Problem 48 - College Physics

Question

Space travel? Travel to the stars requires hundreds or thousands of years, even at the speed of light. Some people have suggested that we can get around this difficulty by accelerating the rocket (and its astronauts) to very high speeds so that they will age less due to time dilation. The fly in this ointment is that it takes a great deal of energy to do this. Suppose you want to go to the immense red giant Betelgeuse, which is about 500 light years away. (A light year is the distance that light travels in one year.) You plan to travel at constance that in a 1000 kg rocket ship (a little over a ton), which, in reality, is far too small for this purpose. In each case that follows, calculate the time for the trip, as measured by people on earth and by astronauts in the rocket ship, the energy needed in joules, and the energy needed as a percent of U.S. yearly use (which is \(1.0 \times 10^{20} \mathrm{J} ) .\) For comparison, arrange your results in a table showing \(v_{\text { Rocket }}, t_{\text { Earth }}, t_{\text { Rocket }}, E(\) in \(\mathrm{J}),\) and \(E\) (as \(\%\) of U.S. use \() .\) The rocket ship's speed is (a) \(0.50 c,\) (b) 0.99\(c\) , and (c) 0.9999\(c .\) On the basis of your results, does it seem likely that any government will invest in such high- speed space travel any time soon?

Step-by-Step Solution

Answer

Such high-speed space travel is unlikely due to immense energy requirements and impracticality.

Step 1: Understanding the problem

We need to calculate the time taken to travel to Betelgeuse, as measured by observers on Earth (\(t_{\text{Earth}}\)), and the time as measured by astronauts on the rocket ship (\(t_{\text{Rocket}}\)), when traveling at different speeds: \(0.50c, 0.99c, \text{and } 0.9999c\). We'll also compute the energy required for such speeds in joules and as a percentage of U.S. yearly energy consumption.

Step 2: Calculating Earth Time

The time taken by the rocket from Earth's perspective (\(t_{\text{Earth}}\)) is calculated using the formula: \(t_{\text{Earth}} = \frac{d}{v}\), where \(d = 500 \text{ light years}\) and \(v\) is the velocity of the rocket in terms of \(c\). For each \(v\) value, calculate \(t_{\text{Earth}}\) accordingly.

Step 3: Calculating Rocket Time using Time Dilation

Time dilation in special relativity gives the time experienced on the rocket ship:\[ t_{\text{Rocket}} = \frac{t_{\text{Earth}}}{\gamma} \]where \(\gamma = \frac{1}{\sqrt{1-v^2/c^2}}\) is the Lorentz factor. Calculate \(t_{\text{Rocket}}\) using the values of \(t_{\text{Earth}}\) from Step 2 and the corresponding \(\gamma\) values for each speed.

Step 4: Calculating Energy Required

The energy required to accelerate the rocket to velocity \(v\) is given by:\[ E = (\gamma - 1)m c^2 \]where \(m = 1000 \text{ kg}\), \(c\) is the speed of light (\(3 \times 10^8 \text{ m/s}\)), and \(\gamma\) is from Step 3. Compute \(E\) for each speed.

Step 5: Energy as a Percentage of U.S. Yearly Use

Convert the energy calculated in Joules to a percentage of the U.S.'s yearly energy usage (\(1.0 \times 10^{20} \text{ J}\)) by using the formula:\[ E(\%) = \left(\frac{E}{1.0 \times 10^{20}}\right) \times 100 \]

Step 6: Analyzing the Results

Create a table to organize results for each speed. Analyze whether such high-speed space travel is feasible considering the vast energy requirements and practical possibilities.

Key Concepts

Speed of Light

The speed of light, denoted as \(c\), is a universal constant valued at approximately \(3 \times 10^8\) meters per second. It's the maximum speed at which all energy, matter, and information in the universe can travel. This speed sets a cosmic speed limit, as it is the fastest anything can move through space.

In the context of space travel, achieving speeds close to the speed of light, even at a fraction like \(0.50c, 0.99c, \) or \(0.9999c\), significantly reduces travel time to distant stars, theoretically making interstellar travel possible. However, due to an increase in energy demand and other physical limits, reaching or sustaining such speeds remains a monumental challenge.

• The faster a spacecraft travels, the more pronounced relativistic effects such as time dilation become.
• Any substantial speed changes close to \(c\) require precise calculations and immense energy, as dictated by Einstein's theory of relativity.

Understanding the ramifications of approaching light speed is crucial for future space endeavors.

Time Dilation

Time dilation is a fascinating effect of Einstein's theory of relativity, describing how time moves at different rates depending on the observer's velocity. As a spaceship accelerates to a significant fraction of the speed of light, passengers on board experience time at a slower rate compared to stationary observers on Earth.

This means astronauts traveling close to the speed of light would age more slowly during their journey. The formula for time dilation, expressed through the Lorentz factor \(\gamma\), shows this relationship:
\[ t_{\text{Rocket}} = \frac{t_{\text{Earth}}}{\gamma} \]

• The Lorentz factor \(\gamma\) increases as the speed approaches \(c\).
• Time dilation becomes more noticeable at speeds close to the speed of light.
• For example, in a journey to Betelgeuse, the onboard time would be much less than the time elapsed on Earth.

Time dilation offers a theoretical explanation for making long-duration space travel more feasible from the astronaut's perspective.

Energy Consumption

Energy consumption for high-speed space travel is immense. To propel a spacecraft to a considerable fraction of the speed of light, vast amounts of energy must be consumed. This energy is calculated using the relativistic formula: \[ E = (\gamma - 1)m c^2 \]
Here, \(E\) is the energy required, \(m\) is the mass of the spacecraft, \(c\) is the speed of light, and \(\gamma\) is the Lorentz factor.

• As velocity increases, \(\gamma\) increases significantly, resulting in higher energy demands.
• For a 1000 kg spacecraft, the energy required at these high speeds becomes astronomically high.
• When compared to current energy production scales, like the annual energy usage of the U.S., this requirement is staggering.

Overcoming this energy barrier presents one of the greatest challenges in realizing the dream of interstellar travel.

Lorentz Factor

The Lorentz factor, represented as \(\gamma\), plays a crucial role in relativity and describes the effects of traveling at high speeds. It is mathematically expressed as: \[ \gamma = \frac{1}{\sqrt{1-v^2/c^2}} \]
This factor determines how much time dilation and energy increase as an object approaches the speed of light. As \(v\) approaches \(c\), the denominator approaches zero, causing \(\gamma\) to increase dramatically.

• In practical terms, \(\gamma\) impacts both the perceived passage of time onboard a fast-moving spacecraft and its required energy consumption.
• A larger Lorentz factor means significant time dilation effects and vast energy needs.
• Understanding \(\gamma\) is essential for calculating and predicting the feasibility of high-speed travel scenarios.

The Lorentz factor is essential for grasping the theoretical underpinnings of traveling near light speed within our universe.