Physics 6C

Special Relativity

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Two Main Ideas – The Postulates of Special Relativity

• Light travels at the same speed in all inertial reference frames.• Laws of physics yield identical results in all inertial reference frames.

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

Two Main Ideas – The Postulates of Special Relativity

• Light travels at the same speed in all inertial reference frames.• Laws of physics yield identical results in all inertial reference frames.

Inertial reference frames refer to observers moving at constant velocity with respect to each other.

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

Two Main Ideas – The Postulates of Special Relativity

• Light travels at the same speed in all inertial reference frames.• Laws of physics yield identical results in all inertial reference frames.

Inertial reference frames refer to observers moving at constant velocity with respect to each other.

If there is a nonzero acceleration, the frames are not inertial, and we would need to use General Relativity. Way too much math for this course – sorry…

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

Two Main Ideas – The Postulates of Special Relativity

• Light travels at the same speed in all inertial reference frames.• Laws of physics yield identical results in all inertial reference frames.

Inertial reference frames refer to observers moving at constant velocity with respect to each other.

If there is a nonzero acceleration, the frames are not inertial, and we would need to use General Relativity. Way too much math for this course – sorry…

The relationship between what is seen in the two reference frames is found via the Lorentz Transformation. We will see the following factor in all of our equations:

22

cv1

1

Here v is the relative speed of the frames, and c is the speed of light.Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

The main results of Special Relativity are the following:1) Time Dilation - if an object is moving, an observer will measure times to be

longer (compared to the frame of the object itself)

0

cv

0 tt1

tt22

t0 refers to the object at rest in its own frame

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

The main results of Special Relativity are the following:1) Time Dilation - if an object is moving, an observer will measure times to be

longer (compared to the frame of the object itself)

2) Length Contraction – if an object is moving, an observer will measure lengths to be shorter in the direction of motion (compared to the frame of the object itself)

0

cv

0 tt1

tt22

t0 refers to the object at rest in its own frame

0

cv

0LL1LL 2

2 L0 refers to the object at rest in its own frame

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

The main results of Special Relativity are the following:1) Time Dilation - if an object is moving, an observer will measure times to be

longer (compared to the frame of the object itself)

2) Length Contraction – if an object is moving, an observer will measure lengths to be shorter in the direction of motion (compared to the frame of the object itself)

3) Addition of velocities is more complicated than in the non-relativistic case. At low speeds, we just add or subtract the relative velocities and it works fine, but near the speed of light we need to be more careful. Here’s a formula:

0

cv

0 tt1

tt22

t0 refers to the object at rest in its own frame

L0 refers to the object at rest in its own frame

v13 is the relative speed between the frames, and v21 and v23 are the velocities of the object in each frame. These formulas are tricky to use, so practice several examples.

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

21321

cvv1321

23 1vvv

0

cv

0LL1LL 2

2

The main results of Special Relativity are the following:1) Time Dilation - if an object is moving, an observer will measure times to be

longer (compared to the frame of the object itself)

2) Length Contraction – if an object is moving, an observer will measure lengths to be shorter in the direction of motion (compared to the frame of the object itself)

3) Addition of velocities is more complicated than in the non-relativistic case. At low speeds, we just add or subtract the relative velocities and it works fine, but near the speed of light we need to be more careful. Here’s a formula:

4) Energy and Mass are equivalent (Erest=m0c2). We can also get formulas for relativistic momentum and total energy.

0

cv

0 tt1

tt22

t0 refers to the object at rest in its own frame

L0 refers to the object at rest in its own frame

22

cv1

vmp

rest

cv

20

total EK1

cmE22

21321

cvv1321

23 1vvv

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

v13 is the relative speed between the frames, and v21 and v23 are the velocities of the object in each frame. These formulas are tricky to use, so practice several examples.

0

cv

0LL1LL 2

2

http://www.youtube.com/watch?v=JQnHTKZBTI4

http://www.youtube.com/watch?v=C2IGMLYyA7U&NR=1

http://www.youtube.com/watch?v=PpFZpqlWiF0

Visual demonstrations of special relativity.

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

Here’s a problem:

In the year 2084, a spacecraft flies over Moon Station III at a speed of 0.800c. A scientist on the moon measures the length of the moving spacecraft to be 140 m. The spacecraft later lands on the moon, and the same scientist measures the length of the now stationary spacecraft. What value does she get?

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

22

cv

0 1LL

Use the length contraction formula with L=140m and v=.8c. We are looking for L0.

Here’s a problem:

In the year 2084, a spacecraft flies over Moon Station III at a speed of 0.800c. A scientist on the moon measures the length of the moving spacecraft to be 140 m. The spacecraft later lands on the moon, and the same scientist measures the length of the now stationary spacecraft. What value does she get?

