Problem 9 - University Physics with Modern Physics

Question

"Maximum Power, Scotty" The starship Enterprise, of television and movie fame, is powered by the controlled combination of matter and antimatter. If the entire \(400-\) kg antimater fuel supply of the Enterprise combines with matter, how much energy is released? How does this compare to the U.S. yearly energy use, which is roughly \(1.0 \times 10^{20} \mathrm{J} ?\)

Step-by-Step Solution

Answer

The energy released is \(7.2 \times 10^{19} \text{ J}\), which is about 72% of the U.S. yearly energy use.

Step 1: Understanding the Concept

When matter and antimatter collide, they annihilate each other and release energy according to the mass-energy equivalence principle given by Einstein's equation: \(E = mc^2\), where \(E\) is energy, \(m\) is mass, and \(c\) is the speed of light in a vacuum \(\approx 3 \times 10^8 \text{ m/s}\).

Step 2: Calculate Energy from Matter-Antimatter

Since the 400 kg of antimatter will completely annihilate an equal mass of ordinary matter, the total mass \(m\) converting to energy is \(2 \times 400\) kg. Thus, \(m = 800\,\text{kg}\). Substitute this into the equation: \[ E = mc^2 = 800 \times (3 \times 10^8)^2 \]

Step 3: Solve for the Total Energy Released

Calculate the energy: \[ E = 800 \times 9 \times 10^{16} = 7.2 \times 10^{19} \text{ J} \] This represents the total energy released from the complete annihilation of 400 kg of antimatter and 400 kg of matter.

Step 4: Compare with U.S. Yearly Energy Use

Now, compare the calculated energy \(7.2 \times 10^{19} \text{ J}\) with the U.S. yearly energy consumption of \(1.0 \times 10^{20} \text{ J}\). The energy released by the matter-antimatter reaction is approximately \(0.72\) times the U.S. yearly energy use. Hence, it's slightly less than the total annual energy consumption in the U.S.

Key Concepts

Matter and Antimatter Annihilation

When matter and antimatter come into contact, a fascinating process known as annihilation occurs. This process involves matter and antimatter particles completely destroying each other and converting their mass into energy. Matter is made up of normal particles like electrons and protons, while antimatter is made up of particles that have opposite charges, such as positrons (positive counterparts to electrons) and antiprotons.
The concept of matter-antimatter annihilation has important implications, particularly in theoretical propulsion systems like those imagined in science fiction, such as the starship Enterprise. During annihilation, 100% of the mass of the particles is transformed into energy, showcasing the incredible potential of matter-antimatter reactions as an energy source.

• The annihilation process releases energy in the form of gamma rays, which are high-energy photons.
• This transformation is a direct application of the mass-energy equivalence principle.
• Annihilation is one of the most efficient energy conversion processes known, hence its allure in theoretical energy applications.

Mass-Energy Equivalence

The principle of mass-energy equivalence is a fundamental concept in physics that demonstrates how matter can be converted into energy. This principle is succinctly captured by Einstein's famous equation, which states that the energy of an object (E) is equal to its mass (m) multiplied by the square of the speed of light (c), denoted as \(E = mc^2\).
This revolutionary idea shows that mass itself is a form of energy, and even a small amount of mass can be transformed into a tremendous amount of energy, due to the large value of the speed of light \( (c \approx 3 \times 10^8 \text{ m/s})\).

• The equation implies that energy and mass are interchangeable, a concept that was initially difficult to grasp but is now foundational in modern physics.
• This principle explains nuclear reactions, fusion, and fission, where small amounts of mass are converted into large amounts of energy.
• In our example from the step-by-step solution, 800 kg of mass yields an enormous energy output of \(7.2 \times 10^{19} \text{ J}\).

Einstein's Equation

Einstein's equation is not just a formula; it is a profound representation of the link between mass and energy. \(E = mc^2\) translates these two seemingly different properties into interrelated aspects of the same entity.
Historically, this equation has been pivotal in advancing both theoretical physics and practical applications. Here’s how it works step by step when applied to energy calculations:

• Determine the mass involved in the process. In our exercise, the mass is the total amount of matter and antimatter combined.
• Substitute this mass into the equation, ensuring the units are consistent, usually in kilograms for mass and meters per second for the speed of light.
• Calculate the energy by squaring the speed of light, multiplying by the mass, and observing the resultant energy value, often presented in joules.
• Use this energy to draw comparisons or applications, such as determining its significance relative to large-scale energy uses, like annual national energy consumption.

Einstein's equation has broadened our understanding of the universe, impacting fields from astrophysics to engineering and beyond.