As you pilot your space utility vehicle at a constant speed toward the moon, a race pilot flies past you in her spaceracer at a constant speed of $0.800c$ relative to you. At the instant the spaceracer passes you, both of you start timers at zero.
(a) At the instant when you measure that the spaceracer has traveled $1.20\times {10}^8$ m past you, what does the race pilot read on her timer?
(b) When the race pilot reads the value calculated in part (a) on her timer, what does she measure to be your distance from her?
(c) At the instant when the race pilot reads the value calculated in part (a) on her timer, what do you read on yours?

Verified step by step guidance

1. Step 1: Understand the problem involves relative motion at relativistic speeds, so we need to use the principles of special relativity, particularly time dilation and length contraction.
2. Step 2: For part (a), use the time dilation formula: \( t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} \), where \( t' \) is the time measured by the race pilot, \( t \) is the time measured by you, \( v = 0.800c \) is the relative speed, and \( c \) is the speed of light. Calculate \( t \) using the distance \( 1.20 \times 10^8 \) m and speed \( 0.800c \).
3. Step 3: For part (b), use the concept of length contraction. The distance you measure is contracted in the race pilot's frame. Use the formula: \( L' = L \sqrt{1 - \frac{v^2}{c^2}} \), where \( L \) is the distance you measure and \( L' \) is the distance measured by the race pilot.
4. Step 4: For part (c), use the time dilation formula again to find the time on your timer when the race pilot's timer reads the value calculated in part (a). Rearrange the time dilation formula to solve for \( t \) in terms of \( t' \).
5. Step 5: Ensure all calculations respect the relativistic effects due to the high speed (0.800c) and verify each step with the appropriate relativistic formulas to ensure consistency in the results.

Key Concepts

Relativity of Simultaneity

The relativity of simultaneity is a principle in Einstein's theory of relativity stating that events perceived as simultaneous in one frame of reference may not be simultaneous in another. This concept is crucial for understanding how different observers measure time and distance differently when moving at high speeds, such as 0.800c, relative to each other.

Time Dilation

Time dilation is a phenomenon predicted by the theory of relativity, where time is observed to pass at different rates for observers in different inertial frames, especially at high velocities close to the speed of light. This concept helps explain why the race pilot's timer reads differently from yours, as her high speed affects her perception of time compared to yours.

Length Contraction

Length contraction is another relativistic effect where the length of an object moving at a significant fraction of the speed of light appears shorter along the direction of motion to a stationary observer. This concept is essential for understanding how the race pilot measures your distance differently, as her high speed alters her perception of spatial dimensions compared to yours.