A proton is moving at a speed of 0.78c. Calculate the relativistic total energy.
A proton is moving at a speed of 0.78c. Calculate the relativistic total energy.
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Question
![**Problem Statement:**
A proton is moving at a speed of 0.78c. Calculate the relativistic total energy.
**Explanation:**
This problem involves calculating the relativistic total energy of a proton when it is moving at a significant fraction of the speed of light (c). The formula for relativistic energy is derived from Einstein’s theory of relativity and extends classical mechanics into high-speed realms.
**Relativistic Energy Formula:**
The relativistic total energy (E) of an object is given by the equation:
\[ E = \gamma mc^2 \]
where:
- \( \gamma \) (gamma) is the Lorentz factor, defined as \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\)
- \( m \) is the rest mass of the object
- \( c \) is the speed of light
- \( v \) is the velocity of the object
In this scenario, the velocity of the proton is given as 0.78c, which denotes 78% of the speed of light.
To solve this problem, one would follow these steps:
1. Calculate the Lorentz factor \(\gamma\).
2. Use the known rest mass of the proton and the calculated \(\gamma\) to find the total energy using the relativistic energy formula.
Understanding and calculating relativistic energy is essential for high-energy physics, especially in contexts like particle accelerators where particles reach speeds close to the speed of light.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F440b7269-170b-4024-8d0e-4b8a2f594bf7%2Ff47fd583-b6c2-492c-b499-7a0145bad9b3%2F1z3wpce_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A proton is moving at a speed of 0.78c. Calculate the relativistic total energy.
**Explanation:**
This problem involves calculating the relativistic total energy of a proton when it is moving at a significant fraction of the speed of light (c). The formula for relativistic energy is derived from Einstein’s theory of relativity and extends classical mechanics into high-speed realms.
**Relativistic Energy Formula:**
The relativistic total energy (E) of an object is given by the equation:
\[ E = \gamma mc^2 \]
where:
- \( \gamma \) (gamma) is the Lorentz factor, defined as \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\)
- \( m \) is the rest mass of the object
- \( c \) is the speed of light
- \( v \) is the velocity of the object
In this scenario, the velocity of the proton is given as 0.78c, which denotes 78% of the speed of light.
To solve this problem, one would follow these steps:
1. Calculate the Lorentz factor \(\gamma\).
2. Use the known rest mass of the proton and the calculated \(\gamma\) to find the total energy using the relativistic energy formula.
Understanding and calculating relativistic energy is essential for high-energy physics, especially in contexts like particle accelerators where particles reach speeds close to the speed of light.
Expert Solution
Concept and Principle:
- The total energy of a particle is the sum of all energies of a particle. The relativistic total energy of a particle depends on the rest mass energy and the velocity of the particle.
- The relativistic total energy is given by,
Here mc2 is the rest mass energy of the particle, v is the speed of the particle, and c is the speed of light.
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