Question

An astronaut on a spaceship traveling at a speed of \(0.50 c\) is holding a meter stick parallel to the direction of motion. a) What is the length of the meter stick as measured by another astronaut on the spaceship? b) If an observer on Earth could observe the meter stick, what would be the length of the meter stick as measured by that observer?

Short answer

Answer: a) The length observed by another astronaut on the spaceship is 1 meter. b) The length of the meter stick as observed by an Earth-bound observer is approximately 0.866 meters.

Step-by-step solution

Part a: Length in the Astronaut's Rest Frame

In this case, the astronaut holding the meter stick and the observer (another astronaut) on the same spaceship are both in the same rest frame. Since they are moving together, there is no relative velocity between them. Hence, the length observed by the second astronaut will be the same as the proper length of the meter stick which is 1 meter. So, the length of the meter stick observed by the second astronaut is 1 meter.

Part b: Length Observed by an Earth-Bound Observer

Now, we need to find out the length of the meter stick as observed by an Earth-bound observer. In this case, the relative velocity between the observer on Earth and the meter stick is 0.50 c; therefore, we can use the length contraction formula given above to find the contracted length of the meter stick as seen by this observer. Plugging in the values into the formula: $$ L = L_0\sqrt{1-\frac{v^2}{c^2} } $$ $$ L = 1\sqrt{1-\frac{(0.50c)^2}{c^2} } $$ $$ L = \sqrt{1-\frac{0.25c^2}{c^2} } $$ $$ L = \sqrt{1-0.25} $$ $$ L = \sqrt{0.75} $$ $$ L \approx 0.866$$ So, the length of the meter stick as measured by the Earth-bound observer is approximately 0.866 meters.

Key concepts

Special Relativity

Special relativity is a fundamental theory in physics introduced by Albert Einstein in 1905. It's crucial for understanding the behavior of objects moving at high speeds, especially those nearing the speed of light. A key principle of special relativity is that the laws of physics are the same for all observers moving at constant speeds relative to each other. This means that measurements of time, space, and length can differ based on the observer's point of view, particularly when moving at different velocities.
Special relativity stands out by incorporating the speed of light as a cosmic speed limit. According to special relativity, nothing can travel faster than light in a vacuum. Furthermore, the speed of light remains constant for all observers, regardless of their own velocities. This has profound implications for our understanding of space and time, leading to phenomena such as time dilation and length contraction.
In essence, relative motion changes the way observers measure space and time. An interesting consequence of this theory is that what one observer perceives as rapid time or an elongated length, another might see as frozen time and contracted length, all dependent on their relative speeds.

Proper Length

The concept of proper length is integral to understanding measurements in special relativity. Proper length refers to the length of an object as measured in the object's rest frame. This means it's the length you would measure if you were stationary relative to the object you're observing.
In the case of the meter stick held by the astronaut on the spaceship, the proper length is 1 meter. This is because the astronauts are at rest relative to the meter stick, and thus they measure its length without the effects of relative motion. Proper length is considered the true length of the object, as it is measured in the observer's own frame of reference where the object is not moving.
The concept becomes particularly significant when comparing it to measurements taken from other frames of reference. When an object moves relative to an observer, the observed length changes—a phenomenon known as length contraction. However, the proper length remains constant and serves as a baseline for calculating such contractions across different frames of reference.

Relative Velocity

Relative velocity is a fundamental concept in understanding the motion of objects as perceived by different observers. It refers to the velocity of an object relative to a particular frame of reference. In the realm of special relativity, determining relative velocity is crucial for interpreting changes in measurements, such as length and time, due to motion.
Imagine two observers: one on a spaceship traveling at 0.50c (half the speed of light), and another stationary on Earth. The relative velocity between them is 0.50c. This relative speed impacts how each observer measures the length of objects, like the meter stick held by the astronaut. While the astronaut measures the stick's proper length, the Earth-bound observer sees it contracted due to their high relative velocity.
In summary, relative velocity helps us understand how motion affects observations. By analyzing how fast one observer is moving concerning another, we can predict how measurements of distance and time will differ in each observer's frame.

Lorentz Factor

The Lorentz Factor, denoted by the Greek letter gamma (\( \gamma \)), is a key element in the mathematical framework of special relativity. It quantifies the amount of time dilation and length contraction experienced by objects moving at significant fractions of the speed of light. The formula for the Lorentz Factor is:

• \[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]

Here, \(v\) represents the relative velocity of the object, and \(c\) is the speed of light. As the relative velocity approaches the speed of light, the Lorentz Factor increases, leading to more pronounced effects of time dilation and length contraction.
In the exercise with the astronaut's meter stick, using the Lorentz Factor helps determine the contracted length as observed by a person on Earth. When the spaceship moves at 0.50c, the Lorentz Factor can be derived to compute how much the length appears to have shrunk from the perspective of the Earth-bound observer.
The Lorentz Factor plays a vital role in ensuring that regardless of the observer, the laws of physics remain consistent and enlightening across various frames of reference. It encapsulates the relativistic effects that come into play at velocities approaching that of light, bridging classical mechanics and modern physics.