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The computer said that messages are being transmitted via laser. Light speed is 299 792 458 m / s. It is supposed to have a delivery time of 19 years. 19 light years are 1,798e+14 kilometers. A reply is expected after 55 years. According to the display that's counting from now on and including Avalon's movement.
How fast was the Avalon moving?
In 55 years, Avalon travelled the same distance that light travels in 17.
17 / 55 = 30.91%
BUT there's a catch.
To be honest, this answer may not be correct since travelling at this speed makes time slower for the passengers and, although Avalon will reach its destination in 120 years, on Earth it will take a different amount of time. The 55 years estimative was calculated using Avalon as reference, but will be different if using Earth as reference and I'm assuming the OP wants to know Avalon's speed if seen from Earth.
An accurate answer to this may be much more complicated than expected, would be interesting to hear a physicist about this one.
Assuming an immediate response, the lasered signals to and from the ship must traverse a total of 55 light years. Since that's 17 light years further than the signals would have to travel if the ship and the message target were stationary, the ship must have traveled 17 light years in the same time that light traveled 55. This gives a relative velocity of 17/55c or 31% of the speed of light as measured by a stationary observer. Note that on the ship the measured time will only be about fifty years due to Special Relativity slowing the onboard clocks.
At the start of the movie, they are 30 years underway. From that position, the signal takes 19 years to reach Earth. That means, with Newtonian physics, they are 19 light years (ly) from Earth. Average speed = 19 ly / 30 years travel, so about 63% of light speed.
For the return signal, arrival is 55 years later (movie is not clear if this is 55 years after arrival at Earth, though). If the signal returns 55 years after being sent, return travel time is 36 years, or 36 ly from Earth. If 55 years from Earth, distance is 55 ly from Earth, but arrives 74 years after being sent. In the first case, it travelled 17 ly in 55 years (31% light), in the second, 36 ly in 74 years (49% light). Both of these answers are inconsistent with the initial condition of travelling 19 ly in 30 years.
At these speeds, relativity kicks in, with shipboard clocks running slow relative to Earth, or conversely, Earth appears closer. At 50% light, for example, the dilation is at 14%. And at this point, I end up confused on how to apply it!
Laurence Fishburne, as Chief Deck Officer Gus Mancuso, tells Aurora that they are traveling at 50% of the speed of light.
This is like calculating the intersection of two lines in a coordinate system, since the speed is constant for both. Sometime around 30 years ago, I did calculations like this at school.
If Avalon's speed is half the speed of light:
•y= (15/30)x+15 (15/30) speed of avalon (Lightyear/year)
•+15 How far avalon is when message left earth (lightyear)
•y= x speed same light speed and message left earth X+0
•X=15/(15/30)
•x=30 Avalon distance lightyear from earth
•30+15=45 45 years later, a reply comes back to Jim's message
If Avalon's speed is 0.63% speed of light:
•y= (19/30)x+19
•y=x
•x= (19/30)x+19
•((30/30)-(19/30))x=19
•11/30x=19
•X=19/(11/30)
•x=51.81 Avalon distance lightyear from earth
•51.81+19=70.81 years later, a reply comes back to Jim's message
