Cyber Dream readers who enjoy math, please read :)

Recently a very helpful reader pointed out that my calculations seemed wonky when it came to ship travel times. I won't spoil for those of you who haven't read that far, but let's just say I might need to do some revising! My friend who initially helped me calculate travel times seems to have made an error, and he's not available at the moment, so I took my problem to ChatGPT. Would any of you care to check the AI's math? Here's our conversation:

Me: Let's calculate some fictional ship travel times. Assume the ship can accelerate at a constant 1G. Assume it's traveling from X to Y and the distance is 1.3 Billion Kilometers. Remember the ship has to decelerate for half the journey. How long will this take?

ChatGPT: To calculate the travel time of the fictional ship, we need to consider that it will accelerate for half the distance and decelerate for the other half. Let's break the problem into two parts: the time it takes to accelerate and the time it takes to decelerate.

First, let's convert the acceleration from 1G to m/s^2. 1G is approximately equal to 9.81 m/s^2.

The total distance is 1.3 billion kilometers, which is 1.3 * 10^9 * 10^3 = 1.3 * 10^12 meters. Half of this distance will be covered during acceleration, and the other half during deceleration, so each part will cover 0.65 * 10^12 meters.

Now, we can use the following kinematic equation to solve for time:

distance = 0.5 * acceleration * time^2

Let's solve for time during acceleration:

0.65 * 10^12 m = 0.5 * 9.81 m/s^2 * time^2

Rearrange to solve for time:

time^2 = (2 * 0.65 * 10^12 m) / 9.81 m/s^2 time^2 ≈ 1.32768 * 10^11 s^2 time ≈ √(1.32768 * 10^11 s^2) ≈ 364740 s

Since the ship takes the same amount of time to decelerate, the total travel time will be twice this value:

total time = 2 * 364740 s ≈ 729480 s

Now, let's convert this to days:

729480 s / 86400 s/day ≈ 8.44 days

So, the fictional ship's journey of 1.3 billion kilometers with a constant 1G acceleration and deceleration would take approximately 8.44 days.

Me: In that example how fast was the ship going before it turned to decelerate?

ChatGPT: To find the maximum velocity of the ship just before it starts decelerating, we can use the following kinematic equation:

v_final^2 = v_initial^2 + 2 * acceleration * distance

Since the ship starts from rest, v_initial = 0. The acceleration is 1G, which is 9.81 m/s^2. The distance covered during acceleration is half of the total distance, which is 0.65 * 10^12 meters.

Now, plug in the values:

v_final^2 = 0 + 2 * 9.81 m/s^2 * 0.65 * 10^12 m v_final^2 ≈ 1.2742 * 10^13 m^2/s^2 v_final ≈ √(1.2742 * 10^13 m^2/s^2) ≈ 3569300 m/s

So, the ship's maximum velocity just before it starts decelerating is approximately 3,569,300 m/s (or about 3.57 km/s).

Me: what percentage of c is that?

ChatGPT: To find the percentage of the speed of light (c) that the ship's maximum velocity represents, we can divide its maximum velocity by the speed of light and multiply by 100.

The speed of light (c) is approximately 299,792,458 m/s.

Using the ship's maximum velocity of 3,569,300 m/s:

Percentage of c = (3,569,300 m/s / 299,792,458 m/s) * 100 Percentage of c ≈ 0.0119 * 100 Percentage of c ≈ 1.19%

So, the ship's maximum velocity is about 1.19% of the speed of light.