Which Nova Happened First in Relation to an Aircraft's Speed?

[EngageEngage]

Homework Statement

Astronomers on the Earth (regarded as an inertial reference frame) see two novas flare up simultaneously. One of the novas is at a distance of 1.0x10^3 lightyears in the constelation Draco; the other nova is at an equal distance in the constellation Tucana in a direction (as seen from Earth) exactly opposite to that of the first nova. According to astronomers aboard an aircraft traveling at 750km/hr along the line from draco to tucana, the novas are not simulatneous. According to these astronomers, which nova happened first/ by how many hours?

Homework Equations

t' = $$\frac{t - \frac{Vx}{c^2}}{\sqrt{1 - \frac{V^2}{c^2}}}$$

The Attempt at a Solution

So, i then have two equations:

t'$$_{1}$$ = $$\frac{t_{1} - \frac{Vx_{1}}{c^2}}{\sqrt{1 - \frac{V^2}{c^2}}}$$

t'$$_{2}$$ = $$\frac{t_{2} - \frac{Vx_{2}}{c^2}}{\sqrt{1 - \frac{V^2}{c^2}}}$$

i know that:
t1 = t2
since the novas flare up at the same time on Earth, and i also set
x1 = 0,
x2 = 2*10^3 lightyears = 2(9.46*10^18m)
V = 750*10^3m/hr

my final equatin looks like so:

$$\frac{t'_{2} - t'_{1} = \frac{Vx_{2}}{c^{2}}}{\sqrt{1-\frac{V^{2}}{c^{2}}}}$$
(The above fraction looks wrong. the fraction should only be on the right side.)

So, plugging in all of my values i get a time of 12.16hrs, which is double the right answer. Can someone please tell me what I could have done wrong? any help at all would be greatly appreciated

 

[Doc Al]

I agree with your answer. So unless we both made the same mistake, the book is wrong.

 

[EngageEngage]

Thank you Doc Al, i appreciate the response. I will ask my professor what he thinks of the problem