer's:

reference frame.

Suppose you are travelling alongside a spacecraft at the same velocity.

In your frame, the spacecraft is:

at rest.

Therefore:

p = 0.

But an observer on Earth may see the spacecraft moving rapidly and therefore measure:

nonzero momentum.

Momentum is therefore:

frame-dependent.

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But Conservation Still Works

Although different inertial observers may measure different individual momenta, each observer finds that total relativistic momentum is conserved when the system is:

isolated.

This consistency is one of the reasons classical momentum must be replaced by:

relativistic momentum at high speeds.

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Momentum in Three Dimensions

For motion in three dimensions:

p⃗ = γm v⃗

Momentum points in the direction of:

velocity.

The components can be written:

pₓ = γmvₓ

pᵧ = γmvᵧ

p_z = γmv_z.

Momentum conservation applies separately in each:

spatial direction.

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Relativistic Momentum and Four-Momentum

In Special Relativity, energy and momentum can be combined into a single mathematical object called:

four-momentum.

Its components include:

energy

and:

three-dimensional momentum.

This mirrors the way space and time combine into:

spacetime.

Four-momentum is especially useful because it transforms consistently between:

inertial reference frames.

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Energy-Momentum Invariant

The relationship:

E² − p²c² = m²c⁴

has the same form in every inertial frame.

This is similar to the invariance of the:

spacetime interval.

It provides a deep connection between:

spacetime geometry

and:

energy-momentum physics.

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Relativistic Momentum in Modern Physics

Relativistic momentum is used extensively in:

• particle accelerators
• cosmic-ray physics
• nuclear physics
• astrophysics
• high-energy collisions
• particle detectors
• radiation physics

Whenever particles move close to c, relativistic momentum becomes:

essential.

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Cosmic Rays

Cosmic rays can produce extremely energetic particles travelling close to:

c.

Their velocities may differ from c by only tiny amounts, yet their momenta can vary enormously.

This is another reason why speed alone is not a good measure of how energetic an ultrarelativistic particle is.

Scientists often focus instead on:

momentum and energy.

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Momentum and Magnetic Fields

Charged particles moving through magnetic fields follow curved paths.

For a charged particle moving perpendicular to a uniform magnetic field:

p = qBr

in the appropriate relativistic treatment.

where:

• p = particle momentum
• q = charge
• B = magnetic field strength
• r = radius of curvature

Particle detectors can therefore determine momentum by measuring:

the curvature of particle tracks.

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Greater Momentum, Less Curvature

For particles with the same charge in the same magnetic field:

larger momentum → larger radius

and:

smaller momentum → tighter curvature.

This allows physicists to use detector images to reconstruct:

particle momenta.

Relativistic momentum therefore becomes something scientists can:

measure experimentally.

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Common Misconception: Mass Increases with Speed

Older explanations sometimes describe relativistic effects using:

"relativistic mass."

Modern physics generally keeps mass as the invariant:

rest mass m.

Instead of saying mass increases with speed, it is clearer to say:

energy and momentum increase according to relativistic equations.

The particle's invariant rest mass does not increase merely because the observer sees it moving faster.

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Common Misconception: Momentum Has a Maximum at c

Velocity has an upper limit for massive particles:

v < c.

Momentum does not have a corresponding finite maximum.

As:

v → c,

γ → ∞

and therefore:

p → ∞.

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Common Misconception: Classical Momentum Suddenly Stops Working

There is no sharp speed at which classical physics suddenly becomes:

wrong.

Instead, the difference gradually increases.

At low speeds, relativistic corrections are:

tiny.

At increasingly high speeds, they become:

significant.

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Common Misconception: Photons Have No Momentum

Photons have zero:

rest mass.

But they possess:

energy and momentum.

For photons:

p = E/c.

This is experimentally observable through phenomena such as:

radiation pressure.

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Common Misconception: Momentum Conservation Changes in Relativity

The principle does not disappear.

Momentum is still:

conserved.

What changes is the equation used to calculate the momentum of a massive high-speed particle:

p = γmv.

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Connecting Momentum and Energy

The previous topic introduced:

E = mc².

Relativistic momentum extends the picture.

For a massive moving particle:

E = γmc²

and:

p = γmv.

These quantities are related through:

E² = p²c² + m²c⁴.

Together, energy and momentum provide a complete description of:

relativistic particle motion.

