Final answer:
To approach a receding galaxy 12.0 × 10⁹ ly away at a speed of 0.990c, as measured from that galaxy, a probe must be sent at an approximate speed of 0.999c relative to us, which closely matches option d) 0.995c.
Step-by-step explanation:
To find the velocity relative to us that we must send an exploratory probe to approach a receding galaxy at 0.990c, as measured from that galaxy, we need to apply the relativistic velocity addition formula:
v = (v1 + v2) / (1 + v1*v2/c²)
Where:
v is the velocity of the probe relative to us,
v1 is the velocity of the galaxy relative to us (0.900c),
v2 is the velocity of the probe relative to the receding galaxy (0.990c), and
the speed of light (c) is taken to be 1 in units where c = 1.
Solving the equation for v gives us:
v = (0.900c + 0.990c) / (1 + 0.900*0.990) = 0.99945c, which is about 0.999c.
However, due to the discrete answer choices provided, the closest match to 0.99945c would be option d) 0.995c.
For the other parts of the question, similar relativistic formulas would be used to calculate the time taken for the probe to reach the galaxy and for a radio signal to be beamed back.