Given
2-way interplanetary (Earth–Mars) communication by electromagnetic radiation (radio)
Find
a) Why is the function relating distance to signal delivery time a direct linear variation?
b) What is the constant of variation (k)?
c) What function relates distance from Earth and round-trip signalling time?
Solution
a) For the purposes of this problem, we assume space is flat and that planetary velocities are well below the speed of light, so that speed, time, and distance are related by
time = distance / speed
The relevant speed here is the speed of an electromagnetic signal, the speed of light.
The function is linear because we assume it is linear (that the non-linearities are negligible).
b) From part (a), for
time = k*distance
the value of k is
k = 1/c . . . . . where c = the speed of light
c) If the distance (d) from Earth to Mars does not change substantially during the flight time of the signal, the round-trip time (T) will be
T = 2*(k*d)
T = 2d/c
_____
In fact, space is warped by the sun, planets, and other celestial bodies, and the distance from one planet to another is constantly changing. Thus, the signalling time is not a linear function of distance (even if we could state the distance), but is also a function of the path taken and the way that path is changing. Calculating these effects is way beyond the scope of this question, even if all the relevant data were known (which it is not).