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A Useful Relationship

Since:

E = γmc²

and:

p = γmv,

divide momentum by energy:

p/E = v/c².

Therefore:

v = pc²/E.

This shows how relativistic:

energy, momentum, and velocity

are directly connected.

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From Newton to Einstein

At low speeds:

p ≈ mv.

At relativistic speeds:

p = γmv.

For photons:

p = E/c.

This progression demonstrates an important feature of modern physics:

classical equations remain useful approximations within their appropriate range, while relativistic equations provide the more general description.

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Check Your Understanding

1. State the classical equation for momentum.

2. State the equation for relativistic momentum.

3. What does γ represent?

4. Write the equation for the Lorentz factor.

5. Why does relativistic momentum approach classical momentum at low speeds?

6. Calculate γ for an object travelling at 0.60c.

7. Calculate γ for an object travelling at 0.80c.

8. A particle of mass 3.0 × 10⁻²⁷ kg travels at 0.80c. Calculate its relativistic momentum.

9. Calculate the classical momentum for the particle in Question 8.

10. Compare the two answers.

11. Why does classical momentum increasingly underestimate momentum as v approaches c?

12. What happens to γ as v approaches c?

13. What happens to relativistic momentum as v approaches c for a massive particle?

14. Explain why a massive particle cannot reach c.

15. Two identical particles travel at equal speeds in opposite directions. What is their total momentum?

16. Does zero total momentum mean that the particles have zero total energy? Explain.

17. State the relativistic energy-momentum relationship.

18. What is the momentum of a photon in terms of its energy?

19. Explain how light can exert pressure despite having zero rest mass.

20. Explain why relativistic momentum is important in particle accelerators.

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Key Terms

• Momentum: Vector quantity describing an object's motion.
• Classical momentum: Momentum calculated using p = mv.
• Relativistic momentum: Momentum of a massive particle calculated using p = γmv.
• Rest mass: Invariant mass of an object.
• Lorentz factor (γ): Relativistic factor that increases as speed approaches c.
• Speed of light (c): Invariant speed of approximately 3.00 × 10⁸ m/s.
• Conservation of momentum: Principle that total momentum remains constant for an isolated system.
• Relativistic energy: Total energy of a particle described by Special Relativity.
• Energy-momentum relation: Relationship E² = p²c² + m²c⁴.
• Photon: Quantum of electromagnetic radiation with zero rest mass.
• Radiation pressure: Pressure produced by momentum transfer from electromagnetic radiation.
• Particle accelerator: Device used to accelerate charged particles to high energies.
• Four-momentum: Relativistic four-vector combining energy and three-dimensional momentum.
• Classical limit: Conditions under which relativistic equations reduce approximately to classical equations.
• Reference frame: Coordinate system from which motion is measured.

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Key Takeaways

• Classical momentum is given by p = mv.
• Classical momentum works extremely well when v ≪ c.
• At relativistic speeds, momentum must be calculated using p = γmv.
• The Lorentz factor is γ = 1/√(1 − v²/c²).
• At low speeds, γ is approximately 1, so relativistic momentum reduces to classical momentum.
• As speed increases, γ becomes increasingly important.
• Classical momentum increasingly underestimates the momentum of a high-speed particle.
• At 0.80c, γ ≈ 1.667.
• At 0.90c, γ ≈ 2.294.
• At 0.99c, γ ≈ 7.09.
• As v approaches c, γ increases without bound.
• The momentum of a massive particle therefore also increases without bound as v approaches c.
• Massive particles cannot be accelerated to exactly c.
• Relativistic momentum remains a vector quantity.
• Total relativistic momentum is conserved in an isolated system.
• Collision problems at relativistic speeds generally require conservation of both energy and momentum.
• Equal and opposite momenta can give a system zero total momentum even when it contains enormous energy.
• Photons have zero rest mass but still possess momentum.
• Photon momentum is given by p = E/c.
• Momentum carried by light produces observable effects such as radiation pressure.
• Relativistic energy and momentum are connected through E² = p²c² + m²c⁴.
• Modern physics normally treats mass as invariant rather than saying that mass increases with speed.
• Relativistic momentum is essential in particle accelerators, nuclear physics, cosmic-ray physics, and astrophysics.
• Momentum and energy together provide a fundamental description of particles in Special Relativity.