# Fermilab

## Inquiring Minds

All About Light Main Page  |  Classical  |  Relativistic  |  Quantum

 Relativistic - Slow Down on the Highway! c and Einstein  |  c Top Speed  |  Speed & Speed  |  Faster than c? Mass Gain  |  E=mc2  |  Age Slower  |  Time Dilation  |  Twin Paradox Dopper Shift  |  Ticketed  |  Simultaneity  |  Competition  |  Fast Strudel Paradox? No!  |  Confusions  |
Where is the famous in this whole business? I want to see Einstein's formula. How does it fit into the discussion of high-speed motion? The answer lies on this and the next page.

Any moving object gains mass!!!

This fact has an important consequence, namely that you should slow down on the highways if you do not want to be too heavy. Wait a second, what do I mean by that?

When we describe the laws of motion in the framework of classical physics (before including relativistic effects), we have a very important parameter in the equations, the REST MASS of the moving object. (Think about Newton's law , or the formula for the kinetic energy , and so on.) How do you measure this rest mass? You weigh an object. Stop the object's motion with respect to the scale, put the object on the scale and there you have it: what you read from the scale is the rest mass. This is the mass in all the equations of classical physics.

However, in the case of the theory of relativity, people noticed that the rest mass always enters into the equations in combination with another factor

(which, as you see, depends on the speed of the object!). They also noticed that the equations of motion arequite different from the classical ones because of this factor gamma. The greatest surprise came when Einsteinintroduced the dynamic mass m by the formula

All of a sudden, many relativistic formulae looked very much like the classical ones, with the understanding, of course, that the mass parameter in those formulae were not the rest mass anymore, but the dynamic mass. Example:

and

Notice that with any nonzero speed, the dynamic mass is greater than the rest mass, which explains the title of this page. Also notice, however (by starting to plug in some speeds v into the formula for the dynamic mass), that does not differ from very much until the speed of the object is very close to the speed of light.

For example if you move on the highway at 50% of the speed of light, all you gain is only 15% of your rest mass. If you move at 99.99% of the speed of light, your mass increases by a factor 70! (Please do not try these experiments, because the police are very sensitive to relativistic speeds, as we will also see when we discuss "How to convince an officer that you are not responsible for running a red light.")

Conclusion: Moving objects gain mass.

But wait, where is this extra mass from? We did not take any glue, and we did not glue anything to the moving object!!!

The answer is hidden in Einstein's famous

formula on the next page.

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