Momentum, Mass & Energy

E = mc² and the cosmic speed limit.

12 minUpdated 22 October 2026relativistic momentummass–energy equivalencerest energy

Lesson at a glanceShow the key points

p = γm₀v (47.7); dp/dt = F (47.8).
E = mc² (47.9); K = (γ − 1)m₀c².
E² = m₀²c⁴ + p²c² (47.10); photons E = pc.
c is the ultimate speed.

If space and time behave differently at high speed, so must momentum and energy. The repaired laws contain a speed limit, a mass that grows with speed, and the most famous equation in physics.

Relativistic momentum

At speeds close to c, F = ma no longer describes motion correctly. We will state the correct laws without deriving them. The momentum of a particle is

Momentum(47.7)

\vec p = \gamma m_0\vec v = \frac{m_0\vec v}{\sqrt{1 - v^2/c^2}}

m₀ is the ordinary (Newtonian) mass, the rest mass. m = γm₀ is called the moving mass; different observers assign it different values.

Law of motion(47.8)

\frac{d\vec p}{dt} = \vec F

With no force, p is constant: relativistic momentum is conserved.

A steady push never reaches c

Start a particle from rest and apply a constant force F. Then p = Ft, so

Speed under a constant force

\frac{m_0 v}{\sqrt{1 - v^2/c^2}} = Ft ;\Rightarrow; v = \frac{Fct}{\sqrt{m_0^2c^2 + F^2t^2}}

For small t this is Ft/m₀, as Newton says; for large t it approaches c but never reaches it.

Worked example

When has the push given momentum m₀c?

A constant force acts on a particle starting from rest. What is its speed at the moment Ft = m₀c? What would Newton’s mechanics have predicted?

Mass is energy

In relativity, matter is a concentrated form of energy. A mass m is equivalent to an energy

Mass–energy equivalence(47.9)

E = mc^2

Doing work on a body adds energy, and therefore mass. Matter can turn into energy and energy into matter.

At rest a particle has rest-mass energy E₀ = m₀c². Moving, its total energy is E = γm₀c². Expanding γ for small v:

Kinetic energy

E = \frac{m_0c^2}{\sqrt{1 - v^2/c^2}} = m_0c^2 + \tfrac12 m_0v^2 + \tfrac38 m_0\frac{v^4}{c^2} + \dots, \qquad K = (\gamma - 1)m_0c^2

At everyday speeds the extra terms are negligible and K reduces to ½m₀v².

Interactive

Why nothing reaches c

Relativistic momentum p = γm₀v and kinetic energy (γ − 1)m₀c² match Newton's ½m₀v² and m₀v at low speed but grow without limit as v approaches c. Push with a constant force and p grows steadily (dp/dt = F), yet the speed creeps towards c and never gets there: every extra joule buys less and less speed.

\gamma

1.667

p / m₀c (relativistic, Newton)1.333, 0.800

K / m₀c² (relativistic, Newton)0.667, 0.320

Plot

Speed

v/c

0.80

Energy and momentum together

Eliminating v between p = γm₀v and E = γm₀c² gives a relation that holds in every frame:

Energy–momentum relation(47.10–47.11)

E^2 = m_0^2c^4 + p^2c^2, \qquad \text{massless particles: } E = pc

For photons m₀ = 0, so p = E/c, the result already used for radiation pressure and the photoelectric effect.

Worked example

One gram of matter

If 1.0 g of matter were converted entirely into energy, how many kilowatt-hours would it give?

Worked example

Tripling an electron's mass

At what speed is an electron’s moving mass three times its rest mass?

Worked example

Lifting adds mass

A 2.0 kg body is raised 5.0 m near the Earth’s surface. By how much does its mass increase?

Worked example

A sticky collision

A body of rest mass m₀ moving at 0.6c hits an identical body at rest, and they stick. Find the speed and rest mass of the combined body.

The ultimate speed

No material particle can be pushed to c, and no signal or information can travel faster than c. If it could, there would be frames in which an effect came before its cause: a bird hit before the gun was fired.

What to remember

1p = γm₀v (47.7) and dp/dt = F (47.8); moving mass m = γm₀.
2Under a constant force v → c but never reaches it.
3E = mc² (47.9); rest energy m₀c²; K = (γ − 1)m₀c² ≈ ½m₀v² at low speed.
4E² = m₀²c⁴ + p²c² (47.10); photons: E = pc.
5c is the ultimate speed for matter and information.

Check yourself

At v = 0.6c, a particle's kinetic energy is…

A photon of energy E has momentum…

Heating a sealed box of gas makes its mass…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

WatchThe real meaning of E = mc²PBS Space Time ReadRelativistic momentumOpenStax University Physics Vol. 3 §5.8 ReadRelativistic energyOpenStax University Physics Vol. 3 §5.9 ReadRelativistic energyHyperPhysics ReadMass–energy equivalenceWikipedia ReadTachyonWikipedia

PreviousLength Contraction & Simultaneity Next The Twin Paradox

Momentum, Mass & Energy

E = mc² and the cosmic speed limit.

