A spaceship is traveling toward the earth from the space colony on Asteroid 1040A. The ship is at the halfway point of the trip, passing Mars at a speed of 0.9c relative to the Mars frame of reference. At the same instant, a passenger on the spaceship receives a radio message from her boyfriend on 1040A and another from her sister on earth. According to the passenger on the ship, were these messages sent simultaneously or at different times?

The messages were sent at different times - from the passenger's reference frame, the message from her sister (on earth) was sent earlier than that from her boyfriend (on asteroid)

Explanation:

The spaceship was travelling from the space colony on Asteroid 1040A towards Earth. It travelled at a speed of 0.9c (relative to Mars). The ship reaches her halfway point in its journey when 2 messages come into a passenger's radio (one from Asteriod 1040A & one from Earth)

Looking at the frame of reference of the ship (not from Mars but from the passenger's frame of reference), we will observe that the message from Earth had to travel over a much longer distance to the spaceship than that from Asteriod 1040A. Since the speed of light is constant in all reference frames and both messages came in at the same time, we can thus safely infer that the message sent by the passenger's sister on Earth was sent first.

Mathematically,

v = x ÷ t

where:

v = velocity, x = distance, t = time

Making t the subject of formula, we have:

t = x ÷ v

v for both Asteroid 1040A & Earth is constant and the same since it is the speed of light we are considering. This means that the message with the greater distance covered was sent earlier as both messages came in together.

Because the distance from Earth to the spaceship is greater, we therefore conclude that the message from the Earth was sent earlier