The document describes calculating the time it takes for sodium streetlamps (emitting light at 589 nm) on Earth to become invisible to astronauts on a spaceship accelerating away from Earth at 3g. It uses the Doppler effect formula to calculate the time shift in wavelength due to the spaceship's motion. The calculation shows it will take about 20 hours for the streetlamps' light to shift outside the visible range for human eyes.

1. 1
 
 
 
2
A spaceship moves radially away
from Earth with acceleration 29.4
m/s about 3g . How much time
does it take for the sodium
streetlamps = 589 nm on Earth
to be invisible
with a powerful telescope to the

human eye of the astronauts? The
range of visible wavelengths
is about 400 to 700 nm.

2. 2
Solution

3. 3
0
0
0
Using
1
1
1
1
1
1
f
f
c
c








 




 


 


4. 4
0
2
0
2
1
1
1
[ ]
1
700 1
[ ]
589 1
1
1.41
1
1.41 1.41 1 0
0.41 2.41 0
0.17
nm
nm


 
 
 




 



 


 


 


 

    
  
 

5. 5
8
8
,
0.17
0.17
0.17 2.998 10
0.51 10
Now
v
c
v c
m
X X
s
m
X
s
  
 



6. 6
8
2
8
2
8
,
0.51 10
29.4
0.51 10
29.4
0.0173469 10 1734690
1734690
20.08
3600 24
f i
Now
v v
a
t
m
X
m s
s t
m
X
s
t
m
s
X s s
days days
X


 
 
 
 