A rocket traveling at 0.50c sets out for the nearest star, Alpha Centauri, which is 4.3 ly away from earth. It will return to earth immediately after reaching Alpha Centauri. What distance will the rocket travel and how long will the journey last according to (a) stay-at-home earthlings and (b) the rocket crew? (c) Which answers are the correct ones, those in part a or those in part b?

Verified step by step guidance

1. Step 1: Understand the problem and identify the key concepts. This problem involves special relativity, specifically time dilation and length contraction. The rocket is traveling at a speed of 0.50c (half the speed of light), and the distance to Alpha Centauri is 4.3 light-years (ly) as measured by observers on Earth. The journey involves two perspectives: (a) the stay-at-home earthlings and (b) the rocket crew.
2. Step 2: Calculate the total distance traveled according to the stay-at-home earthlings. Since the rocket travels to Alpha Centauri and back, the total distance is twice the one-way distance. Use the formula: \( d_{total} = 2 \times d_{one-way} \), where \( d_{one-way} = 4.3 \ \text{ly} \).
3. Step 3: Calculate the time taken for the journey according to the stay-at-home earthlings. The time for a one-way trip is given by \( t_{one-way} = \frac{d_{one-way}}{v} \), where \( v = 0.50c \). The total time is then \( t_{total} = 2 \times t_{one-way} \).
4. Step 4: Calculate the distance and time experienced by the rocket crew. Due to length contraction, the distance to Alpha Centauri as measured by the rocket crew is shorter. The contracted distance is given by \( d' = d_{one-way} \sqrt{1 - \frac{v^2}{c^2}} \). The total contracted distance is \( d'_{total} = 2 \times d' \). The time experienced by the rocket crew is given by \( t' = \frac{d'}{v} \).
5. Step 5: Address part (c) of the problem. The answers in part (a) (stay-at-home earthlings) and part (b) (rocket crew) are both correct, but they are valid in their respective reference frames. This is a fundamental result of special relativity: measurements of time and distance depend on the observer's frame of reference.

Key Concepts

Relativity of Simultaneity

The relativity of simultaneity is a key principle in Einstein's theory of relativity, which states that events that are simultaneous in one frame of reference may not be simultaneous in another. This concept is crucial for understanding how different observers, such as those on Earth and in the rocket, perceive time and distance differently due to their relative velocities.

Time Dilation

Time dilation is a phenomenon predicted by the theory of relativity, where time passes at different rates for observers in different frames of reference. For the rocket crew traveling at a significant fraction of the speed of light (0.50c), time will pass more slowly compared to the stay-at-home Earthlings, leading to discrepancies in the perceived duration of the journey.

Length Contraction

Length contraction is another relativistic effect that states an object in motion will appear shorter in the direction of motion to a stationary observer. In this scenario, while Earthlings measure the distance to Alpha Centauri as 4.3 light-years, the rocket crew will experience a contracted distance due to their high speed, affecting their calculations of the journey's length.