Modern Phys ics

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19.1 SPECIAL RELATIVITY

19.2 THE DISCOVERY OF THE ATOM

19.3 QUANTUM PHYSICS

19.4 NUCLEAR PHYSICS

19.5 KEY FORMULAS 19.6 PRACTICE QUESTIONS 19.7 EXPLANATIONS

Special RelativitySpecial relativity is the theory developed by Albert Einstein in 1905 to explain the

observed fact that the speed of light is a constant regardless of the direction or

velocity of one’s motion. Einstein laid down two simple postulates to explain this

strange fact, and, in the process, derived a number of results that are even stranger.

According to his theory, time slows down for objects moving at near light speeds, and

the objects themselves become shorter and heavier. The wild feat of imagination that

is special relativity has since been confirmed by experiment and now plays an

important role in astronomical observation.

The Michelson-Morley ExperimentAs we discussed in the chapter on waves, all waves travel through a medium: sound

travels through air, ripples travel across water, etc. Near the end of the nineteenth

century, physicists were still perplexed as to what sort of medium light travels

through. The most popular answer at the time was that there is some sort of invisible

ether through which light travels. In 1879, Albert Michelson and Edward Morley

made a very precise measurement to determine at what speed the Earth is moving

relative to the ether. If the Earth is moving through the ether, they reasoned, the

speed of light should be slightly different when hitting the Earth head-on than when

hitting the Earth perpendicularly. To their surprise, the speed of light was the same in

both directions.

For people who believed that light must travel through an ether, the result of the

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Michelson-Morley experiment was like taking a ride in a boat and discovering that

the boat crossed the wave crests at the same rate when it was driving against the

waves as when it was driving in the same direction as the waves.

No one was sure what to make of the Michelson-Morley experiment until 1905, when

Albert Einstein offered the two basic postulates of special relativity and changed

forever the way we think about space and time. He asked all sorts of unconventional

questions, such as, “What would I see if I were traveling at the speed of light?” and

came up with all sorts of unconventional answers that experiment has since more or

less confirmed.

The Basic Postulates of Special RelativitySpecial relativity is founded upon two basic postulates, one a holdover from

Newtonian mechanics and the other a seeming consequence of the Michelson-Morley

experiment. As we shall see, these two postulates combined lead to some pretty

counterintuitive results.

First PostulateThe laws of physics are the same in all inertial reference frames.

An inertial reference frame is one where Newton’s First Law, the law of inertia, holds.

That means that if two reference frames are moving relative to one another at a

constant velocity, the laws of physics in one are the same as in the other. You may

have experienced this at a train station when the train is moving. Because the train is

moving at a slow, steady velocity, it looks from a passenger’s point of view that the

station is moving backward, whereas for someone standing on the platform, it looks

as if the train is moving forward.

Einstein’s first postulate tells us that neither the passenger on the train nor the person

on the platform is wrong. It’s just as correct to say that the train is still and the Earth

is moving as it is to say that the Earth is still and the train is moving. Any inertial

reference frame is as good as any other.

Second Postulate

The speed of light in a vacuum is a constant— m/s—in every

reference frame, regardless of the motion of the observer or the source of

the light.

This postulate goes against everything we’ve learned about vector addition.

According to the principles of vector addition, if I am in a car moving at 20 m/s and

collide with a wall, the wall will be moving at 20 m/s relative to me. If I am in a car

moving at 20 m/s and collide with a car coming at me at 30 m/s, the other car will be

moving at 50 m/s relative to me.

By contrast, the second postulate says that, if I’m standing still, I will measure light to

be moving at m/s, or c, relative to me, and if I’m moving toward the source

of light at one half of the speed of light, I will still observe the light to be moving at c

relative to me.

By following out the consequences of this postulate—a postulate supported by the

Michelson-Morley experiment—we can derive all the peculiar results of special

relativity.

Time DilationOne of the most famous consequences of relativity is time dilation: time slows down

at high speeds. However, it’s important to understand exactly what this means. One

of the consequences of the first postulate of special relativity is that there is no such

thing as absolute speed: a person on a train is just as correct in saying that the

platform is moving backward as a person on the platform is in saying that the train is

moving forward. Further, both the person on the train and the person on the platform

are in inertial reference frames, meaning that all the laws of physics are totally

normal. Two people on a moving train can play table tennis without having to

account for the motion of the train.

The point of time dilation is that, if you are moving relative to me in a very highspeed

train at one-half the speed of light, it will appear to me that time is moving slower on

board the train. On board the train, you will feel like time is moving at its normal

speed. Further, because you will observe me moving at one-half the speed of light

relative to you, you will think time is going more slowly for me.

What does this all mean? Time is relative. There is no absolute clock to say whether I

am right or you are right. All the observations I make in my reference frame will be

totally consistent, and so will yours.

