An astronaut travels to a distant star with a speed of 0.60c relative to Earth. From the astronaut's point of view, the star is 7.0 light-years from Earth. On the return trip, the astronaut travels with a speed of 0.87c relative to Earth.What is the distance covered on the return trip, as measured by the astronaut? Give your answer in light-years.

To determine the distance covered on the return trip as measured by the astronaut, we need to apply some concepts from the theory of relativity, particularly time dilation and length contraction. Let's break this down step by step.

Understanding the Journey

The astronaut travels to a star that is 7.0 light-years away from Earth at a speed of 0.60c (where c is the speed of light). From the astronaut's perspective, the distance to the star is shorter due to the effects of relativity.

Calculating the Time Dilation for the Outbound Trip

First, we need to calculate how long the journey takes from the astronaut's perspective. The time taken for the outbound trip can be calculated using the formula:

• Time = Distance / Speed

For the outbound trip:

• Distance = 7.0 light-years
• Speed = 0.60c

Plugging in the values:

• Time = 7.0 light-years / 0.60c = 11.67 years (as measured by an observer on Earth).

However, due to time dilation, the astronaut experiences less time. The time dilation factor (gamma, γ) is given by:

• γ = 1 / √(1 - v²/c²)

For the speed of 0.60c:

• γ = 1 / √(1 - (0.60)²) = 1 / √(1 - 0.36) = 1 / √0.64 = 1.25.

Now, we can find the time experienced by the astronaut:

• Astronaut's time = Earth time / γ = 11.67 years / 1.25 = 9.34 years.

Calculating the Return Trip

Now, let’s consider the return trip where the astronaut travels at 0.87c. The distance to the star remains the same at 7.0 light-years from Earth's perspective. However, we need to find out how far the astronaut perceives this distance to be.

First, we calculate the new time dilation factor for the return trip:

• γ = 1 / √(1 - (0.87)²) = 1 / √(1 - 0.7569) = 1 / √0.2431 ≈ 2.02.

Next, we calculate the time taken for the return trip from Earth's perspective:

• Time = Distance / Speed = 7.0 light-years / 0.87c ≈ 8.05 years.

Now, we can find the time experienced by the astronaut during the return trip:

• Astronaut's time = Earth time / γ = 8.05 years / 2.02 ≈ 3.98 years.

Determining the Distance Covered by the Astronaut

To find the distance covered by the astronaut on the return trip, we can use the astronaut's speed and the time they experienced:

• Distance = Speed × Time.

Using the astronaut's speed of 0.87c and the time they experienced (3.98 years):

• Distance = 0.87c × 3.98 years.

Since 1 light-year is the distance light travels in one year, we can simplify this to:

• Distance = 0.87 × 3.98 light-years ≈ 3.46 light-years.

Final Result

Thus, the distance covered on the return trip, as measured by the astronaut, is approximately 3.46 light-years.