Problem 5 - College Physics

Question

\(\bullet\) You take a trip to Pluto and back (round trip 11.5 billion km), traveling at a constant speed (except for the turnaround at Pluto) of \(45,000 \mathrm{km} / \mathrm{h}\) . (a) How long does the trip take, in hours, from the point of view of a friend on earth? About how many years is this? (b) When you return, what will be the difference between the time on your atomic wristwatch and the time on your friend's? (Hint: Assume the distance and speed are highly precise, and carry a lot of significant digits in your calculation!)

Step-by-Step Solution

Answer

The trip takes about 29.17 years as seen from Earth. Time dilation effects are negligible.

Step 1: Calculate Total Distance

The total distance for the round trip to Pluto and back is given as 11.5 billion kilometers. This is equal to \(11.5 \times 10^9\) kilometers.

Step 2: Determine Travel Speed

The speed of travel is given as \(45,000 \text{ km/h}\). This will be used to calculate the time taken for the trip.

Step 3: Calculate Time Taken for Trip in Hours

Use the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \) to find the time taken for the trip. Substitute the given values: \[ \text{Time} = \frac{11.5 \times 10^9 \text{ km}}{45,000 \text{ km/h}} \approx 2.55556 \times 10^5 \text{ hours}. \]

Step 4: Convert Hours to Years

To convert the time from hours to years, divide the number of hours by the number of hours in a year (\(8760\) hours). So, \( \frac{2.55556 \times 10^5}{8760} \approx 29.17\) years.

Step 5: Calculate Time Dilation Effect

The time dilation can be calculated using the formula \( \Delta t = t \sqrt{1 - \frac{v^2}{c^2}} \) for special relativity, where \(t\) is the time observed on Earth, \(v\) is velocity, and \(c\) is the speed of light (~\(3 \times 10^8 \text{ m/s}\)). Since \(v\ll c\), \( \Delta t \approx \frac{t v^2}{2 c^2} \). Plug in the values to find \( \Delta t \). However, because \(v\) is very small compared to \(c\), \( \Delta t \) is negligibly small here.

Key Concepts

Special Relativity

Special relativity is a theory developed by Albert Einstein which revolutionizes our understanding of how space and time are intertwined.

• At the heart of this theory is the premise that the laws of physics are the same for all observers, regardless of their velocities, as long as they are not accelerating.
• Time dilation, a key aspect of special relativity, addresses how the time experienced by an observer in motion compared to one at rest differs.
• This means that time can stretch or compress based on the relative speed of an observer.

In the context of the Pluto round trip, special relativity predicts that a clock on the spaceship would measure time differently than one on Earth. It highlights the fascinating consequences of traveling close to the speed of light. Even though the velocities in everyday space travel, like this trip, are much smaller than the speed of light, the concept underscores how space and time are interconnected.

Distance Calculation

Calculating distance is an essential step in understanding journeys like the trip to Pluto. At its core, distance calculation ensures you have an accurate figure for the total length of the journey.To find our total distance:

• Take the round-trip distance to Pluto and back, given as 11.5 billion kilometers.
• This can be expressed in scientific notation as \(11.5 \times 10^9\) kilometers.

This large number highlights the vastness of space and the challenges involved in interplanetary travel.Whether you're calculating for a flight on Earth or a trip to Pluto, understanding the distance helps in planning the time and fuel needed for the trip.

Speed of Light

The speed of light in a vacuum is a critical constant in physics, valued approximately at \(3 \times 10^8\) meters per second.

• This speed is pivotal in special relativity and serves as a universal speed limit.
• For any object, no matter the technology involved, exceeding this speed is impossible according to our current understanding of physics.

When considering space travel, as in the example of a trip to Pluto:- The speed of light sets a benchmark against which our spacecraft speeds are compared.- Even at top conventional speeds (like 45,000 kilometers per hour), we are moving much slower than light.This vast difference in speeds helps us understand why relativistic effects like time dilation are negligible for most space travel.

Velocity Comparison

In the context of traveling to Pluto, comparing velocities is essential for understanding motion and its implications.

• The trip involves a constant travel speed of 45,000 km/h.
• When compared to the speed of light, this velocity, though fast in earthly terms, is minuscule.

This comparison becomes critical in calculating time dilation effects:- The formula for time dilation uses the ratio \( \frac{v^2}{c^2} \).- Given our much smaller speed, the ratio becomes negligibly small.Therefore, although special relativity predicts time differences due to motion, the effects are extremely small for such speeds.Velocity comparison not only provides a scale but also offers insights into the limits and possibilities of human travel beyond our planet.