Chapter 26
The Giancoli onLine Tutor

Special Theory of Relativity

I. Relativistic SpaceTime


I-a. Relativity and Time

Faced with the choice of having time and space absolute and the laws of physics relative or time and space relative and the laws of physics ansolute, Einstein chose the latter. This choice entwines space and time; no longer are space and time separate. We must now operate in the world of space-time. At the left we see the first three ticks of a clock stationary in space as it moves through space-time. In space-time the earths orbit becomes not a circle but a spiral around the time axis. Time itself becomes reative to the observer. To express this we first define

G= SQRT(1-(v/c)2)
and
g= 1/ G
.


Given this we can write

Dt= Dtog

where 'v' is the relative velocity between an observer and another observer or object. For example, if a rocket passes us at a speed v, Dto would be the time on a clock on the rocket and Dt would be our time. But since there is no such thing as absolute motion or absolute rest, the rocket man could as well claim that he was at rest and we would be the one that is moving with velocity -v. Leading to the reverse application of the above formula. This is called Time Dilation. To appreciate this 'play around with the following demo. "I" am the earth observer and you are "rocket man".
My time as seen by me
Your time as seen by me
Your time as seen by you
My time as seen by you
Relative Velocity
You can stop the above at any point and verify the results. To help keep things straight it is useful to utilize a more elaborate notation.
A variable T with repeated subscripts discribes a proper time, time as kept on a clock at rest with respect to an observer. Proper time is always multiplied by the g factor in these calculations. Thus

TE/R= TE/Eg

TR/E= TR/Rg


I-b. Relativity and Space


Space is also affected by the new space-time dynamic. Spatial measurements made by observers with a relative velocity will no longer agree. If the rocket man has measured his rocket ship to be Lo long, we will measure its length, L, to be given by

L= LoG


This is called Length (sometimes Lorentz) Contraction.

Interactive Example


A muon, m, is an atomic found in cosmic rays. It has a lifetime when it is at rest (as seen by itself) of 2.2x10-6 seconds. Given that it is travelling at .8c with respect to our laboratory, answer the following questions.

What is the Proper Time (in microseconds) for its lifetime?
Enter your answer and click the clover for good luck (and the answer) .
 
How long do we see it live (in microseconds)?
Enter your answer and click the clover for good luck (and the answer) .
 
How far do we see it travel?
Enter your answer and click the clover for good luck (and the answer) .
 
From the muons point of view, it is not moving and the laboratory is moving past it at .8c. How much laboratory space passes by it in its lifetime?
Enter your answer and click the clover for good luck (and the answer) .
 

If Lo equals your answer to the third question above what is L?
Enter your answer and click the clover for good luck (and the answer) .
 

II. Relativistic Energy and Momentum


II-a. Relativistic Energy

In Special Relativity we do not deal with potential energy. All we have to deal with is energy of motion and mass energy. The governing formula is
E= moc2g
where mo is the mass of the particle. Note that when v=0, g=1
Eo= moc2 and this represents the energy associated with the mass of the particle.


The kinetic energy is the difference between the total energy E and the rest energy Eo.

KE= moc2[g-1]


Although you generally need to use metric units in working problems. However, commonly energies will be given in units of MeV (million-electron-volts; 1 MeV= 1.6 x 10-13 J). The rest energy of the muon mentioned above is 105.7 MeV. As a general rule you need to use relativistic formulae if a particales KE is greater than .5% its rest energy or its velocity is greater than 10% the speed of light.For relativistic problems involving energy calculations, simplification will occur if both E and moc2 are expressed in MeV. No conversion is necessary because the same units occur on both sides of the equal sign.

II-b. Relativistic Momentum

The momentum of a particle is given by

p= movg



Interactive Examples

A muon has a total energy of 200 Mev what is its kinetic energy (in MeV) ?
Enter your answer and click the clover for good luck (and the answer) .
 

What is the v/c for this particle?
Enter your answer and click the clover for good luck (and the answer) .
 
Show me the details!

Its mass is 1.9x10-28 kg, what is its momentum?
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Show me the details!

II-c. Energy and momentum combined



Note that the formulae for E and p can be combined to form the relationship
E=SQRT((pc)2 + (moc2)2)

This gives you a way to find the momentum of a particle without finding its velocity if you know its energy. NOTE: you must convert E and moc2 to joules when using this formula.

III. Relativistic Velocity Addition


In the figure a flying saucer passes us with a speed v and fires a projectile straight ahead at a speed u', the speed, u, we see for the projectile is given by the formula. NOTE: If the 'saucer' is going to the left v -> -v and if the projectile is going to the left u' -> -u'.

Interactive Examples


If the 'saucer' is travelling at 0.6c with respect to the earth and the projectile is moving at 0.8c with respect to the saucer, what is u/c?
Enter your answer and click the clover for good luck (and the answer) .
 

If the 'saucer' is travelling at 0.6c with respect to the earth and the projectile is moving at 0.8c (to the left) with respect to the saucer, what is u/c?
Enter your answer and click the clover for good luck (and the answer) .
 



finis

© 1999 Carl Adler mailto:Carl@Image-ination.com
















  1. 200=105.7/SQRT(1-(v/c)2)
  2. SQRT(1-(v/c)2)=105.7/200
  3. 1-(v/c)2= (105.7/200)2
  4. (v/c)2= 1- (105.7/200)2
  5. v/c= SQRT(1- (105.7/200)2)
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  1. p= 1.9x10-20*.849*3x108*g
  2. 200=105.7g
  3. g=200/105.7
  4. p= 1.9x10-20*.849*3x108*200/105.7= 9.16x10-20
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