Final answer:
The nearest star, Proxima Centauri, is 4.0 × 10^13 km away from the Earth. It takes approximately 1.33 × 10^8 seconds for a light signal to travel from Earth to Proxima Centauri. A spacecraft traveling at 10^-4 c would take approximately 42 years to reach Proxima Centauri.
Step-by-step explanation:
The nearest star, Proxima Centauri, is 4.0 × 10^13 km away from the Earth. In order to calculate the time it takes a light signal from the Earth to reach Proxima Centauri, we can use the speed of light (c = 3 × 10^8 m/s). The distance from the Earth to Proxima Centauri in meters is 4.0 × 10^13 km x 10^3 m/km = 4.0 × 10^16 meters. The time it takes for a light signal to travel from Earth to Proxima Centauri is the distance divided by the speed of light, so it will be 4.0 × 10^16 meters / (3 × 10^8 m/s) = 1.33 × 10^8 seconds.
To calculate how many years it will take a spacecraft traveling at 10^-4 c to reach Proxima Centauri, we need to divide the distance by the velocity of the spacecraft. The distance in meters is 4.0 × 10^16 meters, and the velocity of the spacecraft is 10^-4 c x 3 × 10^8 m/s = 3 × 10^4 m/s. Thus, it will take 4.0 × 10^16 meters / (3 x 10^4 m/s) = 1.33 × 10^12 seconds, or approximately 42 years.