Mass Distortion Help (page 2)
Another interesting effect of relativistic speeds is an increase in the masses of objects as they move faster and faster. This increase occurs to the same extent as the decrease in length and the slowing down of time.
Point Of View: Mass
If we travel inside a space ship, regardless of its speed, the masses of all the objects in the ship with us appear normal as long as our ship is not accelerating. However, from the vantage point of Earth, the mass of the ship and the masses of all the atoms inside it increase as its speed increases.
Let m be the mass of the moving ship as a multiple of its mass when it is stationary relative to an observer. Let u be the speed of the ship as a fraction of the speed of light. Then
m = 1/(1 − u 2 ) 1/2
= (1 − u 2 ) −1/2
This is the same as the factor k that we defined a little while ago. It is always greater than or equal to 1.
Look again at Fig. 20-4. As the space ship moves faster, it “scrunches up.” Imagine now that it also becomes more massive. The combination of smaller size and greater mass is a “double whammo” in regard to the density of the ship.
Suppose that the rest mass (the mass when stationary) of our ship is 10 metric tons. When it speeds by at half the speed of light, its mass increases to a little more than 11 metric tons. At 80 percent of the speed of light, its mass is roughly 17 metric tons. At 95 percent of the speed of light, the ship masses about 32 metric tons. At 99.9 percent of the speed of light, the ship’s mass is more than 220 metric tons. And so it can go indefinitely. As the speed of the ship approaches the speed of light, its mass grows larger and larger without limit.
Speed Is Self-limiting
It is tempting to suppose that the mass of an object, if it could be accelerated all the way up to the speed of light, would become infinite. After all, as u approaches 1 (or 100 percent), the value of m in the preceding formula increases without limit. However, it’s one thing to talk about what happens as a measured phenomenon or property approaches some limit; it is another matter entirely to talk about what happens when that limit is reached, assuming that it can be reached.
No one has ever seen a photon at rest. No one has ever seen a space ship moving at the speed of light. No finite amount of energy can accelerate any real object to the speed of light, and it is because of relativistic mass increase that this is so. Even if it were possible, the mass-increase factor, as determined by the preceding formula, would be meaningless. We would have to divide by zero to calculate it, and division by zero is not defined in mathematics.
The more massive a speeding space ship becomes, the more powerful is the rocket thrust necessary to get it moving faster. As a space ship approaches the speed of light, its mass becomes gigantic. This makes it harder and harder to give it any more speed. Using integral calculus, astronomers and physicists have proven that no finite amount of energy can propel a space ship to the speed of light.
You’ve heard expressions such as electron rest mass , which refers to the mass of an electron when it is not moving relative to an observer. If an electron is observed whizzing by at relativistic speed, it has a mass greater than its rest mass and thus will have momentum and kinetic energy greater than is implied by the formulas used in classical physics. This fact, unlike spatial distortion, is more than mere fodder for “mind experiments.” When electrons move at high enough speed, they attain properties of much more massive particles and acquire some of the properties of x-rays or gamma rays such as are emitted by radioactive substances. There is a name for high-speed electrons that act this way: beta particles .
Physicists take advantage of the relativistic effects on the masses of protons, helium nuclei, neutrons, and other subatomic particles. When these particles are subjected to powerful electrical and magnetic fields in a device called a particle accelerator , they get moving so fast that their mass increases because of relativistic effects. When the particles strike atoms of matter, the nuclei of those target atoms are fractured. When this happens, energy can be released in the form of infrared (IR), visible light, ultraviolet (UV), x-rays, and gamma rays, as well as a potpourri of exotic particles.
If astronauts ever travel long distances through space in ships moving at speeds near the speed of light, relativistic mass increase will be a practical concern. While their own bodies won’t seem to be more massive from their own point of view and the things inside the ship will appear normal to them, the particles whizzing by outside will become more massive in a real and dangerous way. It is scary enough to think about what will happen when a 1-kg meteoroid strikes a space ship traveling at 99.9 percent of the speed of light. However, that 1-kg stone will mass more than 22 kg when u = 0.999. In addition, every atom outside the ship will strike the vessel’s “prow” at relativistic speed, producing deadly radiation of the same sort that occurs in high-energy particle accelerators.
Relativistic time dilation and mass increase have both been measured under controlled conditions, and the results concur with Einstein’s formulas stated earlier. Therefore, these effects are more than mere tricks of the imagination.
To measure time dilation, a superaccurate atomic clock was placed on board an aircraft, and the aircraft was sent up in flight to cruise around for awhile at several hundred kilometers per hour. Another atomic clock was kept at the place where the aircraft took off and landed. Although the aircraft’s speed was only a tiny fraction of the speed of light and the resulting time dilation therefore was exceedingly small, it was large enough to measure. When the aircraft arrived back at the terminal, the clocks, which had been synchronized (when placed right next to each other, of course!) before the trip began, were compared. The clock that had been on the aircraft registered a time slightly earlier than the clock that had been resting comfortably on Earth.
To measure mass increase, particle accelerators are used. It is possible to determine the mass of a moving particle based on its known rest mass and the kinetic energy it possesses as it moves. When the mathematics is done, Einstein’s formula is always shown to be correct.
Mass Distortion Practice Problem
Suppose that a small meteoroid, whose mass is 300 milligrams (300 mg), strikes the shell of a space vessel moving at 99.9 percent of the speed of light. What is the apparent mass of the meteoroid?
Use the preceding formula for relativistic mass increase, considering u = 0.999. Then the mass is multiplied by a factor m as follows:
m = (1 − 0.999 2 ) −1/2
= (1 − 0.998) −1/2
= 0.002 −1/2
= 1/(0.002) 1/2
The mass of the meteoroid when it strikes the vessel is 300 × 22.4 mg, or 6.72 grams (g).
Practice problems of these concepts can be found at: Relativity Theory Practice Test
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