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

Use the length contraction formula with L=140m and v=.8c. We are looking for L0.

2

2

22

cc8.

0

cv0

1Lm140

1LL

Here’s a problem:

In the year 2084, a spacecraft flies over Moon Station III at a speed of 0.800c. A scientist on the moon measures the length of the moving spacecraft to be 140 m. The spacecraft later lands on the moon, and the same scientist measures the length of the now stationary spacecraft. What value does she get?

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

m233L

6.0Lm1401Lm140

1LL

0

0

cc8.

0

cv

0

22

22

Notice that c2 cancels out. This usually happens when you use speeds written in terms of c.

Our result is consistent with the concept of length contraction. The ship is measured to be shorter when it is moving.

Here’s a problem:

In the year 2084, a spacecraft flies over Moon Station III at a speed of 0.800c. A scientist on the moon measures the length of the moving spacecraft to be 140 m. The spacecraft later lands on the moon, and the same scientist measures the length of the now stationary spacecraft. What value does she get?

Use the length contraction formula with L=140m and v=.8c. We are looking for L0.

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

Here’s a problem:

Inside a spaceship flying past the earth at ¾ the speed of light, a pendulum is swinging.a) If each swing takes 1.5 s as measured by an astronaut performing an experiment inside the spaceship, how long will the swing take as measured by a person at mission control on earth who is watching the experiment?b) If each swing takes 1.5 s as measured by a person at mission control on earth, how long will the swing take as measured by an astronaut inside the spaceship?

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

We will be using the time dilation formula. Notice the difference between part a) and part b) – In part a) the time as measured on the spaceship is given. This is Δt0 because the pendulum is at rest relative to the ship.

Here’s a problem:

Inside a spaceship flying past the earth at ¾ the speed of light, a pendulum is swinging.a) If each swing takes 1.5 s as measured by an astronaut performing an experiment inside the spaceship, how long will the swing take as measured by a person at mission control on earth who is watching the experiment?b) If each swing takes 1.5 s as measured by a person at mission control on earth, how long will the swing take as measured by an astronaut inside the spaceship?

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

We will be using the time dilation formula. Notice the difference between part a) and part b) – In part a) the time as measured on the spaceship is given. This is Δt0 because the pendulum is at rest relative to the ship.

Here’s a problem:

Inside a spaceship flying past the earth at ¾ the speed of light, a pendulum is swinging.a) If each swing takes 1.5 s as measured by an astronaut performing an experiment inside the spaceship, how long will the swing take as measured by a person at mission control on earth who is watching the experiment?b) If each swing takes 1.5 s as measured by a person at mission control on earth, how long will the swing take as measured by an astronaut inside the spaceship?

22

cv

01

tt

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

We will be using the time dilation formula. Notice the difference between part a) and part b) – In part a) the time as measured on the spaceship is given. This is Δt0 because the pendulum is at rest relative to the ship.

Here’s a problem:

Inside a spaceship flying past the earth at ¾ the speed of light, a pendulum is swinging.a) If each swing takes 1.5 s as measured by an astronaut performing an experiment inside the spaceship, how long will the swing take as measured by a person at mission control on earth who is watching the experiment?b) If each swing takes 1.5 s as measured by a person at mission control on earth, how long will the swing take as measured by an astronaut inside the spaceship?

s3.2t

1

s5.1t1

tt2

222

cc75.

cv

0

The people on earth measure a longer

(dilated) time for each swing, as expected.

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

We will be using the time dilation formula. Notice the difference between part a) and part b) – In part a) the time as measured on the spaceship is given. This is Δt0 because the pendulum is at rest relative to the ship.

Here’s a problem:

Inside a spaceship flying past the earth at ¾ the speed of light, a pendulum is swinging.a) If each swing takes 1.5 s as measured by an astronaut performing an experiment inside the spaceship, how long will the swing take as measured by a person at mission control on earth who is watching the experiment?b) If each swing takes 1.5 s as measured by a person at mission control on earth, how long will the swing take as measured by an astronaut inside the spaceship?

s3.2t

1

s5.1t1

tt2

222

cc75.

cv

0

The people on earth measure a longer

(dilated) time for each swing, as expected.

Part b) uses the same formula, but now we are given Δt instead.

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

We will be using the time dilation formula. Notice the difference between part a) and part b) – In part a) the time as measured on the spaceship is given. This is Δt0 because the pendulum is at rest relative to the ship.

