Problem 7 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding Matter-Antimatter Annihilation

Matter and antimatter annihilation converts their mass into energy, following the mass-energy equivalence principle which is given by Einstein's equation: \(E=mc^2\). Here, \(m\) is the mass and \(c\) is the speed of light (\(c = 3.0 \times 10^8 \ \text{m/s}\)).

Step 2: Calculating Total Mass Involved

The total mass involved in the reaction is the sum of the matter and antimatter mass. The antimatter mass is \(400\ \text{kg}\), and it combines with an equal mass of matter, so total mass \(m = 400 \ \text{kg} + 400 \ \text{kg} = 800 \ \text{kg}\).

Step 3: Calculating Released Energy

Apply the formula \(E = mc^2\) to find the energy released. Substitute \(m = 800\ \text{kg}\) and \(c = 3.0 \times 10^8 \ \text{m/s}\): \[E = 800 \ \text{kg} \times (3.0 \times 10^8 \ \text{m/s})^2 = 7.2 \times 10^{19} \ \text{J}\].

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

The energy released \(7.2 \times 10^{19} \ \text{J}\) can be compared with the U.S. yearly energy use. Calculate the ratio of released energy to U.S. energy use: \[\frac{7.2 \times 10^{19}}{1.0 \times 10^{20}} = 0.72\]. This means the released energy is 72% of the U.S. yearly energy consumption.

Key Concepts

Mass-Energy Equivalence

In the realm of physics, understanding how mass translates into energy is crucial. This idea is beautifully encapsulated in Einstein's famous equation, \(E=mc^2\).

• \(E\) represents the energy produced.
• \(m\) is the mass undergoing transformation.
• \(c\) is the speed of light, approximately \(3.0 \times 10^8 \, \text{m/s}\).

This equation tells us that mass can be converted into energy, and the conversion factor is \(c^2\). This is an incredibly large number, indicating that even a small amount of mass can result in a tremendous amount of energy. In the context of matter-antimatter annihilation, when 400 kg of antimatter combines with 400 kg of matter, both are converted into pure energy. This direct conversion exploits the mass-energy equivalence at its fullest potential.

Relativistic Physics

Relativistic physics extends classical mechanics to include what happens when objects move close to the speed of light.

• In such scenarios, we need to account for increases in mass and the time-slowing effects described by Einstein.
• In our case, we're considering mass as a form of energy storage.

Even when stationary, mass has energy content, demonstrated by its potential conversion into other energy forms. When both matter and antimatter are present, they can annihilate, releasing energy as per the relativistic concepts. This ensures that the laws of physics, including conservation of energy, hold true even at cosmic or subatomic scales. The annihilation process adheres to these principles, showing that relativistic physics governs not only motion but energy conservation and transformation as well.

Energy Calculation

Calculating the energy released from matter-antimatter annihilation involves straightforward application of \(E = mc^2\). Here's a step-by-step breakdown:

• First, identify the total mass involved, which is the sum of the masses of matter and antimatter. In this example, it's \(400 \, \text{kg} + 400 \, \text{kg} = 800 \, \text{kg}\).
• Next, substitute this mass and the speed of light into the equation: \[E = 800 \, \text{kg} \times (3.0 \times 10^8 \, \text{m/s})^2\]

The result is a massive \(7.2 \times 10^{19} \, \text{J}\), showing significant energy release. To put this in perspective, it is 72% of the total energy used in an entire year by all of the U.S. Understanding this calculation highlights the potential power of matter-antimatter reactions and the stunning efficiency of mass-energy conversion.