The regular old low-velocity expression for momentum

doesn't work when objects move with relativistic speeds.
Why not?
Well, consider a proton, which has **m = 1.67 x 10^(-27) kg**.
Suppose that we put the proton into an accelerator and
push it forward until it gains some momentum.
How fast will it be going?

Q: If we give the proton a momentum p = 4 x 10^(-19) kg*m/s what will its speedvbe? Express in terms ofc. Q: If we give the proton a momentum p = 8 x 10^(-19) kg*m/s what will its speedvbe? Express in terms ofc.

Whoops! That's a problem. It should not be possible to accelerate objects with mass to the speed of light, let alone to superluminal speeds.

It turns out that there is a slightly different expression
for momentum which must be used when objects are moving
at relativistic speeds.
I won't derive it here; you can look at
Taylor and Wheeler's book *Spacetime Physics*
for the details if you wish.
The result is

Q: Now use the relativistic expression for momentum. If we give the proton a momentum p = 8 x 10^(-19) kg*m/s what will its speedvbe? Express in terms ofc. (Having trouble isolating the speedv? Maybe you should peek at a bit of algebra. )

If a particle is moving at very high speeds,
so that its velocity **v** is *almost* equal to **c**,
then
the relativistic expression for momentum
will be approximately

Note how similar these two expressions are.

It appears that we can *almost*
turn momentum into energy, using **c**.

How good is this approximation?
Well, it depends on how relativistic the particle happens to be.
Consider an electron, which has mass
**m = 9.1 x 10^(-31) kg = 0.511 MeV / c^2. **

Q: Two electrons are sent through an accelerator. Electron A has total energy A: 1 MeV = 1.6 x 10^(-13) J but electron B has total energy B: 1 GeV = 1.6 x 10^(-10) J For each electron, what is itsgammafactor? what is its velocity? what is its momentum? How close does (momentum timesc) really come to the total energy?

The connection between momentum and total energy

works best for particles which are moving at
speeds very close to **c**,
so that their **gamma** factors are very large.
Under these hyper-relativistic conditions,
the total energy

is almost completely kinetic energy,
with only a tiny contribution from rest energy.
The closer the speed is to **c**,
the larger the contribution from kinetic energy,
and the closer the relationship is to perfect equality.

Suppose that we take it to the limit: what if a particle was moving with the speed of light? In that case, we should expect that the total energy would be ENTIRELY kinetic, and the momentum should be EXACTLY

"Wait a minute!" you might cry. "That's impossible! Objects with mass cannot travel at the speed light!"

Correct. Objects with mass can't travel at the speed of light.
But what about **objects without mass?**
In particular,
what about **PHOTONS?**

- photons have zero rest mass
- photons travel at the speed of light
- photons have energy

And so there is a perfect relationship between the energy of a photon and its momentum. Yes, photons DO have momentum, even though they have no mass!

As you will learn in a modern physics course,
the energy of a photon is related to its wavelength
and frequency via
**Planck's constant h = 6.626 x 10^(-34) kg*m ^{2}/s **.

Q: A photon of yellowish light has wavelengthlambda = 550 nm, in the middle of the human visual range. What is the energy of the photon? What is the momentum of the photon?

Hmmmm. That doesn't sound like much momentum.
However, photons are cheap;
the Sun produces many, many, MANY photons each second.
A pretty large number of those photons zip through space
past the Earth every second.
In fact,
if one goes up above the Earth's atmosphere,
very roughly
**3.8 x 10^(21) ** yellowish photons
will pass through an area
one meter on a side oriented
face-on to the Sun.

Joe places a very thin square of black plastic into space. The plastic is1 meter longby1 meter wide, and absorbs every photon which strikes it. It has a mass of0.1 kg.Q: How much momentum is transferred to Joe's plastic in one second? How much momentum is transferred to Joe's plastic in one hour? Q: (Ignore the force of gravity for now) If the plastic square was originally motionless, what is its velocity after one hour?

The momentum of photons from the Sun
provide the thrust for **solar sails,**
which may be used in future space missions.
Solar sails provide very, very small
values of thrust, so they can accelerate
payloads only gradually.
However, they don't require any fuel.

The Japanese Aerospace Exploration Agency (JAXA) launched a solar sail called "IKAROS" in 2010.

A proper treatment of solar sails does require the inclusion of gravitational forces. If you wish, you may try an extra credit project on solar sails.

An **invariant quantity** is one that
has the same value, no matter which observer is making measurements.
The space-time interval

is an example of an invariant quantity.

Is the relativistic version of momentum an invariant quantity? Let's find out.

The Blue Man throws a ball (**m = 0.2 kg **)
at a speed of **v = 0.5 c** to the right.

Q: What is the momentum of the ball, according to the Blue Man?

But the Blue Man and the ball are all travelling across
the landscape at a speed of **w = 0.8 c** to the right,
as measured by the Red Men.

Q: What is the momentum of the ball, as measured by the Red Men?

- You might look at the notes to one of the other courses I teach, Modern Physics for more details about kinetic energy and momentum in the non-relativistic regime.
- See Taylor and Wheeler, Spacetime Physics . In Chapter 7, they discuss relativistic expressions for energy and momentum, which become entwined in a quantity they call "momenergy."
- Interested in the physics of rocket propulsion? You might look at a lecture on rockets and momentum from another course of mine.
- The Solar Sails wiki has links to lots of information on solar sails.
- The IKAROS project is a solar sail recently launched by the Japanese space science agency JAXA. This technical presentation provides a great deal of information on the mission, and includes plans for a followup mission with a larger sail in the future. This conversation between Ikaros and its deployable cameras (in Japanese) is really cute. You can read an English translation on the Planetary Society blog .
- The Planetary Society is building a small satellite to test a light sail of its own.

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.