vce physics

VCE Physics

Unit 3

Einstein’s Relativity

Revision Questions

Einstein’s Relativity RevisionQuestion Type:

According to the postulates of special relativity, certain properties are dependent on the reference frame inwhich they are observed.

The mass of an electron measured at rest

The time interval between two given events

The distance between two given events

Column 1 Column 2

Q: In column 2 of the table above, indicate whether the entry in column 1 is always the same (S), or may sometimesbe different (D).

S

D

D

Relativity postulates


Q: A Year 12 physics class is studying Einstein’s special relativity.The teacher postulates a thought experiment:Imagine you are travelling at a speed of 3 × 108 ms-1 alongside a beam of light. What would you measure the speed of a beam of light to be?

Two students put up their hands to offer an answer.Hilary says: You would measure the beam of light to be moving away from you at 3 ×108 ms-1.Ryan says: You would measure the beam of light to be at rest with respect to yourself: that is, its speed would be 0 ms-1.

Q: Which student’s answer is consistent with Einstein’s special theory of relativity? Explain your reasoning.

A: Hillary is correct The speed is light is invariant in all inertial frames.

Light Speed Invariance


A container inside a rocket ship is observed through a window by Sam, an astronaut, floating freely in space.Sam observes the rocket ship travelling past at a constant speed VR = 0.2 C. The dimensions of the container, as measured by astronauts inside the rocket ship, are shown in Figure 1, and are. the proper length L0 (parallel to the direction of motion of the rocket ship). the proper width W0 (perpendicular to the direction of motion of the rocket ship). the proper height H0 (perpendicular to the direction of motion of the rocket ship).

Q: Which of the options (A to D) best describes its dimensions as observed by Sam?A. L < L0, W < W0 , H < H0

B. L < L0, W = W0 , H = H0

C. L > L0, W = W0 , H = H0

D. L < L0, W < W0 , H = H0Answer:B. L < L0, W = W0 , H = H0

Length Contraction


Michelson and Morely hypothesised: if Earth moves around the Sun then it must travel though the ether. Since the medium through which light travels is the ether, there should be a difference in the measured speed of light depending on whether the light is travelling parallel to or perpendicular to the direction of Earth’s movement through the ether.

A simplified plan of their equipment is shown in Figure 2. The apparatus was set up so that light travelling towards mirror 2 was travelling perpendicular to the motion of Earth around the Sun, and light travelling towards mirror 1 was in the direction of Earth’s motion in its orbit. Michelson and Morely observed a fringe pattern at the detector resulting from interference between the two light beams. The measurement was then repeated with the apparatus turned through an angle of 90°, and no change was seen in the interference-fringe pattern after the rotation of the apparatus.

Q: Explain the significance of this null observation.

A: • the speed of light is the same in both directions

• no motion relative to the ether • no evidence to support the existence of the ether • light does not need a medium • there is no absolute frame of reference.

The Ether


One of the basic particles of nature is the tau meson, which can be created using beams of high energy particles from an accelerator. When created, the tau meson has a very high velocity of 0.998749 c, which means it has a Lorentz factor of 20. However it only exists for a period of 6.10 × 10-12 s as measured by the scientists at the accelerator laboratory. After this time it decays into two other particles. During this time it is observed to travel a distance d. Figure 3 shows the creation and decay of the tau meson in the reference frame of the scientists.

Q: What is the lifetime of the tau meson as measured in its own frame of reference?

A: t = γto

6.10 x 10-12 s = 20(to) to = 3.05 x 10-13 s

Time dilation


When created, the tau meson has a very high velocity of 0.998749 c, which means it has a Lorentz factor of 20. However it only exists for a period of 6.10 × 10-12 s as measured by the scientists at the accelerator laboratory.

Q: What is the distance d in Figure 3, as measured by the scientists?

A: v = d/t d = v.t = (0.998749)(3 x 10-8)(6.1 x 10-12) = 1.83 x 10-3 m

Constant Velocity Motion


When created, the tau meson has a very high velocity of 0.998749 c, which means it has a Lorentz factor of 20. However it only exists for a period of 6.10 × 10-12 s as measured by the scientists at the accelerator laboratory. During this time it is observed to travel a distance d. Figure 3 shows the creation and decay of the tau meson in the reference frame of the scientists.

