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If a particle with rest-mass were to, in theory, travel at the speed of light ,would its mass actually be infinite, or just very, very, very, large, just like it would supposedly take an infinite amount of energy to accelerate the particle to the speed of light in the first place? How can you calculate this?
Question Date: 2007-06-21
Answer 1:

The mass of the particle approaches infinity as the speed approaches that of light in a vacuum. These however are limits and are not achieved by particles with non zero rest mass.

Answer 2:

It is impossible for a particle with mass to reach the speed of light. AtFermilab, for example, protons are accelerated close to the speed oflight, and it takes a huge amount of energy.

The rest mass does not change - by definition, it is the mass, orequivalent energy, of a particle AT REST. The total energy is theparticle's rest mass energy PLUS its kinetic energy. Einstein discoveredthat the total energy of a particle moving at speeds close to the speed oflight (relativistic speeds) is given as mc2/((1-(v2/c2)1/2). The totalenergy - rest energy plus kinetic energy - changes, and that is what you,as an "external observer" of a relativistic particle, can measure. You canonly measure rest mass if you are at rest relative to the particle.

Just from this formula, you can see that as v approaches c, thedenominator approaches zero, so the total energy becomes undefined.

How do you get a particle to accelerate? You know from your study ofphysics that you have to apply a force. For something like a proton, youcan apply a constant force, thus increasing its acceleration, bycirculating it in a magnetic field. The magnetic field is external to theparticle, so it can keep doing work on the particle. This is how they doit at particle accelerators - the protons (or antiprotons, or whatever)circulate round and round, accelerating to .99999c. They get arbitrarilyclose to the speed of light, but theoretically nothing can actually get tothe speed of light.

Interesting effects happen - if you could actually travel at c, time wouldstop, and the distances in front of you would shrink to nothing, so thatyou would be everywhere simultaneously! Now, how bizarre is THAT!

A good book to read is The Physics of Star Trek by Lawrence Krauss. Youcan get this at Borders, or on Amazon.


Answer 3:

We calculate that the relativistic form of the energy of a particle is E = mc2 / sqrt(1-v2/c2). Where m is the rest mass, and v is the velocity with respect to your frame, and this E is the energy with respect to your frame. So if you set v = 0, then E = mc2 as expected. And if you set v = c, you get E = mc2 / sqrt(0). So basically you're dividing by zero. This is undefined.

However, you can talk about the limit as v goes to c. even though the actual value is not defined as anything, if we make v closer and closer to c, then E goes higher and higher and approaches infinity. So we consider it impossible to come up with enough energy to get something with rest mass up to the speed of light.

click_to_graph has a grapher you can use. I advise you to plot 1/sqrt(1-x), and see how it behaves as x -> 1 (which is equivalent to v->c). Note that if you go above 1, you get an imaginary answer (as in imaginary numbers like sqrt(-1)) which are related to the tachyonic solutions I mentioned before (which can also never get to c).

So as far as our current laws of physics are concerned it is an impossibility to bring v up to c. The idea of infinity is not an actual number, but rather a concept. There is no such number as infinity. It literally means something that goes impossibly and arbitrarily high as you approach some limit. So it's not like giving a particle as much energy as you like will bring it up to c. It is that you can put as much energy as you like, and it will never get to c.

So talking about a rest mass particle with velocity c doesn't make any sense physically. It's like asking someone about the physics of how ghosts move through walls: as far as physics is concerned (to our current knowledge) those things simply don't exist.

If you want to calculate physical mass, then you set E = m'c2, where m' is the physical mass. Then we get that m' = m/sqrt(1-v2/c2), and you will see that this number gets bigger and bigger as v goes up.

Answer 4:

A particle with non-zero rest-mass cannot be accelerated to the speed of light. Put in other terms, the energy of a moving particle with rest-mass m equals E=(r-1)mc2, where the factor r=1/sqrt(1-(v/c)2), with v the speed of the particle and c the speed of light. You can use this formula in an Excel sheet to try different values of rest-mass m and speed v. This equation tells you that you need an infinite amount of energy to accelerate a particle to (exactly) the speed of light, however, you can always take it to, say 99.99999% the speed of light with a finite (but huge) amount of energy. Enjoy!

Answer 5:

Here's what we know from the best physics we have available, namely Einstein's special relativity. A particle which has a finite (nonzero) rest mass would indeed have a infinite apparent mass if it could travel *exactly at* the speed of light. If it were *almost* at the speed of light, then it would "only" be very large. Anything with an infinite mass would suck in the rest of the universe at light speed, but don't worry about that happening, because you would have to supply an infinite amount of energy to make the bugger in the first place!

The equation to calculate relativistic mass from rest mass is M = m / sqrt(1 - v2 / c2),where M is the relativistic mass, m is the rest mass, v is the velocity, c is the speed of light, and sqrt(...) means the square root of everything in parentheses. If you plug in numbers where v is much smaller than c (for an object traveling much slower than the speed of light), then 1-v2/c2 is approximately 1, so M=m: we only see the rest mass. On the other hand, if v=c (traveling *at* the speed of light), then M=m/sqrt(0)=m/0. If you divide any finite number by zero, you get infinity. You can try other values with a calculator or spreadsheet.

Now if you were traveling along with the object, even at the speed of light, it would appear to have its ordinary, rest mass. It's only to the outside world that it appears to have a greater mass. In other words, it's only if the object is moving *relative to you* that you see a difference--which is why we call it relativity. If you're moving along with the object, we say that you are in the object's reference frame. A stationary observer has a different reference frame.

Of course, it may be possible that special relativity is wrong, and something else happens at extremely high velocities. But big particle accelerators like at Stanford and in Switzerland use special relativity every day, and it's been perfectly correct even for the fastest particles we can accelerate.

Good questions! If you can get it, check out the Mechanical Universe video series sometime. It's a lot of fun, and it's where I first learned physics. Some libraries carry the videotapes. It looks like there might be free video-on-demand here:
It requires a free registration, so have a parent/guardian fill that part out. (Can't trust who's on the other end on the Internet!) The lectures on relativity start around Lecture #41.

Have fun!

Answer 6:

It's rest mass would be undefined, because you would be dividing by zero. Effectively, this means infinite.

Here's the equation:M = m / ((1 - ((v2)/(c2))(1/2))

(I can't use special characters here, but the expression in the parentheses is the square root of one minus v-squared over c-squared)

M = relativistic mass (mass of the moving particle)
m = rest mass
v = velocity
c = speed of light

So, you see, if you plug v = c in here, then v2/c2is 1, 1 - 1 = 0, the square root of 0 is 0, so you ultimately wind up with M = m/0, so anything with a finite m is going to have an infinite M.

The only way that a particle can be moving the speed of light and not be infinitely massive is if it has no rest mass, in which case you wind up with M = 0/0,which is also undefined, but when you take the limit as m goes to 0 and v goes to c, then you can come up with a finite value. This is why photons can still have energy.

This branch of physics is called special relativity.You can do some more algebra with these equations and come up with the famous E = mc2 equation, and what that means is that energy (E) is equivalent to mass(m), and the proportionality constant is the square of the speed of light (c). If you want to calculate the amount of energy needed to get a particle up to a certain speed, just plug in the value of v that you want and calculate the difference between M and m(that's the kinetic energy component of the particle's mass), and then plug that in for m in the mass-energy equivalence equation, and calculate E. For the record,one gram of matter converted to energy comes out to about 20,000 tons of TNT - about the size of the atomic bomb that was dropped on Hiroshima.

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