A rocket ship flies past the earth at 91.0% of the speed of light. Inside, an astronaut who is undergoing a physical examination is having his height measured while he is lying down parallel to the direction in which the ship is moving. (a) If his height is measured to be 2.00 m by his doctor inside the ship, what height would a person watching this from the earth measure? (b) If the earth-based person had measured 2.00 m, what would the doctor in the spaceship have measured for the astronaut’s height? Is this a reasonable height?

Verified step by step guidance

1. Step 1: Understand the concept of length contraction. Length contraction is a phenomenon predicted by Einstein's theory of special relativity, where the length of an object moving relative to an observer is measured to be shorter than its proper length (the length measured in the object's rest frame). The formula for length contraction is: $L=L$₀1-v2c2, where $L$ is the contracted length, $L$₀ is the proper length, $v$ is the relative velocity, and $c$ is the speed of light.
2. Step 2: Solve part (a). The astronaut's height is measured as 2.00 m inside the ship, which is the proper length $L$₀. To find the height as measured by a person on Earth, use the length contraction formula. Substitute $L$₀ = 2.00 m, $v$ = 0.91$c$, and calculate $\sqrt{1-\frac{v^2}{c^2}}$ to find the contracted length $L$.
3. Step 3: Solve part (b). If the Earth-based person measures the astronaut's height to be 2.00 m, this is the contracted length $L$. To find the proper length $L$₀ as measured by the doctor inside the ship, rearrange the length contraction formula to solve for $L$₀: $L$₀=L1-v2c2. Substitute $L$ = 2.00 m and $v$ = 0.91$c$ to calculate $L$₀.
4. Step 4: Interpret the results. Compare the values obtained in parts (a) and (b). Discuss how the measurements differ depending on the observer's frame of reference and why this is consistent with the principles of special relativity.
5. Step 5: Reflect on the reasonableness of the height. Consider whether the calculated proper length in part (b) is realistic for a human astronaut. If the proper length is significantly larger than typical human heights, discuss how relativistic effects can lead to such results and why this is a theoretical outcome rather than a practical concern.

Key Concepts

Relativity of Length Contraction

Length contraction is a phenomenon predicted by Einstein's theory of special relativity, which states that an object in motion will appear shorter in the direction of its motion to an observer at rest. This effect becomes significant as the object's speed approaches the speed of light. For example, if an astronaut measures his height as 2.00 m while traveling at 91.0% of the speed of light, an observer on Earth would measure a shorter height due to this contraction.

Reference Frames

A reference frame is a perspective from which measurements are made, and it plays a crucial role in understanding motion and relativistic effects. In this scenario, the astronaut's measurements are taken in the rocket's reference frame, while the Earth observer's measurements are taken in a stationary reference frame. The differences in measurements arise because the two observers are in different states of motion, highlighting the relativity of simultaneity and measurement.

Lorentz Factor

The Lorentz factor is a mathematical factor that quantifies the effects of time dilation and length contraction in special relativity. It is defined as γ = 1 / √(1 - v²/c²), where v is the object's velocity and c is the speed of light. At 91.0% of the speed of light, the Lorentz factor becomes significant, allowing us to calculate how much the astronaut's height appears contracted to an Earth observer, and vice versa, providing a basis for understanding the measurements in the problem.