Problem 7 - College Physics

Question

\(\bullet\) An alien spacecraft is flying overhead at a great distance as you stand in your backyard. You see its searchlight blink on for 0.190 s. The first officer on the craft measures the searchlight to be on for 12.0 ms. (a) Which of these two measured times is the proper time? (b) What is the speed of the spacecraft relative to the earth, expressed as a fraction of the speed of light, c?

Step-by-Step Solution

Answer

(a) 12.0 ms is the proper time; (b) Speed is approximately 0.997c.

Step 1: Understanding Proper Time

Proper time is the time interval between two events as measured by an observer who sees these two events occurring at the same location. In this problem, the first officer on the spacecraft measures the time interval of the searchlight turning on and off while being in the same location as the searchlight. Thus, the first officer's time, 12.0 ms, is the proper time (\(\Delta t_0\)).

Step 2: Converting Milliseconds to Seconds

To maintain consistency in units, convert the officer’s measurement from milliseconds to seconds: \(12.0 \text{ ms} = 12.0 \times 10^{-3} \text{ s} = 0.012 \text{ s}\).

Step 3: Relativity Time Dilation Formula

The time measured by a stationary observer (you in the backyard), denoted by \(\Delta t\), is related to the proper time by the time dilation formula: \(\Delta t = \gamma \Delta t_0\), where \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\) is the Lorentz factor and \(v\) is the speed of the spacecraft.

Step 4: Calculate Lorentz Factor \(\gamma\)

Rearrange the time dilation formula to find \(\gamma\):\[\gamma = \frac{\Delta t}{\Delta t_0} = \frac{0.190}{0.012} \approx 15.8333.\]

Step 5: Determine the Speed as a Fraction of the Speed of Light

Solve for \(v\) using the Lorentz factor:\[\frac{1}{\gamma^2} = 1 - \frac{v^2}{c^2}\]\[\frac{v^2}{c^2} = 1 - \frac{1}{\gamma^2}\]\[v^2 = c^2\left(1 - \frac{1}{\gamma^2}\right)\]\[v = c \sqrt{1 - \frac{1}{\gamma^2}} = c \sqrt{1 - \frac{1}{15.8333^2}} \approx 0.997 c.\]

Step 6: Conclusion

The proper time is the time measured by the spacecraft's officer: 12.0 ms. The speed of the spacecraft relative to Earth is approximately 0.997 times the speed of light, \(c\).

Key Concepts

Proper Time

Proper time is a crucial concept in understanding time dilation in special relativity. It refers to the time interval between two events as measured by an observer who is located at the same position where both events occur.
This concept is fundamental because it is the shortest possible duration that can be measured for an event sequence. In our exercise, the searchlight on the alien spacecraft is turned on and off while the officer is onboard. Thus, the officer's measurement of 12.0 milliseconds is considered the proper time (\(\Delta t_0\)). Knowing this allows us to understand various phenomena in high-speed physics.

• Proper time is observer-specific, tied to the frame of reference moving with the observed object.
• In the twin paradox, the traveling twin experiences proper time while the stay-at-home twin does not, emphasizing its relevance in relativity studies.

Lorentz Factor

The Lorentz factor, denoted as \(\gamma\), is a pivotal element of Einstein's theory of special relativity. It describes how much time, length, and relativistic mass change for an object while it is moving.
In essence, it quantifies the effects of time dilation and length contraction experienced by an object moving at a significant fraction of the speed of light. The formula for the Lorentz factor is \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\).

• As an object's velocity approaches the speed of light, the Lorentz factor increases significantly, indicating greater effects of relativity.
• In our example, a calculated Lorentz factor of approximately 15.8333 suggests the spaceship is moving very close to light speed, affecting how time is experienced aboard the craft compared to on Earth.

Speed of Light

The speed of light, denoted by \(c\), is a fundamental constant in physics, with a value of approximately 299,792 kilometers per second. It is the ultimate speed limit in the universe, according to the principles of special relativity.
In time dilation problems, like the one in our exercise, the speed of light serves as a comparative benchmark for measuring the velocity of fast-moving objects, such as the alien spacecraft.

• Any object moving at speeds approaching the speed of light will experience time dilation effects, as noticeable in our calculation where the spacecraft's speed is about 0.997 times \(c\).
• This constant ties together various relativistic equations, demonstrating its critical role in the framework of special relativity.

Special Relativity

Special relativity is a foundational theory in modern physics proposed by Albert Einstein in 1905. It revolutionized how we understand space, time, and motion, particularly at high velocities close to the speed of light.
It introduces groundbreaking concepts, such as the relativity of simultaneity, time dilation, and length contraction, altering classical notions of absolute time and space.

• Special relativity affects phenomena we observe when objects move quickly relative to each other, as seen with our speeding spacecraft example.
• It is essential for the accurate description of the behavior of particles in accelerators and the propagation of electromagnetic fields.
• Through its principles, we understand that time and space are intertwined into a single continuum known as spacetime, fundamentally altering physics' perspectives since the early 20th century.