We can express time dilation mathematically. If I were carrying a stopwatch and

measured a time interval, , you would get a different measure, t , for the amount of

time I had the stopwatch running.

The relation between these measures is:

So suppose I am moving at one-half the speed of light relative to you. If I measure 10

seconds on my stopwatch, you will measure the same time interval to be:

This equation has noticeable effects only at near light speeds. The difference between

t and is only a factor of . This factor—which comes up so frequently

in special relativity that it has been given its own symbol, —is very close to 1 unless

v is a significant fraction of c. You don’t observe things on a train moving at a slower

rate, since even on the fastest trains in the world, time slows down by only about

0.00005%.

Time Dilation and SimultaneityNormally, we would think that if two events occur at the same time, they occur at the

same time for all observers, regardless of where they are. However, because time can

speed up or slow down depending on your reference frame, two events that may

appear simultaneous to one observer may not appear simultaneous to another. In

other words, special relativity challenges the idea of absolute simultaneity of events.

EXAMPLE

A sp aceship of alien sp orts enthusiasts p asses by the Earth at a sp eed of 0.8c,watching the final minute of a basketball game as they zoom by . Though the clock onEarth measures a minute left of p lay , how long do the aliens think the game lasts?

Because the Earth is moving at such a high speed relative to the alien spaceship, time

appears to move slower on Earth from the aliens’ vantage point. To be precise, a

minute of Earth time seems to last:

Length Contraction

Not only would you observe time moving more slowly on a train moving relative to

you at half the speed of light, you would also observe the train itself becoming

shorter. The length of an object, , contracts in the direction of motion to a length

when observed from a reference frame moving relative to that object at a speed v.

EXAMPLE

You measure a t rain at rest to have a length of 100 m and width of 5 m. When y ouobserve this t rain t raveling at 0.6c (it ’s a very fast t rain), what is it s length? What isit s width?

WHAT IS ITS LENGTH?We can determine the length of the train using the equation above:

WHAT IS ITS WIDTH?The width of the train remains at 5 m, since length contraction only works in the

direction of motion.

Addition of VelocitiesIf you observe a person traveling in a car at 20 m/s, and throwing a baseball out the

window in the direction of the car’s motion at a speed of 10 m/s, you will observe the

baseball to be moving at 30 m/s. However, things don’t quite work this way at

relativistic speeds. If a spaceship moving toward you at speed u ejects something in

the direction of its motion at speed relative to the spaceship, you will observe that

object to be moving at a speed v:

EXAMPLE

A sp aceship fly ing toward the Earth at a sp eed of 0.5c fires a rocket at the Earth thatmoves at a sp eed of 0.8c relat ive to the sp aceship . What is the best ap p roximat ion forthe sp eed, v, of the rocket relat ive to the Earth?

(A) v > c

(B) v = c

(C) 0.8c < v < c

(D) 0.5c < v < 0.8c

(E) v < 0.5c

The most precise way to solve this problem is simply to do the math. If we let the

speed of the spaceship be u = 0.5c and the speed of the rocket relative to the spaceship

be = 0.8c, then the speed, v, of the rocket relative to the Earth is

As we can see, the answer is (C). However, we could also have solved the problem by

reason alone, without the help of equations. Relative to the Earth, the rocket would

be moving faster than 0.8c, since that is the rocket’s speed relative to a spaceship that

is speeding toward the Earth. The rocket cannot move faster than the speed of light,

so we can safely infer that the speed of the rocket relative to the Earth must be

somewhere between 0.8c and c.

Mass and EnergyMass and energy are also affected by relativistic speeds. As things get faster, they

also get heavier. An object with mass at rest will have a mass m when observed to

be traveling at speed v:

Kinetic EnergyBecause the mass increases, the kinetic energy of objects at high velocities also

increases. Kinetic energy is given by the equation:

You’ll notice that as v approaches c, kinetic energy approaches infinity. That means it

would take an infinite amount of energy to accelerate a massive object to the speed of

light. That’s why physicists doubt that anything will ever be able to travel faster than

the speed of light.

Mass-Energy EquivalenceEinstein also derived his most famous equation from the principles of relativity. Mass

and energy can be converted into one another. An object with a rest mass of can

be converted into an amount of energy, given by:

We will put this equation to work when we look at nuclear physics.

Relativity and GraphsOne of the most common ways SAT II Physics tests your knowledge of special

relativity is by using graphs. The key to remember is that, if there is a dotted line

representing the speed of light, nothing can cross that line. For instance, here are two

graphs of kinetic energy vs. velocity: the first deals with normal speeds and the

second deals with relativistic speeds:

In the first graph, we get a perfect parabola. The second graph begins as a parabola,

but as it approaches the dotted line representing c, it bends so that it constantly

approaches c but never quite touches it, much like a y = 1/x graph will constantly

approach the x-axis but never quite touch it.

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