
If a particle with restmass 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: 20070621   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
mc^{2}/((1(v^{2}/c^{2})^{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. Cheers,
  Answer 3:
We calculate that the relativistic form of the
energy of a particle is E = mc^{2} /
sqrt(1v^{2}/c^{2}). 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 = mc^{2} as expected. And if you set v
= c, you get E = mc^{2} / 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(1x), 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'c^{2}, where m' is the physical mass.
Then we get that m' =
m/sqrt(1v^{2}/c^{2}), and you
will see that this number gets bigger and bigger
as v goes up.
  Answer 4:
A particle with nonzero restmass cannot be
accelerated to the speed of light. Put in other
terms, the energy of a moving particle with
restmass m equals E=(r1)mc^{2}, 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 restmass 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  v^{2} / c^{2}),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
1v^{2}/c^{2} 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 differencewhich 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
videoondemand here: videoondemand 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 
((v^{2})/(c^{2}))^{(1/2)}) (I
can't use special characters here, but the
expression in the parentheses is the square root
of one minus vsquared over csquared) 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 v^{2}/c^{2}is 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
= mc^{2} 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 massenergy 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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