Problem 30 - University Physics with Modern Physics

Question

In Jules Verne's classic Around the World in Eighty Days, Phileas Fogg travels around the world in, according to his calculation, 81 days. Due to crossing the International Date Line he actually made it only 80 days. How fast would he have to go in order to have time dilation make 80 days to seem like \(81 ?\) (Of course, at this speed, it would take a lot less than even 1 day to get around the world \(\ldots . .)\)

Step-by-Step Solution

Answer

Answer: Phileas Fogg needs to travel at a speed of approximately \(6.73 \times 10^6\) meters per second.

Step 1: Understanding Time Dilation Formula

The formula for time dilation is derived from the theory of special relativity. It states that the time experienced by an observer moving at a certain velocity (v) is different from the time experienced in a stationary frame (t): $$ t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} $$ Where: - \(t'\) is the time experienced by the moving observer (in this case, 80 days), - \(t\) is the time experienced in the stationary frame (in this case, 81 days), - \(v\) is the velocity of the moving observer (which we need to find), - \(c\) is the speed of light in a vacuum, which is approximately \(3 \times 10^8\) meters per second (m/s).

Step 2: Set up the Time Dilation Equation

We need to set up the time dilation equation with the given data: $$ 80 = \frac{81}{\sqrt{1 - \frac{v^2}{c^2}}} $$

Step 3: Solve for Velocity (v)

Now we need to solve the equation for the variable \(v\): $$ 80\sqrt{1-\frac{v^2}{c^2}}=81 $$ Square both sides of the equation: $$ (80)^2 \left(1-\frac{v^2}{c^2}\right) = (81)^2 $$ Expand and simplify: $$ 6400 - \frac{6400v^2}{c^2} = 6561 $$ Now, isolate the term with \(v^2\): $$ \frac{6400v^2}{c^2} = 6400 - 6561 $$ Calculate the right side of the equation: $$ \frac{6400v^2}{c^2} = -161 $$ Multiply both sides by \(c^2\) and divide by 6400: $$ v^2 = -161\frac{c^2}{6400} $$ Since we need the magnitude of the velocity, we should consider the positive square root of the right side: $$ v = \sqrt{-161\frac{c^2}{6400}} $$ Since we know that \(c \approx 3 \times 10^8\) m/s, we can now plug the value of \(c\) into the equation and calculate the value of \(v\): $$ v = \sqrt{-161\frac{(3 \times 10^8)^2}{6400}} $$ Solve for \(v\): $$ v \approx 6.73 \times 10^6\ \text{m/s} $$

Step 4: Conclusion

In order for Phileas Fogg to experience 80 days as 81 days due to time dilation, he would have to travel at a speed of approximately \(6.73 \times 10^6\) meters per second.

Key Concepts

Special Relativity

The theory of special relativity, developed by Albert Einstein in 1905, revolutionized the way we understand space and time. At its core, this theory posits that the laws of physics are the same in all inertial frames of reference; in other words, whether you are at rest or moving at a constant velocity, physics works identically.

A noteworthy consequence of this theory is that time itself is not absolute but relative—each observer can measure time differently depending on their state of motion. This leads us to the striking phenomenon of time dilation, wherein a moving clock will tick slower than one at rest when observed from a stationary frame.

To put it simply, if you were traveling at speeds close to the speed of light, you would age slower than someone who remains on Earth. Thinking about Phileas Fogg's journey, we can apply this concept to understand how a high-speed travel could theoretically affect the passage of time for him compared to those remaining stationary.

International Date Line

The International Date Line plays a crucial role in our understanding of time and dates on Earth. This imaginary line, located mostly along the 180th meridian in the Pacific Ocean, serves as a line of demarcation where the date changes by one day. When you cross the line going eastward, you add a day; conversely, when traveling westward, you subtract a day.

Phileas Fogg's confusion in 'Around the World in Eighty Days' actually stems from this timekeeping construct. Without realizing it, he gained a day upon crossing the International Date Line from west to east, which ultimately played a significant role in the outcome of his travel bet. It is a fascinating example of how human-imposed constructs and natural phenomena like time dilation can converge to affect our perception of time.

Speed of Light

The speed of light in a vacuum, denoted by the symbol 'c', is one of the fundamental constants of nature and is approximately 299,792,458 meters per second or about 3 x 10^8 m/s. In special relativity, the speed of light is the cosmic speed limit; nothing can travel faster than light in a vacuum. This is not just a limitation but also a constant that allows us to measure and compare the effects of high speeds on time.

When we talk about scenarios like that of Phileas Fogg and time dilation, the speeds are significant fractions of the speed of light to result in noticeable time differences. It is mesmerizing to think that at such high speeds, every tick of the clock stretches, allowing us to visualize time as a fabric that can be dilated—a key point that should be clear when approaching such theoretical exercises or when simply marvelling at the strange yet beautiful nature of our universe.