Q: As measured in the reference frame of the tau meson, what would be the distance d ?

A: L = Lo/γ = (1.83 x 10-3)/20 = 9.15 x 10-5 m

Length Contraction


According to Einstein’s special theory of relativity, mass and energy are related. The mass of an electron whenit is at rest is 9.1 × 10-31 kg.

Q: Show that this is equivalent to an energy of 8.20 × 10-14 J.

A: E = mc2

= (9.1 x 10-31)(3.0 x 108)2

= 8.2 x 10-14 J

Mass Energy Equivalence


The electron accelerator at the ARPANSA laboratory at Yallambie, near Melbourne, can accelerate an electron to a speed such that its mass increases by a factor of 22.

Q: What is the value of the Lorentz factor for an electron as it leaves the accelerator?

A: Since the mass increases by 22, the Lorentz factor is 22.

Lorentz Factor


Which of the following (A to D) gives the kinetic energy of the electron as it leaves the accelerator?

A. 8.20 × 10-15 JB. 1.72 × 10-12 JC. 1.80 × 10-13 JD. 5.11 × 10-6 J A:

KE = mc2 – moc2 = 22moc2 – moc2

= 21moc2

= (21)(9.1 x 10-31)(3.0 x 108)2

= 1.72 x 10-12 J Alternative B is correct

The electron accelerator at the ARPANSA laboratory at Yallambie, near Melbourne, can accelerate an electron to a speed such that its mass increases by a factor of 22.

Relativistic Kinetic Energy


Val, Pat and Bruce are discussing the meaning of Einstein’s famous equation E = mc2, when applied to an electron with mass m.Val says that an electron will transform its mass m into an amount of pure energy E, when it is travelling at the speed of light (c).Pat disagrees, and says that if it were moving at a high velocity inside a cathode ray tube it would convert its mass m into a light photon of energy E when it hits the glass face.Bruce, on the other hand, thinks that just by its existence, the electron of mass m has an energy of E.

Q: Write the name of the student with the best explanation of the equation.

A: Bruce

Mass Energy Equivalence


In 1861 James Clerk Maxwell proved that light was an electromagnetic wave with a speed of 3 × 108 ms–1. Following Maxwell’s predictions, physicists had some concerns.

Q: Which one or more of the statements (A to D) below outlines one of these concerns?A. All waves propagate in a medium, but there was no medium in empty space.B. Unlike the speed of other waves, experiments showed that the speed of light did not depend on the observer or source speed.C. If light was a wave, it would not diffract.D. The speed of light predicted by Maxwell did not agree with the measured speed.

A: Alternatives A & B

History of Relativity


Imagine two students travelling in a spaceship toward the Sun at speed v (Figure 1). They plan to measure the speed of light in two experiments, as a test of the prediction of James Clerk Maxwell and that of Galilean relativity.In the first experiment, they determine the speed of light (c) within the rocket ship by measuring the time for a short pulse of light emitted from a flashbulb at point A to reach point B. They got the accepted value for c. In the second experiment they determined the speed of light for a beam of light from the Sun, which passed through a porthole in the rocket nose. This was done by measuring the time for the light to travel between point A and point B.


Q: Which one of the following statements (A to D) is consistent with the predictions of Maxwell, and of Galilean relativity, concerning the value of the speed of light obtained in this second measurement?A. Both predict the speed to be c.B. Maxwell’s theory predicts the speed to be c, while Galilean relativity predicts a value of (c + v).C. Maxwell’s theory predicts a speed of (c + v), while Galilean relativity predicts a value of c.D. Both predict the speed to be (c + v).

A: Alternative B

In the second experiment they determined the speed of light for a beam of light from the Sun, which passed through a porthole in the rocket nose. This was done by measuring the time for the light to travel between point A and point B.

Light Speed Invariance


In an experiment done at Massachusetts Institute of Technology, electrons were given a series of known kinetic energies (KE) by accelerating them across a range of electric potentials, and measuring the electron’s velocity, v, for each value of KE

The solid curve in Figure 2 shows the variation of v2 as a function of KE, as measured in the experiment. The dashed curve is the value of v2 calculated using the Newtonian expression for kinetic energy.


