Question

As observed on earth, a certain type of bacteria is known to double in number every 24 hours. Two cultures of these bacteria are prepared, each consisting initially of one bacterium. One culture is left on earth and the other placed on a rocket that travels at a speed of 0.746c relative to the earth. At a time when the earthbound culture has grown to 32 bacteria, how many bacteria are in the culture on the rocket, according to an earth-based observer? Please explain each step and use (delta)t = (delta)to/sqrt(1-(v^2/c^2))

Answer 1

Doubling of population P every 24 h means P = 2^N , where N = 1 , 2 , 3, ... ,
with the total elapsed time
(1) ..... Te = N*(24 h)
Given P = 32 = 2^N, we solve for
N in the following way:
.......... ln(32) =ln(2^N) = N*[ln(2)]
(2) ..... N = ln(32)/ln(2) = 5
The total elapsed time according to (1) is then
(3) ..... Te = N*(24 h) = 5*(24 h) = 120 h
Since there is no relative motion between the clock used in measuring the above
time and the culture of bacteria whose population is being studied, that time is
the proper time To (according to an earth-based observer), so that
(4) ..... To = Te = 120 h
Using the time dilation formula
(5) ..... T = To/sqrt [ 1 – (V/C)^2 ]
where V = 0.746*C, we get
(6) ..... sqrt [1 – (V/C)^2 ] = sqrt [1 – (0.746)^2 ] = 0.6659
(7) ..... T = To/sqrt [ 1 – (V/C)^2 ] = (120 h)/(0.6659) = 180.21 h
for the elapsed time on the rocket as measured by an observer on earth.
According to (1), Te = N*(24 h) = 180.21 h , gives
(8) ..... N = (180.21 h)/(24 h) = 7.5 or N = 7 only
It follows that according to the earth-based observer, the number of bacteria in
the culture on the rocket is
(9) ..... P = 2^(7) = 128 <<<<< ANSWER
However, according to a rocket-based observer, Te in (3) is not the proper time but
an apparent time T. Substituting T = Te , from (3), in (5), we get [using (6)]
(10) ... To = T*sqrt [ 1 – (V/C)^2 ] = (120h)*( 0.6659) = 79.908 h
According to (1), Te = N*(24 h) = 79.908 h , gives
(11) ... N = ( 180.21h)/(24 h) = 7.5 or N = 7 only
It follows that according to the rocket-based observer, the number of bacteria in
his culture is only
(12) ... P =2^(7) = 128
.