Ch 44: Particle Physics and Cosmology

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 44: Particle Physics and Cosmology

Problem 7

Chapter 44, Problem 7

The starship Enterprise, of television and movie fame, is powered by combining matter and antimatter. If the entire $400$ -kg antimatter 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 $1.0 \times 10^{20}$ J?

Verified step by step guidance

Step 1: Understand the concept of matter-antimatter annihilation. When matter and antimatter combine, they annihilate each other completely, converting their mass into energy according to Einstein's equation:

E = mc^2

, where E is the energy released, m is the total mass, and c is the speed of light (approximately 3.0 × 10^8 m/s).

Step 2: Calculate the total mass involved in the annihilation. Since the problem states that 400 kg of antimatter combines with an equal amount of matter, the total mass is

m = 400 \, \(\text{kg}\) + 400 \, \(\text{kg}\) = 800 \, \(\text{kg}\)

Step 3: Substitute the total mass and the speed of light into Einstein's equation. The energy released is given by

E = (800 \, \(\text{kg}\)) \(\cdot\) (3.0 \(\times\) 10^8 \, \(\text{m/s}\))^2

. Perform the squaring of the speed of light and multiply it by the total mass to find the energy.

Step 4: Compare the energy released to the U.S. yearly energy use. The problem states that the U.S. yearly energy use is approximately

1.0 \(\times\) 10^{20} \, \(\text{J}\)

. Divide the energy released from the matter-antimatter annihilation by this value to determine how many times greater the released energy is compared to the U.S. yearly energy use.

Step 5: Interpret the result. The energy released from the annihilation of 800 kg of matter and antimatter is expected to be significantly larger than the U.S. yearly energy use, demonstrating the immense energy potential of matter-antimatter reactions.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Mass-Energy Equivalence

Mass-energy equivalence is a principle articulated by Einstein's famous equation E=mc², which states that mass can be converted into energy and vice versa. In this context, when matter and antimatter annihilate each other, their entire mass is converted into energy, resulting in a significant energy release. This concept is fundamental for calculating the energy produced from the annihilation of the antimatter fuel.

Annihilation Reaction

Annihilation reactions occur when a particle and its corresponding antiparticle collide, resulting in their complete conversion into energy, typically in the form of gamma rays. In the case of the Enterprise, the 400-kg of antimatter would annihilate with an equal mass of matter, producing energy according to the mass-energy equivalence principle. Understanding this reaction is crucial for determining the total energy output from the antimatter fuel.

Comparative Energy Analysis

Comparative energy analysis involves evaluating the energy output from a specific reaction or process against a known benchmark, such as national energy consumption. In this scenario, comparing the energy released from the annihilation of 400 kg of antimatter to the U.S. yearly energy use (approximately 1.0 * 10^20 J) provides context for the magnitude of energy produced. This analysis helps illustrate the potential of antimatter as a powerful energy source.

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