I beg anybody to rephrase it understandingly with using some units similar to football fields. Is it possible to launch a little cheap rocket with a transmitter just to correct Voyager's position?
If it's 15 billion miles away (sorry for my Freedom Units), it is 22 light-hours away, or 0.0026 light years away (unless my Google-fu is way off). If we could move at the speed of light, which we can't, it would still take nearly a day to get there. So if we were on the Enterprise moving at Warp 1, it would take a day to get there and reorient it back towards Earth. If we could move at Warp 10, we'd have already been there and fixed it.
More realistically, even if say we did have a light speed capable craft, we'd be limited by the forces for accel and decel to have a starting speed of 0 relative to Earth, and ending speed of 0 relative to Voyager. The limits would be determined by the withstanding capability of the bodies of the pilots, or the mechanical strength if it were another drone.
I don't feel like doing the math properly, so I asked GPT4 this question:
> let's say we wanted to move a human astronought in a spacecraft to the current position of Voyager 2 which is 22 light-hours away. We want to get there as fast as possible, and arrive at a co-moving frame with Voyager, which has a relative velocity of 32000 miles per hour moving away from the Earth. Assuming the astronaut can withstand a g-force maximum on their body of 6G, and using that to inform the maximum accelleration and deceleration for the voyage, how long does it take to reach the Voyager 2 co-moving frame?
Response:
Firstly, let's convert everything to consistent units. Here we'll use hours for time, miles for distance, and miles per hour^2 for acceleration.
1 G is approximately equal to 22 miles/hour^2. Therefore, 6G is approximately 132 miles/hour^2. The distance to Voyager 2 in miles is 22 light-hours * 670616629 miles/light-hour = approximately 14753561838 miles.
Assuming we accelerate at 6G until halfway, then decelerate at 6G for the second half, we can use the kinematic equation:
d = 0.5 * a * t^2
Solving for t when d equals half the total distance gives:
t = sqrt((2 * d) / a)
Substituting in the values:
t = sqrt((2 * (14753561838 miles / 2)) / 132 miles/hour^2) = 84331.5 hours
So, the total time to reach Voyager 2 while maintaining a maximum acceleration/deceleration of 6G is approximately 84331.5 hours * 2 = 168663 hours, or about 19.25 years.
This is a rough estimation and doesn't take into account the relative velocity of Voyager 2 or relativistic effects at high speeds.
It also assumes constant acceleration and deceleration, which wouldn't be feasible with today's technology due to fuel constraints.
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So our 22 hour lightspeed trip would still take 19 years if it had a human onboard.
The biggest limitation with sci-fi travel isn't breaking relativity, it's finding a way to accelerate squishy meat-bags. This person will be living at an effective weight of nearly 1000lbs for 20 years, which will really mess them up.
For the sake of the longevity of our pilot, I asked to re-run the calculation limiting the on-board gravity to 1G. It now takes 114 years to reach Voyager's current position, which by then will be much farther. It turns out we will never catch up, and reaching it is impossible if we limit our craft to earth like conditions necessary for human health.