# Energy of motion at high speed

1. Jul 5, 2005

### lechiffre

Now I don't pretend to be anything but a layman so 'm not sure about this but here goes anyway.........the theory contends that a body's apparent mass increases as it accelerates on account of an increase in its energy of motion. Is this energy thermal energy? If so what is its source? I read in Einsteins paper that if you approached a star at lightspeed the frequency of light would be infinite (from the star). Does this mean QED that your temperature, absorbed energy and thus apparent mass would all become very great (ie infinite)? Would this account for the inertia which prevents further acceleration? Would the direction of travel then be irrelevant, since high speed motion would foreshorten everything ahead of the traveller? I've puzzled over this matter of 'energy of motion' a lot but never seem to have found a reference to it that made any sense to me. Am I just a dud at science? Surely not altogether? Better barking up the wrong tree than at the moon I suppose..................

2. Jul 5, 2005

### Staff: Mentor

No, the added energy is kinetic energy, which comes directly from the work done by the force that accelerates the body. It's possible to derive the relativistic equation for kinetic energy,

$$K = \frac {mc^2}{\sqrt{1 - v^2 / c^2 }} - mc^2$$

(where $m$ is what many people call "rest mass") directly from the work-energy theorem, basically by taking time-dilation into account.

Re-arrange the above equation a bit and you get

$$E = E_0 + K$$

where

$$E = \frac {mc^2}{\sqrt{1 - v^2 / c^2 }}$$

is the (total) energy of the body,

$$E_0 = mc^2$$

is the "rest energy" that the body has because of its ("rest") mass, and $K$ is the kinetic energy as defined above.

3. Jul 5, 2005

### pervect

Staff Emeritus
that's "relativistic mass"
No. It's kinetic energy.
I guess it depends on how perverse your imagination is. As you approach the speed of light, the energy of the blueshifted light from in front of you does increases without limit. However, I would not say that your temperature would become infinite, but rather that your cooling systems, which would certainly try to keep your temperature to a survivable level, would eventually fail, and you would vaporize along with your ship.

With a sufficiently perverse imagination I suppose you could imagine some sort of ideal substance that would heat up indefinitely, and I suppose such a substance would heat up indefinitely, having an infinte temperature. But such a substance wouldn't be possible in actuality, any physical substance will vaporize when it gets hot enough. I think it's likely that assuming such an idealized substance exists will eventually lead to a contradiction (just as assuming that rigid rods exist leads to contradictions). If you are goind to make assumptions, why not assume that your cooling system keeps working.
[/quote]

You don't need inertia to account to prevent further acceleration. Nor do you need the rocket to "heat up" to prevent further acceleration. With idealized cooling systems, it could remain perfectly cool and comfortable on the rocket, and it STILL wouldn't be able to reach the speed of light.

The fact that one's acceleration from the point of view of an inertial observer decreases with one's velocity is due to the way the velocity addition formula works in relativity.

Using the formula

v_sum = (v1+v2)/(1+v1*v2/c^2)

one can add any number of velocities, v, together, and as long as each individucal velocity is less than c, the sum will be less than c Try it!

If you want a proof, look at the relativistic sum of c+c. It is equal to c. It's also true the the relativistic sum of any two numbers increases monotionically with v1 and v2. Thus the sum of any two numbers v1,v2 < c is <c.

While this failure to reach 'c' can also be explained in terms of relativistic mass, such an approaach is not necessary. One can explain it without any recourse to relativistic mass whatsoever.

Certainly everything in front of the traveller would be greatly foreshortened, but I'm not sure why you think that means the direction of trvael would be "irrelevant". If he maintains his heading and accelerates in the same direction all the time, things will continue to foreshorten. If he accelreates perpendicularly to his velocity, his direction will change, and the amount of foreshorteining (his gamma factor) will remain constant.

You might be interested in the actual equations which describe the position and velocity a rocket would take if it could accelerate at 1g (measured by an accelerometer mounted on the rocket) indefinitely. These are known as the relativistic rocket equations, and are descriped at

http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

4. Jul 8, 2005

### lechiffre

Hhhhhhmmmmmm.................think I'll stick to woodwork. I know for sure of only two ways to increase the size (inertial mass) of a body:

A) Bolt an extra bit on
B) Heat it up

I wouldn't mind betting there's a mathematical correlation between inertial mass, rest mass and temperature. Ijust wish I had the maths to figure it out

5. Jul 8, 2005

### Meir Achuz

"Is this energy thermal energy?"
It can't be thermal energy, because thermal energy is a property of an object in its rest system, so no motion of the object can change this.
Your problem is retaining the early, and confusing, notion (no longer in fashion)
that mass increases with velocity. As Bell's first post shows, the KE increases, but the mass doesn't change. I would strengthen his attribution of "rest mass" to call the m in his equation "invariant mass". Then your confusion about thermal energy would never arise. To get technical: Mass is a relativistic invariant because it is the invariant length of the momentum-enrgy four-vector.