12 minUpdated 22 October 2026relativistic momentummass–energy equivalencerest energy

Lesson at a glanceShow the key points

p = γm₀v (47.7); dp/dt = F (47.8).
E = mc² (47.9); K = (γ − 1)m₀c².
E² = m₀²c⁴ + p²c² (47.10); photons E = pc.
c is the ultimate speed.

If space and time behave differently at high speed, so must momentum and energy. The repaired laws contain a speed limit, a mass that grows with speed, and the most famous equation in physics.

Relativistic momentum

At speeds close to c, F = ma no longer describes motion correctly. We will state the correct laws without deriving them. The momentum of a particle is

Momentum(47.7)

\vec p = \gamma m_0\vec v = \frac{m_0\vec v}{\sqrt{1 - v^2/c^2}}

m₀ is the ordinary (Newtonian) mass, the rest mass. m = γm₀ is called the moving mass; different observers assign it different values.

Law of motion(47.8)

\frac{d\vec p}{dt} = \vec F

With no force, p is constant: relativistic momentum is conserved.

A steady push never reaches c

Start a particle from rest and apply a constant force F. Then p = Ft, so

Speed under a constant force

\frac{m_0 v}{\sqrt{1 - v^2/c^2}} = Ft ;\Rightarrow; v = \frac{Fct}{\sqrt{m_0^2c^2 + F^2t^2}}

For small t this is Ft/m₀, as Newton says; for large t it approaches c but never reaches it.

Worked example

When has the push given momentum m₀c?

A constant force acts on a particle starting from rest. What is its speed at the moment Ft = m₀c? What would Newton’s mechanics have predicted?

Mass is energy

In relativity, matter is a concentrated form of energy. A mass m is equivalent to an energy

Mass–energy equivalence(47.9)

E = mc^2

Doing work on a body adds energy, and therefore mass. Matter can turn into energy and energy into matter.

At rest a particle has rest-mass energy E₀ = m₀c². Moving, its total energy is E = γm₀c². Expanding γ for small v:

Kinetic energy

E = \frac{m_0c^2}{\sqrt{1 - v^2/c^2}} = m_0c^2 + \tfrac12 m_0v^2 + \tfrac38 m_0\frac{v^4}{c^2} + \dots, \qquad K = (\gamma - 1)m_0c^2

At everyday speeds the extra terms are negligible and K reduces to ½m₀v².

Interactive

Why nothing reaches c

\gamma

1.667

p / m₀c (relativistic, Newton)1.333, 0.800

K / m₀c² (relativistic, Newton)0.667, 0.320

Plot

Speed

v/c

0.80

Energy and momentum together

Eliminating v between p = γm₀v and E = γm₀c² gives a relation that holds in every frame:

Energy–momentum relation(47.10–47.11)

E^2 = m_0^2c^4 + p^2c^2, \qquad \text{massless particles: } E = pc

For photons m₀ = 0, so p = E/c, the result already used for radiation pressure and the photoelectric effect.

Worked example

One gram of matter

If 1.0 g of matter were converted entirely into energy, how many kilowatt-hours would it give?

Worked example

Tripling an electron's mass

At what speed is an electron’s moving mass three times its rest mass?

Worked example

Lifting adds mass

A 2.0 kg body is raised 5.0 m near the Earth’s surface. By how much does its mass increase?

Worked example

A sticky collision

A body of rest mass m₀ moving at 0.6c hits an identical body at rest, and they stick. Find the speed and rest mass of the combined body.

The ultimate speed

What to remember

1p = γm₀v (47.7) and dp/dt = F (47.8); moving mass m = γm₀.
2Under a constant force v → c but never reaches it.
3E = mc² (47.9); rest energy m₀c²; K = (γ − 1)m₀c² ≈ ½m₀v² at low speed.
4E² = m₀²c⁴ + p²c² (47.10); photons: E = pc.
5c is the ultimate speed for matter and information.

Check yourself

At v = 0.6c, a particle's kinetic energy is…

A photon of energy E has momentum…

Heating a sealed box of gas makes its mass…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

PreviousLength Contraction & Simultaneity Next The Twin Paradox

Relativistic momentum#

A steady push never reaches c

Mass is energy#

Energy and momentum together

The ultimate speed#

Check yourself

Keep exploring

Relativistic momentum#

A steady push never reaches c

Mass is energy#

Energy and momentum together

The ultimate speed#

Check yourself

Keep exploring

Relativistic momentum

Mass is energy

The ultimate speed

Relativistic momentum

Mass is energy

The ultimate speed