Here’s a problem:

Inside a spaceship flying past the earth at ¾ the speed of light, a pendulum is swinging.a) If each swing takes 1.5 s as measured by an astronaut performing an experiment inside the spaceship, how long will the swing take as measured by a person at mission control on earth who is watching the experiment?b) If each swing takes 1.5 s as measured by a person at mission control on earth, how long will the swing take as measured by an astronaut inside the spaceship?

s3.2t

1

s5.1t1

tt2

222

cc75.

cv

0

The people on earth measure a longer

(dilated) time for each swing, as expected.

Part b) uses the same formula, but now we are given Δt instead.

s0.1t

1

ts5.11

tt 0

cc75.

0

cv

0

22

22

Again the people on earth measure a longer time because the clock is moving relative to them.

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

Here’s a problem:

Two particles are created in a high-energy accelerator and move off in opposite directions. The speed of one particle, as measured in the laboratory, is 0.65c, and the speed of each particle relative to the other is 0.95c. What is the speed of the second particle, as measured in the laboratory?

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

Here’s a problem:

Two particles are created in a high-energy accelerator and move off in opposite directions. The speed of one particle, as measured in the laboratory, is 0.65c, and the speed of each particle relative to the other is 0.95c. What is the speed of the second particle, as measured in the laboratory?

12

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

21321

cvv1321

23 1vvv

Pictures will probably help here(the laboratory is object #3)

v23=? v13=0.65c

This is what you see in the laboratory. Particle 1 is moving at 0.65c, and Particle 2 is moving the other direction.

Here’s a problem:

Two particles are created in a high-energy accelerator and move off in opposite directions. The speed of one particle, as measured in the laboratory, is 0.65c, and the speed of each particle relative to the other is 0.95c. What is the speed of the second particle, as measured in the laboratory?

12v21=-0.95c

This is the same scenario in the reference frame of Particle 1. Particle 2 is moving away at -0.95c, and Particle 1 is at rest in its own frame.(If we were to put the laboratory in this picture we would see it moving to the left at -0.65c.)

12

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

21321

cvv1321

23 1vvv

Pictures will probably help here(the laboratory is object #3)

v23=? v13=0.65c

This is what you see in the laboratory. Particle 1 is moving at 0.65c, and Particle 2 is moving the other direction.

Here’s a problem:

Two particles are created in a high-energy accelerator and move off in opposite directions. The speed of one particle, as measured in the laboratory, is 0.65c, and the speed of each particle relative to the other is 0.95c. What is the speed of the second particle, as measured in the laboratory?

12This is the same scenario in the reference frame of Particle 1. Particle 2 is moving away at -0.95c, and Particle 1 is at rest in its own frame.(If we were to put the laboratory in this picture we would see it moving to the left at -0.65c.)

12

c78.0v6175.01

c3.0v

1c65.0c95.0v

2cc95.0c65.0

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

21321

cvv1321

23 1vvv

Pictures will probably help here(the laboratory is object #3)

v23=? v13=0.65c

This is what you see in the laboratory. Particle 1 is moving at 0.65c, and Particle 2 is moving the other direction.

v21=-0.95c

Here’s a problem:

Two particles are created in a high-energy accelerator and move off in opposite directions. The speed of one particle, as measured in the laboratory, is 0.65c, and the speed of each particle relative to the other is 0.95c. What is the speed of the second particle, as measured in the laboratory?

12This is the same scenario in the reference frame of Particle 1. Particle 2 is moving away at -0.95c, and Particle 1 is at rest in its own frame.(If we were to put the laboratory in this picture we would see it moving to the left at -0.65c.)

12

c78.0v6175.01

c3.0v

1c65.0c95.0v

2cc95.0c65.0

So in the lab, Particle 2 looks like it is moving to the left at speed 0.78c.

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

21321

cvv1321

23 1vvv

Pictures will probably help here(the laboratory is object #3)

v23=? v13=0.65c

This is what you see in the laboratory. Particle 1 is moving at 0.65c, and Particle 2 is moving the other direction.

v21=-0.95c

Here’s a sample problem:

The sun produces energy by nuclear fusion reactions, in which matter is converted to energy. The rate of energy production is 3.8 x 1026 Watts. How many kilograms of mass does the sun convert to energy each second?

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

Here’s a sample problem:

The sun produces energy by nuclear fusion reactions, in which matter is converted to energy. The rate of energy production is 3.8 x 1026 Watts. How many kilograms of mass does the sun convert to energy each second?

We only need to use Einstein’s E=mc2 for this one.

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB

Here’s a sample problem:

The sun produces energy by nuclear fusion reactions, in which matter is converted to energy. The rate of energy production is 3.8 x 1026 Watts. How many kilograms of mass does the sun convert to energy each second?

We only need to use Einstein’s E=mc2 for this one.

Remember, a Watt is a Joule per second kg102.4m

103mJ108.39

2sm826

Prepared by Vince ZacconeFor Campus Learning Assistance Services at UCSB