Problem 69 - University Physics with Modern Physics

Question

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 a year.) You plan to travel at constant speed in a \(1000-\mathrm{kg}\) 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 percentage of U.S. yearly use (which is 1.0 \(\times 10^{19} \mathrm{J} ) .\) For comparison, arrange your results in a table showing \(v_{\text { rocket }},\) \(t_{\text { earth }},\) \(\mathbf{t}_{\text { rouket }} \boldsymbol{E}\) \((\text { in } \mathrm{J}),\) and \(E\) (as \(\%\) of U.S. use). The rocket ship's speed is (a) \(0.50 \mathrm{c} ;\) (b) 0.99 \(\mathrm{c}\) (c) \(0.9999 \mathrm{c} .\) 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

Space travel to Betelgeuse at high speeds requires enormous energy, making it highly impractical with current technology.

Step 1: Understanding Time Dilation

To solve the problem, we need to understand the concept of time dilation in special relativity. Time dilation means that time passes at a different rate for observers moving at high speeds compared to those at rest. The formula to calculate the time experienced by astronauts (\(t_{rocket}\)) is:\[t_{rocket} = t_{earth} \times \sqrt{1 - \left(\frac{v}{c}\right)^2}\]where \(v\) is the speed of the rocket, \(c\) is the speed of light, and \(t_{earth}\) is the time measured by people on Earth, which is given by \(distance/speed\).

Step 2: Calculating Earth and Rocket Time for Each Speed

For each speed given, calculate \(t_{earth}\) and \(t_{rocket}\). 1. \(v = 0.50c\): - \(t_{earth} = \frac{500}{0.50} = 1000\) years. - \(t_{rocket} = 1000 \times \sqrt{1 - 0.25} = 1000 \times 0.866 = 866\) years.2. \(v = 0.99c\):- \(t_{earth} = \frac{500}{0.99} = 505.05\) years.- \(t_{rocket} = 505.05 \times \sqrt{1 - 0.9801} = 505.05 \times 0.141 = 71.2\) years.3. \(v = 0.9999c\):- \(t_{earth} = \frac{500}{0.9999} = 500.05\) years.- \(t_{rocket} = 500.05 \times \sqrt{1 - 0.99980001} = 500.05 \times 0.0141 = 7.055\) years.

Step 3: Calculating Energy Required

The energy required is calculated using the relativistic kinetic energy formula:\[E = \left(\frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} - 1\right) m c^2\]where \(m = 1000 \text{ kg}\), and \(c = 3 \times 10^8 \text{ m/s}\).1. \(v = 0.50c\): - \(E = (1.1547 - 1) \times 1000 \times (3 \times 10^8)^2 = 6.97 \times 10^{19}\) J.2. \(v = 0.99c\):- \(E = (7.0888 - 1) \times 1000 \times (3 \times 10^8)^2 = 4.38 \times 10^{20}\) J.3. \(v = 0.9999c\):- \(E = (70.71 - 1) \times 1000 \times (3 \times 10^8)^2 = 4.24 \times 10^{21}\) J.

Step 4: Calculating Energy as a Percentage of U.S. Yearly Use

Calculate the energy needed as a percentage of the U.S. yearly use.1. For \(v = 0.50c\):- \(\frac{6.97 \times 10^{19}}{1.0 \times 10^{19}} = 697\%\)2. For \(v = 0.99c\):- \(\frac{4.38 \times 10^{20}}{1.0 \times 10^{19}} = 4380\%\)3. For \(v = 0.9999c\):- \(\frac{4.24 \times 10^{21}}{1.0 \times 10^{19}} = 42400\%\)

Step 5: Tabulating Results

Create a table using the results from the previous steps:\[\begin{array}{|c|c|c|c|c|}\hlinev_{\text{rocket}} & t_{\text{earth}} (\text{years}) & t_{\text{rocket}} (\text{years}) & E (\text{J}) & E (\% \text{of U.S. use}) \hline0.50c & 1000 & 866 & 6.97 \times 10^{19} & 697\0.99c & 505.05 & 71.2 & 4.38 \times 10^{20} & 4380\0.9999c & 500.05 & 7.05 & 4.24 \times 10^{21} & 42400\\hline\end{array}\]

Key Concepts

Special Relativity

Special Relativity is a cornerstone of modern physics, introduced by Albert Einstein in 1905. One of its fascinating implications is time dilation. According to this theory, the faster an object moves, the more time "stretches" for it compared to a stationary observer. In our exercise, time dilation is crucial to understanding how astronauts can experience time differently during interstellar journeys. For example, as a spacecraft speeds towards a distant star like Betelgeuse, time for the astronauts on board slows down compared to people on Earth. This difference can be calculated using the formula: \[t_{\text{rocket}} = t_{\text{earth}} \times \sqrt{1 - \left(\frac{v}{c}\right)^2}\] Here, \(v\) is the spaceship's velocity, and \(c\) represents the speed of light. Despite appearing theoretical, these calculations show real effects like those experienced by astronauts on high-speed travels.

Energy Calculations in Physics

Energy calculations in physics often consider the conversion and conservation of energy under various conditions. In high-speed travel, such as attempting to reach near-light speeds, relativistic kinetic energy becomes significant. This is because traditional mechanics does not sufficiently describe energies at such velocities. The energy required to propel a spacecraft at high speeds is given by: \[E = \left(\frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} - 1\right) m c^2\] In this equation, \(m\) is the mass of the spaceship, \(v\) is its velocity, and \(c\) is the speed of light. By substituting the values for different speeds like \(0.50c, 0.99c, \text{and } 0.9999c\), we find that the energy required increases dramatically. The results show enormous energy demands, far surpassing current U.S. annual energy usage. This illustrates the immense challenge of achieving high-speed interstellar travel with our current capability.

Interstellar Travel

Interstellar travel refers to the journey between stars, often spanning light-years—distances light takes years to traverse. While such journeys have captured imaginations, implementing them presents vast challenges. Considerations include the speed required to reach destinations like Betelgeuse, 500 light-years away, within a human lifetime. Accelerating a spacecraft to speeds such as \(0.50c, 0.99c, \text{or } 0.9999c\) would reduce the perceived travel time for astronauts due to time dilation. However, the technological and energy requirements remain currently unattainable. Even if we assume a 1000 kg spaceship, the energy demands for each proposed speed are staggering—which, as calculated, exceed U.S.'s yearly energy use by thousands of percent. Thus, while intriguing, current interstellar travel plans face significant practical obstacles.