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chit-chat about space travel math
19.7.23 23:23
User:
If one would travel 4.3 billion kilometers from Earth in 72 days and the same distance back in the same time what would be the age difference between the traveler and those who stayed on Earth upon his return?

Chatter:
To calculate the age difference between the traveler and those who stayed on Earth, we need to take into account the effects of time dilation due to special relativity.

The formula for time dilation is:

Δt' = Δt * sqrt(1 - v^2/c^2)

where:
Δt' = time experienced by the moving observer (traveler)
Δt = time measured by a stationary observer (on Earth)
v = relative velocity between the two observers (in this case, the speed of the traveler)
c = speed of light in a vacuum (approximately 299,792 kilometers per second)

Let's break it down step by step:

Calculate the relative velocity (v) of the traveler:
The traveler travels a total distance of 4.3 billion kilometers from Earth and then returns. So, the total distance covered is 2 * 4.3 billion kilometers = 8.6 billion kilometers.
Since this distance is traveled in 72 days, we can calculate the average speed (relative velocity) of the traveler:

v = distance / time = 8.6 billion kilometers / (72 days * 24 hours/day * 3600 seconds/hour)

Remember to convert days to seconds:
v = 8.6 * 10^9 km / (72 * 24 * 3600 s) ≈ 1061.98 km/s

Now, let's calculate the time dilation factor (γ) for the traveler:
γ = 1 / sqrt(1 - v^2/c^2)
γ = 1 / sqrt(1 - (1061.98 km/s)^2 / (299,792 km/s)^2)

γ ≈ 1.000001242

Calculate the time experienced by the traveler (Δt'):
Δt' = γ * Δt
Since the traveler spends 72 days on the journey, Δt = 72 days * 24 hours/day * 3600 seconds/hour ≈ 6,220,800 seconds.

Δt' ≈ 1.000001242 * 6,220,800 s ≈ 6,220,800.7 seconds

Convert the time experienced by the traveler back to days:
Δt' ≈ 6,220,800.7 seconds / (24 hours/day * 3600 seconds/hour) ≈ 72.01 days
So, the time experienced by the traveler is approximately 72.01 days. Since the traveler spends the same amount of time going back to Earth, the total time experienced by the traveler would be around 144.02 days.

Now, to find the age difference between the traveler and those who stayed on Earth, we need to subtract the travel time from 144.02 days:

Age difference ≈ 144.02 days - 72 days ≈ 72.02 days

The age difference between the traveler and those who stayed on Earth would be approximately 72.02 days.

User:
I believe there is an error in your calculation of velocity v = distance / time = 8.6 billion kilometers / (72 days * 24 hours/day * 3600 seconds/hour). Because it is 4.3 billion km in 72 days away and 4.3 billion km in 72 days on the way back. Please recalculate.

Chatter:
You are correct, my apologies for the oversight. Let's recalculate the velocity of the traveler:

Distance traveled away from Earth = 4.3 billion kilometers
Time taken for the outward journey = 72 days

Distance traveled on the way back = 4.3 billion kilometers (same as the outward journey)
Time taken for the return journey = 72 days

Total distance covered = 4.3 billion km + 4.3 billion km = 8.6 billion kilometers
Total time taken = 72 days + 72 days = 144 days

Now, let's calculate the velocity (relative speed) of the traveler:

v = total distance / total time
v = 8.6 billion kilometers / (144 days * 24 hours/day * 3600 seconds/hour)

v = 8.6 * 10^9 km / (144 * 24 * 3600 s) ≈ 659.72 km/s

Now that we have the correct velocity, we can proceed with the time dilation calculation:

Calculate the time dilation factor (γ) for the traveler:
γ = 1 / sqrt(1 - v^2/c^2)
γ = 1 / sqrt(1 - (659.72 km/s)^2 / (299,792 km/s)^2)
γ ≈ 1.00000000217

Calculate the time experienced by the traveler (Δt'):
Δt' = γ * Δt

Since the traveler spends 144 days on the journey, Δt = 144 days * 24 hours/day * 3600 seconds/hour ≈ 12,441,600 seconds.

Δt' ≈ 1.00000000217 * 12,441,600 s ≈ 12,441,600.053 seconds

Convert the time experienced by the traveler back to days:
Δt' ≈ 12,441,600.053 seconds / (24 hours/day * 3600 seconds/hour) ≈ 144.0019 days
So, the time experienced by the traveler is approximately 144.0019 days. Since the traveler spends the same amount of time going back to Earth, the total time experienced by the traveler would be around 288.0038 days.

Now, to find the age difference between the traveler and those who stayed on Earth, we need to subtract the travel time from 288.0038 days:

Age difference ≈ 288.0038 days - 144 days ≈ 144.0038 days

The age difference between the traveler and those who stayed on Earth would be approximately 144.0038 days, which is roughly 144 days and 0.0048 days or about 3.67 hours.
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