Question

To measure the speed of light, Galileo and a colleague stood on different mountains with covered lanterns. Galileo uncovered his lantern and his friend, seeing the light, uncovered his own lantern in turn. Galileo measured the elapsed time from uncovering his lantern to seeing the light signal response. The elapsed time should be the time for the light to make the round trip plus the reaction time for his colleague to respond. To determine reaction time, Galileo repeated the experiment while he and his friend were close to one another. He found the same time whether his colleague was nearby or far away and concluded that light traveled almost instantaneously. Suppose the reaction time of Galileo's colleague was \(0.25 \mathrm{s}\) and for Galileo to observe a difference, the complete round trip would have to take 0.35 s. How far apart would the two mountains have to be for Galileo to observe a finite speed of light? Is this feasible?

Short answer

The mountains would need to be about 14,989 kilometers apart, which is not feasible on Earth.

Step-by-step solution

Analyze Required Time for Observation

The problem states that Galileo could only observe a difference when the total time for the light to make a round trip exceeds 0.35 s. This means that the light round trip time needs to be greater than the colleague's reaction time if Galileo were to notice any difference. We are given \(0.25\) s as the reaction time of the colleague.

Determine Needed Round Trip Time for Light

In order to observe a finite speed of light, the round trip of light must exceed his friend's reaction time by \(0.10\) s, which leads to the required round trip time for light as \(0.35\) s. Hence, the time required for the light alone is \(0.35 - 0.25 = 0.10\) s.

Relate Round Trip Time to Distance

Use the known speed of light \(c = 299,792,458\) m/s. Since \(0.10\) s is required for the round trip (back and forth travel), we calculate the one-way travel time as \(0.10 / 2 = 0.05\) s.

Calculate the One-Way Distance

The one-way distance \(d\) that light can travel in \(0.05\) s is computed using the formula \(d = c \times t\), where \(t\) is the one-way time. Thus, \(d = 299,792,458 \times 0.05 = 14,989,622.9\) meters, which is approximately \(14989.62\) kilometers (or about \(9317.68\) miles).

Evaluate Feasibility

The calculated distance between the two mountains that would allow Galileo to detect the finite speed of light exceeds any practical mountain-to-mountain distance on Earth. Thus, such a measurement would not be feasible with the technology available to Galileo and his colleague.

Key concepts

Galileo's Experiment

Galileo's experiment to measure the speed of light was quite simple in concept but presented practical challenges. Imagine standing on two separate mountains with your friend. Both of you have lanterns that you can cover or uncover, and the idea is that when one person uncovers their lantern, the other sees the light and uncovers theirs in response.

The aim for Galileo was to measure how quickly he could see the return signal after he uncovered his lantern. His process involved timing the interval between uncovering the lantern and seeing his colleague’s signal. It was ingenious because it attempted to measure the incredibly fast speed of light with the rudimentary tools available.

• The main challenge was the immense speed of light compared to human reaction times.
• Galileo hoped that any measurable delay would give him insight into the speed of light.
• His setup required large distances to have any chance of success.

Ultimately, Galileo concluded that light traveled too quickly to be measured effectively with his experimental setup, estimating it to be almost instantaneous.

Reaction Time Measurement

Understanding reaction time was crucial for Galileo’s experiment because it added an unavoidable delay to the timing. Reaction time refers to how quickly someone can respond to a stimulus, like seeing a flash of light and reacting by uncovering their lantern.

In Galileo's setup, the reaction time of his colleague when he saw the initial lantern uncovering was significant. Galileo tested this by performing the experiment both at a close range and at the intended greater distance.

• Galileo realized that with a reaction time of about 0.25 seconds, it posed a challenge.
• The experiment would only reveal the speed of light if this reaction time was finely measurable against the speed of light’s travel time.

The fact that no difference in elapsed time was observed whether the experiment was conducted nearby or from a distance suggested the speed of light far exceeded their ability to measure it with reaction time in play.

Feasibility Analysis

The feasibility of measuring the speed of light with Galileo's method largely depended on whether the light's travel time could exceed the reaction time of his colleague. In the context of the problem, it was necessary for the light to take more than the observer's reaction time of 0.25 seconds, preferably reaching 0.35 seconds, to see any discernible lag caused by the light's travel.

The calculated required distance for Galileo's method posed significant challenges. They needed a practical situation where:

• The light's round-trip time would be greater than 0.35 seconds, which would require astronomical distances.
• The setup needed a level of precision unachievable at the time in terms of clock technology.

This meant that the idea, while brilliant, was not feasible with the distances available and the technology of Galileo's day. Ultimately, the experiment was insightful but flawed in practical execution due to limitations in natural and man-made constraints.

Distance Calculation

Calculating the distance was a pivotal part of Galileo's experiment because it directly related to how and if the speed of light could be inferred. Essentially, the idea was that by knowing the time light took to travel, one could calculate the distance it had moved.

Using the speed of light in modern measures, which is approximately 299,792,458 meters per second, calculating realistic times like 0.05 seconds (one-way) or 0.10 seconds (round-trip) for this velocity allows us to estimate distances.

• The problem required computing the one-way distance light could travel in 0.05 seconds.
• Using the formula \(d = c \times t\), it was found that light would travel 14,989,622.9 meters, translating to approximately 14,989.62 kilometers in that time.

However, such vast distances between mountains were impossible on Earth, confirming the infeasibility of results gained from the proposed experimental circumstances. Galileo's approach, while conceptually fascinating, highlights the challenges of measuring such large-scale and fast-moving phenomena with the tools of the time.