space fight spacecraft of equal rest length pass very, very close to each other as they travel in...
TRANSCRIPT
Space Fight
Spacecraft of equal rest length pass very, very close to each
other as they travel in opposite directions at a relative speed of
3/5 c.
F
F’
Ali, the captain of ship F has a laser cannon at his tail that he
plans to fire at the nose of Brenda’s F’ ship when their nose is
lined up with her tail.
F’
F
It is only supposed to be a warning shot across nose and he figures it won’t hit because her F’ length is
contracted.
F’
F
However, his co-pilot says that the shot will hit because
Brenda sees that the F length is shortened.
F’
F
Who is right?
Draw it on a spacetime diagram. Label the axes with a scale of
one square to 10 ns.
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The ships have a length of 100 nm.
Put Ali’s nose at the origin. Draw the position of Ali’s ship at
t = -50, 0, 50, 100 and 150 ns.
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Put Brenda’s tail at the origin. Draw the position of her ship at
t’ = -80, -40, 0, 40 and 80 ns.
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You can see the nose of Ali’s ship and the tail of Brenda’s ship at the
same spacetime point. This is event #1
At this time t=0, the laser cannon at Ali’s tail is fired. This happens at a different point and is event #2
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Event #1 Event #2
You can’t show the path of the laser shot on the diagram
because it moves
A) too fastB) too slowC) parallel to the pageD) perpendicular to the
page
You can’t show the path of the laser shot on the diagram
because it moves into another dimension that is perpendicular
to the page.t
x
y
F
F’
However, the ships are very close and the shot will cross the
distance in almost no time. The important question is, is the nose of Brenda’s ship nearby when the
shot is fired?
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This line marks all the positions ever occupied by the nose of Brenda’s ship.
At t = 0 ns, where does Ali see the nose of Brenda’s
ship?
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At t = 0 ns, Ali sees the nose of Brenda’s ship at x = 80 ns.
At t’ = 0 ns, where doesBrenda say her nose is?
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At t’ = 0 ns, Brenda says her nose is at x’ = 100 ns.
When does Brenda say that her nose was closest
to the laser cannon?
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At t’ = -32 ns, Brenda sees her nose beside the cannon, well before it is fired.
Ali and Brenda disagree because of a difference in simultaneity.
The simultaneity of two events depends on the frame of
reference .
This is Ali’s view.
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The shot is fired when the nose and tail are lined up. The ships are in the position shown.
This is Brenda’s view.
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At t’ = 0, the two ships are in this position, with the nose and tail lined up.
The two ships are in this position when the shot is fired.
The shot does not occur at the same time as the tail and nose being lined up. The events are not simultaneous.
Not everything is relative.
They both agree that the shot misses.
They just don’t agree as to why.
Reality Check#6: Muons are formed at the top of the atmosphere by cosmic rays. Muons that are at rest have a half-life of 2.2 x 10-6 s.
However, when they are streaming toward the Earth at 0.995 c, we see them live 10 times longer!
The muons would disagree. Their time feels completely normal.
They would say that the muons at rest in the Earth lab were living 10
times longer.
How could the muons explain the fact that they managed to make it
through the 10-kilometer thick atmosphere?
This diagram shows the problem with the speed slowed down.
Event #1: Muon enters atmosphere
Event#2: Muon hits Earth
Atmosphere Earth
t t’
The times between events are different.
x
x’
The distances between events are also different.
t t’
We measure the muons living ten times longer.
x
x’
The muons measure an atmosphere that is ten times shorter.
The Earth measurement of length is the largest any frame will measure.
We and our ruler are at rest with both positions.
This is called the ‘proper’ length.
t
The muon’s time interval is the shortest any frame will measure.
They and their clock are at rest with both events.
This is called the ‘proper’ time.
t t’
x
x’
Not everything is ‘relative’.
We agree on our relative speeds and the speed of light.
c = x’ /t’
= x /t
We just don’t agree on the explanation of the two events.
We also agree on the number of muons getting the Earth.