# Fermilab

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 Questions About Physics

Fast object mass increase?

You Wrote:
Hi, Some time ago, I did some, amateur, research into the speed of light and why things cannot travel faster than the speed of light. dv=dt/ds being the equation for velocity at standard "normal" speeds, what is the equation for velocity approaching that of the speed of light.

Hi Ken,

well, actually the speed is defined by the formula v= ds/dt always. No matter how fast an object moves, this is the definition of its speed. Just to raise your curiosity, I tell you, that in relativistic physics, i.e. when you have objects moving very fast, 1 plus 1 is never 2!!! What do I mean by that? By that I mean the following. If you ride your bike with a speed 15 miles/h, and you have an apple in your hand, and you throw it to your friend standing in front of you with a speed 5 miles/h with respect to you, your friend will be hit by the apple moving with the speed

15+5=20 miles/h.

This addition of speeds works only for low speed objects, where low means far from the speed of light.

If you were moving with a speed close to the speed of light, and you were able to throw the apple with a speed again close to the speed of light, your friend would not see the apple coming with

c + c = 2c,

but according to Einstein's theory of relativity, the right formula would be

c + c = c !!!!!!!!

Isn't it interesting?

Also, the equation E=mc^2 is modified in Einstein's special theory of relativity with the addition of (sqrt(1-v^2/c^2)). I understand this is from Lorenz or Lorenz's theorem. He tried to prove there was an "ether" in space which allowed light to travel similar to air and sound waves. Is this correct.

Yes, there is certainly a factor of 1/sqrt(1-v^2/c^2). Let us see what it means. The E in the above formula always means the TOTAL energy of the object. That is the maximum energy, that one can get from an object. ( for example by annihilation). The c is the speed of light. The parameter m is the one, we have to talk a little bit about. If an object does not move, m stands for for the rest mass of the object, i.e. you take your object and put it onto a scale and m is what the scale shows.

If the object moves, the situation is different. Its total energy is obviously different then the rest mass multiplied by c squared, because it has an additional part, called the kinetic energy. In order to reflect this kinetic energy in the total energy, we have to modify the equation E=mc^2 by a factor 1/sqrt(1-v^2/c^2), where v is the speed of the moving object. So it will look like

E=m/sqrt(1-v^2/c^2)*c^2, where m is the REST mass of the object.

Some people thought, this formula is ugly, and they decided to introduce a new mass, called the dynamic mass M, defined by

M=m/sqrt(1-v^2/c^2)*c^2

and then the Einstein's formula will look nice again,

E=Mc^2.

This trick will make it easier to use many of the fundamental formulae from classical mechanics in Einstein's theory of relativity, just by simply exchanging the rest mass m for the dynamical mass M. ( Also it is easy to show, that for very low speeds, compared to the speed of light, m=M.)

This factor 1/sqrt(1-v^2/c^2) indeed comes from the measurements related to the experiments, in which scientists tried to prove the existence of an absolute frame of reference( ether), in which the light moves. They made different kind of measurements, and the failure to find the absolute frame of reference forced them to explain the experimental results in many ways. For example, Lorenz believed in ether so much, that he did not mind to introduce an idea of contraction of objects moving with respect to the ether, just to keep the theory of ether "alive". His contraction factor was precisely 1/sqrt(1-v^2/c^2). Then Einstein came, said NO ETHER, wrote his theory and all over the sudden, many questions got answers.

The research I did was more along the lines of my own curiosity and was not part of an academic endeavor and I am not totally sure any of this is correct. I took notes but can no longer find them. My curiosity has been peeked with a article I received via email..."the physics of Santa clause".

Dear Ken, thank you for your questions and your comment at the end. It gives me a chance to tell you, how important in our field the curiosity is. CURIOSITY IS ACTUALLY THE POWER, THAT ROTATES THE WHEELS OF FUNDAMENTAL SCIENCE WE DO AT FERMILAB. We are curious, and we are looking for answers, and in this process we find many new questions, many new answers ... . So please be curious, and keep wondering.

- bye,
Arnold Pompos

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 last modified 1/20/1999   physicsquestions@fnal.gov

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