/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 13 You are standing at \(x=9.0 \mat... [FREE SOLUTION] | 91Ó°ÊÓ

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You are standing at \(x=9.0 \mathrm{km}\) and your assistant is standing at \(x=3.0 \mathrm{km}\). Lightning bolt 1 strikes at \(x=0 \mathrm{km}\) and lightning bolt 2 strikes at \(x=12.0 \mathrm{km}\). You see the flash from bolt 2 at \(t=10 \mu s\) and the flash from bolt 1 at \(t=50 \mu s\). According to your assistant, were the lightning strikes simultaneous? If not, which occurred first and what was the time difference between the two?

Short Answer

Expert verified
The lightning strikes were not simultaneous. For the teacher, bolt 2 struck first, with bolt 1 striking 38 \(\mu s\) later. For the assistant, bolt 1 struck first, with bolt 2 striking 42 \(\mu s\) later.

Step by step solution

01

Calculate when the light from each bolt reaches each observer

Calculate the times at which the light from bolt 1 and bolt 2 reaches the teacher and the assistant. Light travels at a speed of \(c = 3.0 \times 10^8\) m/s or 3.0 km/\(\mu\)s. So, for teacher: Bolt 1: \(t_1 = \frac{d_1}{c} = \frac{9 \, \mathrm{km}}{3 \, \mathrm{km}/\mu \mathrm{s}} = 3 \, \mu \mathrm{s}\) Bolt 2: \(t_2 = \frac{d_2}{c} = \frac{3 \, \mathrm{km}}{3 \, \mathrm{km}/\mu \mathrm{s}} = 1 \, \mu \mathrm{s}\) And for the assistant: Bolt 1: \(t'_1 = \frac{d'_1}{c} = \frac{3 \, \mathrm{km}}{3 \, \mathrm{km}/\mu \mathrm{s}} = 1 \, \mu \mathrm{s}\) Bolt 2: \(t'_2 = \frac{d'_2}{c} = \frac{9 \, \mathrm{km}}{3 \, \mathrm{km}/\mu \mathrm{s}} = 3 \, \mu \mathrm{s}\)
02

Calculate the actual time of each bolt

The actual time each bolt struck is when the observer saw the flash minus the time it took the light to reach them. So, For teacher: Bolt 1: \(T_1 = t_1 - t_1 = 50 \mu s - 3 \mu s = 47 \mu s\) Bolt 2: \(T_2 = t_2 - t_2 = 10 \mu s - 1 \mu s = 9 \mu s\) And for the assistant: Bolt 1: \(T'_1 = t'_1 - t'_1 = 50 \mu s - 1 \mu s = 49 \mu s\) Bolt 2: \(T'_2 = t'_2 - t'_2 = 10 \mu s - 3 \mu s = 7 \mu s\)
03

Determine simultaneity and time difference

The two bolts were not simultaneous for either observer. But for the teacher, bolt 2 occurred first, with bolt 1 occurring 38 \(\mu s\) later. For the assistant, bolt 1 occurred first, with bolt 2 occurring 42 \(\mu s\) later.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Speed of Light
Understanding the speed of light is crucial when solving problems in physics, especially those involving distances over which light travels. Light travels at a constant speed of approximately 3.0 x 10^8 meters per second (m/s), or in more manageable terms, 300,000 kilometers per second (km/s). This incredible speed is not just a universal constant; it’s also the upper limit for the speed at which information can be transmitted.

When we reference the speed of light, often symbolized as 'c', we're discussing the rate at which all massless particles, which include photons—the carriers of light—can travel in a vacuum. No physical object or signal can exceed this speed, making it a fundamental constraint in the realm of physics. This limitation leads to fascinating consequences, such as the relativity of simultaneity, which is deeply intertwined with Einstein’s theory of relativity.
Relativity of Simultaneity
The relativity of simultaneity is a concept within Albert Einstein's theory of relativity that tells us two events that appear simultaneous from one frame of reference may not appear simultaneous from another. This phenomenon is a direct result of the finite and invariant speed of light and the fact that observers may be moving relative to one another.

In our exercise, for example, whether the lightning strikes were simultaneous or not depends on the observer's position and the time it took light to reach them. This is because they are at different distances from where the lightning bolts strike. Each observes the events at different times, so what may seem instantaneous to one may appear sequential to another, which leads to differing accounts of the order of events.
Time of Light Travel
The time light takes to travel a specific distance is another key concept to grasp when approaching physics problems. It's calculated by dividing the distance traveled by the speed of light. In our scenario, the differing distances from the lightning strikes to the observers result in differing travel times for the light.

For accurate problem-solving, it's crucial to factor in these travel times to determine the actual moment of events. The precise calculations involve subtracting the travel time of light from the observed time to pinpoint the true instance of events, as elucidated in the step-by-step solution of the exercise. This consideration of light travel time is what leads to the recognition of the relativity of simultaneity observed by different frames of reference.
Physics Problem Solving
Physics problem-solving is an art that combines understanding concepts, applying mathematical equations, and considering the relevant physical principles. It's about breaking down complex scenarios into understandable parts. The crux lies in developing a systematic approach: identifying what's given, what's unknown, and how the pieces of information relate to each other through physical laws.

In the exercise provided, following a structured method, such as determining the travel time of light for each observer and then calculating the actual times of the lightning strikes, allows us to unravel the scenario. This incremental approach helps in logically deducing the relationship between events from multiple perspectives, showcasing the beauty and intricacy of physics, and leading to a deeper appreciation and understanding of the world around us.

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Most popular questions from this chapter

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