Transcript Document

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.
120
150
100
80
120
80
50
50
100
150
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.
120
150
100
120
50
50
100
150
Put Brenda’s tail at the origin.
Draw the position of her ship at
t’ = -80, -40, 0, 40 and 80 ns.
120
150
100
120
50
50
100
150
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
120
150
100
120
Event #1
50
Event #2
50
100
150
You can’t show the path of the laser
shot on the diagram because it moves
A) too fast
B) too slow
C) parallel to the page
D) 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
F’
y
F
x
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?
120
150
100
120
50
50
100
This line marks
all the positions
ever occupied by
the nose of
150
Brenda’s ship.
At t = 0 ns, where does Ali see
the nose of Brenda’s ship?
120
150
100
120
50
50
= 0 ns, Ali
100 At t150
sees the nose of
Brenda’s ship at
x = 80 ns.
At t’ = 0 ns, where does
Brenda say her nose is?
120
150
100
120
50
50
At t’ = 0 ns, Brenda
says her nose is at
100 x’ =150
100 ns.
When does Brenda say that
her nose was closest to the
laser cannon?
120
150
100
120
50
50
At100
t’ = -32 150
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.
120
150
100
120
50
50
100
The shot is fired when
the nose and tail are
lined up. The ships are
in the position shown.
150
This is Brenda’s view.
120
150
The shot does not
occur at the same
time as the tail and
120
nose being lined up.
The events are not
simultaneous.
At t’ = 0, the two ships
are in this position,
100
with the nose and tail
lined up.
50
50
100
150
The two ships are in this
position when the shot is
fired.
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?
Event#2:
Muon hits
Earth
This diagram
shows the
problem with
the speed
slowed down.
Event #1:
Muon enters
atmosphere
Atmosphere
Earth
The times
between
events are
different.
The distances
between
events are
also different.
Dt
Dt’
Dx’
Dx
We measure
the muons
living ten
times longer.
The muons
measure an
atmosphere
that is ten
times shorter.
Dt
Dt’
Dx’
Dx
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.
The muon’s time
interval is the
shortest any frame
Dt
will measure.
They and their
clock are at rest
with both events.
This is called the
‘proper’ time.
Not everything
is ‘relative’.
Dt
We agree on
our relative
speeds and the
speed of light.
c = Dx’ /Dt’
= Dx
/Dt
Dt’
Dx’
Dx
We also agree
on the number
of muons
getting the
Earth.
We just don’t
agree on the
explanation of
the two events.