Announcements

• Special Relativity Test on class period after lab– Last grade of the year– Review packet will be handed out tomorrow– You will actually need to study!

Length Contraction, Simultaneity and Velocity Addition

Meet the Muon

Muons are formed when high-energy protons from the solar wind hit Carbon nuclei in our atmosphere.

After being formed, they fly outward at speeds of up to 99.5% of the speed of light!

However, muons are very unstable particles. A muon at rest exists for only about 2.2 x 10-6 seconds before it decays.

Muon Whiteboard: Part 1A muon is formed by a nuclear reaction in the high atmosphere 5 km above Earth’s surface. The muon travels straight downward at a speed of 0.995c. From the reference frame of the muon, how far will it travel in the 2.2 x 10-6 seconds before it decays?

According to classical physics, the muon will decay long before it reaches Earth’s surface!

However, muons from the high atmosphere can be regularly found striking the

surface of the Earth!

How is this possible?

Muon Whiteboard: Part 2A muon travels straight downward at a speed of 0.995c. From the reference frame of an observer on Earth, how far will it travel in the 2.2 x 10-6 seconds before it decays?

Note: The time for the muon to decay applies within the reference frame of the muon itself – not to the observer!

Muons are Evidence of Special Relativity!

Time elapsed according to the rest frame

(muon’s ref frame)

Time elapsed according to an outside observer

t = 22.0 x 10-

6 s

d = 6,570 m!

From the reference frame of the Earth, the muon has plenty of time to reach the

ground!

But hey, wait a second…

How can it be possible that the muon travels only 650 m in its own reference frame, but travels a whole 6,500 m in our reference frame?

It either hits the ground or it doesn’t!

…right?

As it turns out, the 650 m that the muon travels in its own reference frame is equivalent to the 6,500 m

that it travels in ours.

Time is not the only quantity that is relative

to the observer.

Lengths are also relative

In formulating special relativity, Einstein

showed that space and time are linked in the

most fundamental way.

Another Thought Experiment

In the year 2500, an astronaut takes a trip to Vega - a distant star. The trip is

a distance of 25.3 light-years, as measured by an observer on Earth. The astronaut travels at a speed of

0.99c

How will the astronaut see this?

From the astronaut’s reference frame, Earth and Vega are moving at 0.99c,

and their ship is at rest.

The astronaut and the Earth observer will agree on their relative velocity, but that’s

about all they will agree upon!

Whiteboard: Length Contraction!

a) How much time will the trip take, according to each of the observers?

b) What is the distance between Earth and Vega, according to each of the observers?

Two Different Stories – Both Correct!

Time:25.6 years

Time:3.61 years

The only thing they will agree upon is their relative velocity

Length Contraction

Length measurement of an observer moving relative to the object being measured

Length measurement of an observer at rest relative to the object being measured

Rel. speed of object/observer

Speed of light

Lengths are shorter to observers who are moving relative to the object being measured.

Lengths only contract along the direction of motion

Both observers will measure the other’s lengths

to be contracted (and both will be correct!)

Whiteboard: Length Contraction

How fast would a meter stick have to move for it to become a half-meter stick

from your reference frame?

v = 0.866c

Whiteboard: Laying DownAn astronaut is resting on a bed inclined at an angle theta above the floor of a spaceship.

From the reference frame of an observer who is moving to the right with a speed close to c, is the angle that the bed makes with the floor (a) greater than, (b) less than, or (c) equal to the angle as observed by the astronaut?

SolutionLengths contract along the direction of relative motion. This will cause x to contract, while y is constant.

Therefore, the moving observer will measure a larger theta than the astronaut!

Final Whiteboard of… Ever!

By combining the concepts of time dilation and length contraction, describe the journey of a muon traveling downward at 0.995c from the reference frame of…

a)an Earth observerb)the muon

“It takes the muon 22.0 μs to hit the

ground, which is 5 km away.”

“It takes the ground 2.2 μs to hit me,

starting from just 500 m away.”

“When you’re talking to a pretty girl, an hour feels like a second. When you put your hand on a red-hot ember, a second feels like an hour. That’s relativity.”