Question

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 ?

Short answer

The energy released is \( 7.2 \times 10^{19} \) J, which is 0.72 times the U.S. yearly energy use.

Step-by-step solution

Understand the Concept of Matter-Antimatter Annihilation

In matter-antimatter annihilation, when antimatter meets matter, they annihilate each other, converting their entire mass into energy according to Einstein's mass-energy equivalence principle, given by the equation \[ E = mc^2 \]. This means the released energy \( E \) can be found by multiplying the combined mass \( m \) by the speed of light squared \( c^2 \).

Calculate the Combined Mass

The problem states that the starship has a 400-kg antimatter fuel supply that completely annihilates with an equal amount (400 kg) of matter. Therefore, the total mass \( m \) getting converted into energy is \[ m = 400\, \text{kg} + 400\, \text{kg} = 800\, \text{kg}. \]

Apply Einstein's Equation

Insert the total mass and the speed of light into the equation \( E = mc^2 \). The speed of light \( c \) is approximately \( 3.0 \times 10^8 \) m/s, so the energy is \[ E = 800\, \text{kg} \times (3.0 \times 10^8\, \text{m/s})^2. \]

Perform the Calculation

Calculate the energy released by computing: \[ E = 800 \times 9.0 \times 10^{16} = 7.2 \times 10^{19}\, J. \]

Compare with U.S. Yearly Energy Use

The energy released by the annihilation process is \( 7.2 \times 10^{19} \) J. Comparing this with the U.S. yearly energy use of \( 1.0 \times 10^{20} \) J shows that the energy from the matter-antimatter annihilation is slightly less — about \( 0.72 \) times — than the total energy used by the U.S. in a year.

Key concepts

Einstein's mass-energy equivalence

Einstein's mass-energy equivalence is a fascinating breakthrough that revolutionized our understanding of physics. It tells us that mass and energy are two forms of the same thing and can be converted into each other. This principle is encapsulated in the famous equation - \[ E = mc^2 \] - where:

• \( E \) represents energy in joules (J)
• \( m \) symbolizes mass in kilograms (kg)
• \( c \) denotes the speed of light in meters per second (m/s), specifically \( 3.0 \times 10^8 \ \text{m/s} \)

This means if you convert any mass fully into energy, its energy output is the mass times the speed of light squared. In scenarios like matter-antimatter annihilation, both the matter and antimatter are completely transformed, unleashing massive amounts of energy.

Energy calculation

To find out how much energy the starship \(Enterprise\) can release, we need to apply Einstein's equation correctly. We begin by understanding we have both matter and antimatter each weighing 400 kg. When they combine, they form an 800 kg mass that undergoes annihilation. Therefore, our combined mass is - \[ m = 400 \, \text{kg} + 400 \, \text{kg} = 800 \, \text{kg} \].- Next, we plug this total mass into Einstein's formula:

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

The key point is calculating the speed of light squared, giving \( 9.0 \times 10^{16} \). Our final energy calculation turns into:

• \( E = 800 \times 9.0 \times 10^{16} = 7.2 \times 10^{19} \, \text{J} \).

This calculation tells us that a huge amount of energy, \( 7.2 \times 10^{19} \ \text{J} \), could be unleashed through this process.

Comparison with energy consumption

To put this tremendous release of energy into perspective, compare it to the yearly energy consumption of the United States. The U.S. uses approximately \( 1.0 \times 10^{20} \) joules each year. Observing this, the energy from the matter-antimatter reaction is slightly less than what the entire country consumes annually.- Since the energy provided is \( 7.2 \times 10^{19} \ \text{J} \), dividing this by \( 1.0 \times 10^{20} \ \text{J} \) tells us how significant this energy output is.

• The reaction provides about 72% of what the U.S. uses yearly.

Such a comparison highlights how even a relatively small amount of matter and antimatter can produce an enormous quantity of energy, showcasing the unparalleled potential for power if matter-antimatter reactions could be controlled for practical use.