Problem 53 - College Physics

Question

The starship Enterprise, of television and movie fame, is powered by the controlled combination of matter and antimatter. If the entire 400 \(\mathrm{kg}\) antimatter fuel supply of the Enterprise combines with matter, how much energy is released?

Step-by-Step Solution

Answer

The energy released is \(7.2 \times 10^{19}\) Joules.

Step 1: Understanding the Problem

The problem involves calculating the energy released when 400 kg of antimatter combines with an equal amount of matter. This is given by Einstein's mass-energy equivalence principle.

Step 2: Applying Einstein's Equation

Use the formula for mass-energy equivalence, which is given by: \[ E = mc^2 \] where \( m \) is the total mass and \( c \) is the speed of light in vacuum (approximately \(3 \times 10^8 \ m/s\)). Since the antimatter combines with an equal amount of matter, the total mass \( m = 400 \, kg + 400 \, kg = 800 \, kg \).

Step 3: Calculating Energy

Plug the total mass and the speed of light into the equation: \[ E = 800 \, kg \times (3 \times 10^8 \, m/s)^2 \] Calculating this gives: \[ E = 800 \, kg \times 9 \times 10^{16} \, m^2/s^2 \]\[ E = 7.2 \times 10^{19} \, Joules \]

Key Concepts

Einstein's Equation

Einstein's equation, also known as the mass-energy equivalence principle, is a cornerstone of modern physics. This equation is famously written as \( E = mc^2 \). Here, \( E \) represents energy, \( m \) is mass, and \( c \) is the speed of light in vacuum. The equation suggests that mass can be converted into energy and vice versa. This principle helps us understand phenomena in physics where mass disappears but energy is conserved.

• \( E \) is energy in joules.
• \( m \) is mass in kilograms.
• \( c \) is the speed of light, approximately \(3 \times 10^8 \) meters per second.

Einstein's insight into the relationship between mass and energy has profound implications, explaining how astronomical events and particle interactions work fundamentally.

Matter-Antimatter Annihilation

Matter-antimatter annihilation is a process where matter and antimatter particles collide and destroy each other, releasing energy. Each particle of antimatter corresponds to a matter particle, with opposite charge and properties. Annihilation is not just theoretical; it's used in practical applications like PET scans in medical imaging.

• When they meet, they destroy each other and release energy.
• This energy release is calculated using Einstein's equation.
• In our example, 400 kg of antimatter and 400 kg of matter are involved.

This annihilation is complete, converting the entire mass into energy, which is why it is so powerful. It highlights the massive energy potential stored in even small amounts of matter.

Energy Calculation

Calculating energy released in a process like matter-antimatter annihilation involves the application of Einstein’s equation. Given a total mass and the speed of light, we compute the energy output. This kind of calculation demonstrates the immense energy stored in matter.

• We have a mass \( m = 800 \) kg, since both matter and antimatter contribute equal mass.
• The speed of light \( c \) is \(3 \times 10^8 \) m/s.
• Energy is calculated as \( E = 800 \, kg \times (3 \times 10^8 \, m/s)^2 \).
• Resulting in \( E = 7.2 \times 10^{19} \) Joules.

This huge energy result underscores why matter-antimatter annihilation is a subject of interest for energy production and theoretical space travel possibilities.

Speed of Light

The speed of light, denoted as \( c \), is a fundamental constant in physics. It is approximately \(3 \times 10^8 \) meters per second and is crucial in calculations involving relativity and mass-energy equivalence.

• Light speed is a universal constant.
• It signifies the maximum speed at which information or energy can travel.
• In Einstein’s equation, it emphasizes how even tiny amounts of mass can convert to tremendous energy.

This constant forms the basis for understanding critical theories in physics like relativity, influencing our comprehension of space, time, and energy. Its unchanging nature across the universe also serves as a link across various phenomena, from cosmic events to subatomic reactions.