Problem 13 - College Physics

Question

A rocket ship flies past the earth at 85.0\(\%\) of the speed of light. Inside, an astronaut who is undergoing a physical exami- nation is having his height measured while he is lying down parallel to the direction the rocket ship is moving. (a) If his height is measured to be 2.00 \(\mathrm{m}\) by his doctor inside the ship, what height would a person watching this from earth measure for his height? (b) If the earth-based person had measured \(2.00 \mathrm{m},\) what would the doctor in the spaceship have measured for the astronaut's height? Is this a reasonable height? (c) Sup- pose the astronaut in part (a) gets up after the examination and stands with his body perpendicular to the direction of motion. What would the doctor in the rocket and the observer on earth measure for his height now?

Step-by-Step Solution

Answer

(a) 1.05 m, (b) 3.62 m (not reasonable), (c) 2.00 m; 2.00 m.

Step 1: Understand the Concept of Length Contraction

Length contraction occurs in special relativity when an object is moving relative to an observer. The formula for length contraction is given by \( L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \), where \( L \) is the length measured by an observer, \( L_0 \) is the rest length, \( v \) is the velocity of the object, and \( c \) is the speed of light.

Step 2: Identify Given Values for Part (a)

In this part, the height of the astronaut measured by the doctor inside the spaceship is \( L_0 = 2.00 \, \mathrm{m} \). The speed at which the spaceship is moving is \( v = 0.85c \). We need to find \( L \), the height measured by an observer on Earth.

Step 3: Calculate Length Measured from Earth in Part (a)

Using the formula for length contraction: \( L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \). Substitute \( L_0 = 2.00 \, \mathrm{m} \) and \( v = 0.85c \):\[L = 2.00 \sqrt{1 - (0.85)^2}\]. Calculate: \[L = 2.00 \sqrt{1 - 0.7225} = 2.00 \sqrt{0.2775} \approx 1.05 \, \mathrm{m} \].

Step 4: Identify Given Values for Part (b)

In this part, the height of the astronaut measured by the Earth observer is given as \( L = 2.00 \, \mathrm{m} \). We need to find \( L_0 \), the height measured by the doctor inside the spaceship.

Step 5: Calculate Rest Length Measured in Spaceship in Part (b)

Rearrange the length contraction formula to solve for \( L_0 \): \( L_0 = \frac{L}{\sqrt{1 - \frac{v^2}{c^2}}} \). Substitute \( L = 2.00 \, \mathrm{m} \) and \( v = 0.85c \):\[L_0 = \frac{2.00}{\sqrt{1 - (0.85)^2}} \]. Calculate: \[L_0 = \frac{2.00}{\sqrt{0.2775}} \approx 3.62 \, \mathrm{m} \]. This value is not reasonable for human height.

Step 6: Understand Orientation Effect in Part (c)

When the astronaut stands perpendicular to the direction of motion, there is no length contraction as relativity only affects dimensions parallel to the motion. Therefore, both the doctor and the Earth observer will measure the height as \(2.00 \, \mathrm{m} \).

Key Concepts

Special Relativity

Special Relativity is a theory of physics proposed by Albert Einstein. It describes how time and space are connected for objects moving close to the speed of light. The core idea is that the laws of physics are the same for all non-accelerating observers. This theory introduces several concepts that seem counterintuitive. Among these are time dilation, length contraction, and the equivalence of mass and energy. When objects travel near the speed of light, these factors become significant. For instance, an object moving at high speeds will appear shorter in the direction of motion when observed by someone at rest relative to the object. This is known as length contraction. These changes are not noticeable in everyday life but become significant at speeds approaching that of light.

Speed of Light

The speed of light, denoted by the symbol \(c\), is approximately \(299,792,458\) meters per second. It is considered the ultimate speed limit in the universe. According to Einstein's Special Relativity, nothing with mass can move faster than light. The speed of light plays a crucial role in the theory of Special Relativity. It provides the benchmark against which all speeds are measured. Because light's speed is constant for all observers, regardless of their motion relative to the light source, it creates interesting effects for moving objects, like time dilation and length contraction. For example, a rocket moving at a significant fraction of the speed of light will experience length contraction along its direction of motion. This means that, to an outside observer, objects within the rocket would appear shorter.

Relative Motion

Relative motion refers to the concept that movement is observed and measured in relation to another object. In our exercise, the astronaut on the spaceship and the observer on Earth experience phenomena differently due to their relative motion. In terms of length contraction, the astronaut will measure his height as 2.00 meters because he is at rest relative to himself. However, for someone on Earth, who sees the spaceship moving, the astronaut appears shorter along the direction of motion. This is because the lengths parallel to the direction of motion contract as per the formula given by \(L = L_0 \sqrt{1 - \frac{v^2}{c^2}}\). Thus, an Earth-bound observer will measure the astronaut’s height as 1.05 meters when he is lying parallel to the spaceship's motion. Understanding how measurements change due to relative motion is essential in grasping Special Relativity.

Measurement Accuracy

Measurement accuracy in the context of Special Relativity can be quite complex. As observed, measurements can drastically change depending on the observer's frame of reference due to effects like length contraction. It's crucial for observers to carefully note who is making a measurement and their respective speeds and positions relative to each other. For instance, an astronaut measures his static height without considering any relative motion. However, an Earth-based observer makes a different measurement due to the high velocity of the spaceship. As the speed approaches a significant percentage of the speed of light, even small discrepancies can lead to large errors in calculations if not accurately accounted for. In scientific observations, it's vital to apply the correct relativistic formulas to ensure accurate results after considering the relative motions involved.