On earth, a certain type of bacteria doubles 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 is put on a rocket travelling at .866c relative to the earth. By the time the earth bound bacteria have grown to 256 bacteria, how many bacteria are in the culture on the rocket, according to earth-based observers?

Question

On earth, a certain type of bacteria doubles 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 is put on a rocket travelling at .866c relative to the earth. By the time the earth bound bacteria have grown to 256 bacteria, how many bacteria are in the culture on the rocket, according to earth-based observers?

Answer 1

Time on the rocket goes more slowly, as seen by earth-based observer. The elapsed time on the rocket is lower by a factor sqrt[1-(v/c)^2] = sqrt(0.25) = 1/2

2^8 = 256 is the number of bacteria that are produced on earth during the time interval you are discussing.

The rocket's bacteria will have doubled half as many times, so there will be 2^4 = 16 bacteria

Answer 2

To solve this problem, we need to consider time dilation, which is a phenomenon observed in special relativity. Time dilation occurs when there is relative motion between observers in different reference frames.

Let's start by understanding the time dilation equation:

Δt' = γ * Δt

Where:
Δt' is the time interval measured in the observer's frame of reference (on the rocket)
γ is the Lorentz factor, given by γ = 1/√(1 - v^2/c^2)
Δt is the time interval measured in the stationary frame of reference (on Earth)
v is the relative velocity between the two frames of reference
c is the speed of light

In this case, the relative velocity of the rocket (v) is given as 0.866c, where c is the speed of light.

Now, let's solve the problem step by step:

1. In the Earth frame of reference, the bacteria double in number every 24 hours. So, to reach 256 bacteria, it will take (256-1)*24 hours = 6136 hours.

2. We need to find the corresponding time interval on the rocket (Δt'). Using the time dilation equation:

Δt' = γ * Δt

Substituting the values:
Δt' = γ * 6136 hours

3. Now, let's calculate the Lorentz factor (γ):

γ = 1/√(1 - v^2/c^2)
γ = 1/√(1 - (0.866c)^2/c^2)
γ = 1/√(1 - 0.75) ≈ 2.0

4. Substituting the value of γ into the equation:

Δt' = 2.0 * 6136 hours
Δt' = 12272 hours

Therefore, according to Earth-based observers, by the time the Earth-bound bacteria have grown to 256 bacteria, there will be approximately 12272 bacteria in the culture on the rocket.