Q: According to Newtonian mechanics, what would be the speed of an electron with a KE of 10 × 10–13 J? The mass of an electron is 9.1 × 10–31 kg.

A: KE = ½mv2 v = √2KE/m = √2(10 x 10-13)/ (9.1 x 10-31) = 1.5 x 109 ms-1

Non Relativistic KE


Q: Using the experimental data shown in Figure 2, indicate which one of the statements below gives the best estimate of the measured speed of an electron with a KE of 10 × 10–13 J.A. It is approximately 9 × 1016 ms–1.B. It is approximately 3 × 108 ms–1.C. It is approximately 9.1 × 10–31 ms–1.D. It cannot be estimated.

2. Then read off v2 value of 8.5 x 1016 m2s-2.

Relativistic Kinetic Energy

A: 1. Need to extrapolate graph out to 10 x 10-13 J

This gives v = 2.9 x 108 ms-1

Alternative B is correct.


In order to explain the propagation of a light wave, it was assumed in the late 19th century that all of space was filled with ‘ether’, which not only provided a medium for the wave, but also an absolute spatial reference frame. In order to check this, Michelson and Morley performed their famous experiment. A simplified plan of their equipment is shown below in Figure 3. The arrows show the direction in which the light is travelling. The intensity at the detector relied on interference between the light travelling the different paths. The apparatus was mounted on a rigid stand so that it could be rotated.


Q: Which one of the statements (A to D) specifies the critical basis of the apparatus?A. The half-silvered mirror must reflect exactly half the light.B. Mirror 1 and mirror 2 must be identical.C. The distance travelled by light using either path must be equal.D. The components must not move relative to each other.

A: Alternative D is correct

Michelson Morley Experiment


The apparatus was set up so that the light travelling towards mirror 2 was travelling perpendicular to the motion of Earth around the Sun, and the light travelling towards mirror 1 was in the direction of Earth’s motion in its orbit. The measurement was then repeated with the apparatus turned through an angle of 90°.

Q: Explain why this was done, and what results Michelson and Morley obtained.

A: One mark was given for indicating the predicted difference in travel times in the two perpendicular directions OR that they were attempting to measure the speed of the Earth with respect to the ether OR that they were trying to compare the speed of light parallel to and perpendicular to the motion of the Earth through the ether. The second mark was for explaining that rotating the apparatus was expected to produce a shift in the interference pattern. The third mark was given for stating that no shift in the interference fringes was observed OR that the speed of light was the same in both directions OR that the experiment produced no evidence to support the idea of motion through the ether.

Michelson Morley Experiment


Jason and Kylie are sitting at the northern and southern ends respectively of a train carriage travelling north at a high speed. Each holds a torch that they turn on and off. Harold is standing on a platform beside the train. As the midpoint of the carriage passes Harold, he observes simultaneous light flashes from both Jason and Kylie.

Q: Which one of the following statements is true?A. To an observer inside the carriage, located at its midpoint, Jason and Kylie turned on their torches at the same time.B. To an observer inside the carriage, located at its midpoint, Jason turned on his torch before Kylie.C. To an observer inside the carriage, located at the midpoint, Kylie turned on her torch before Jason. D. It does not make any sense to ask in which order Jason and Kylie turned on their torches, because Einstein showed that time is relative.

A: The correct answer was C. Since Kylie was at the rear of the carriage she would have been further from Harold than Jason and so would have had to turn her torch on first.

Simultaneity


The electron accelerator at Stanford University is 3.2 km long (Figure 4). Electrons reach a velocity of 0.9999995 c, which means the Lorentz factor is 1000.

Q: For an electron travelling the length of the accelerator at this velocity, what would be the length of the accelerator in the electron’s reference frame?

A: L = Lo/γ = (3.2 x 103)/1000 = 3.2 m

Length Contraction


Q: As measured by a scientist at the accelerator laboratory, how long would the electron take to travel the length of the accelerator?

A: v = d/t t = d/v = 3200/(0.9999995)(3.0 x 108) = 1.06 x 10-5 s


Constant Velocity Motion


Q: How long would the electron measure its time of travel to be?

A: t = γto

to = t/γ = (1.06 x 10-5)/1000 = 1.06 x 10-8 s


Understanding Time Dilation

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