What is the speed of a proton that has total energy 1000 1000 GeV?

Welcome back, everyone. We are making observations about protons that are traveling at a very high speed. Now, we are told that the total energy of a proton is 980 giga electron volts and we are tasked with finding what is the speed of our proton here. Well, what we can say is that our total energy of our proton is equal to gamma times MC squared. And we actually have for gamma, that gamma is equal to one divided by the square root of one minus our desired speed divided by C squared. So what we can do, let me look at this equation on the left here, I'm gonna divide both sides by MC squared. And what we get is we get that gamma is equal to E divided by MC squared. So let's go ahead and solve for that. Well, we have gamma is equal to our total energy of 980 giga electron volts. We actually want this in mega electron volts. So let's go ahead and convert here real quick. We know in one giga electron volt that there is 10 to the third mega electron volts which gives us 980 times 10 to the third mega electron volts. Wonderful. So our game is gonna be equal to 980 times 10 to the third mega electron volts divided by MC squared. Well, MC squared is just the less energy of a proton. And we know this to be 938 mega electron volts. So what this gives us on the denominator, denominator is 938 mega electron volts. When we divide this out, what we get is 1044.84 hour gamma. So now let me change colors here real quick. We're gonna take a look at this second equation for gamma here. And what we actually want to solve for is our desired speed. So I'm going to square both sides of our equation. What we get is gamma squared is equal to one divided by one minus V squared over C squared. And what I can do is I can switch the left side of the equation with the denominator of the right side of the equation. And what we get is that one minus V squared over C squared is equal to one over gamma squared. Now, what I'm gonna do is I'm gonna subtract one from both sides and then multiply both sides by negative one. What we get is we get that V squared over C squared is equal to one minus one divided by gamma squared. Now let's go ahead and solve for the right hand side of our equation here. What we get is one minus one divided by 1044.8 squared, giving us that V squared over C squared is equal to 0.999999083. Now, what I'm gonna do is I'm gonna multiply both sides by C squared here. And you'll see on the left hand side, our C square is canceled out and all we are left with is that V squared is equal to that long number times C squared. Finally, to isolate our speed variable. I'm gonna take the square root of both sides giving us that our speed of our proton is equal to 0. 542 times the speed of light, which corresponds directly with our answer choice. B thank you all so much for watching. I hope this video helped. We will see you all in the next one.

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Problem 13a

Textbook Question

What is the speed of a proton that has total energy $1000$ GeV?

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Understand the problem: The total energy of the proton is given as 1000 GeV. The total energy (E) of a particle is related to its rest energy and kinetic energy. The relationship between total energy, rest energy, and momentum is given by the relativistic energy-momentum relation.

Write the relativistic energy-momentum relation: $ E^2 = (pc)^2 + (m_0c^2)^2 $, where $ E $ is the total energy, $ p $ is the momentum, $ c $ is the speed of light, and $ m_0 $ is the rest mass of the proton.

Rearrange the equation to solve for the momentum $ p $: $ p = \sqrt{\frac{E^2 - (m_0c^2)^2}{c^2}} $. Substitute the given total energy $ E = 1000 \text{ GeV} $ and the rest energy of the proton $ m_0c^2 = 0.938 \text{ GeV} $.

Relate the momentum $ p $ to the speed $ v $ of the proton using the relativistic momentum formula: $ p = \frac{m_0v}{\sqrt{1 - \frac{v^2}{c^2}}} $. Solve for $ v $ by substituting the value of $ p $ obtained in the previous step.

Simplify the expression for $ v $ and solve numerically if needed. The speed $ v $ will be very close to the speed of light $ c $, as the total energy is much larger than the rest energy.

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Key Concepts

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

Total Energy in Relativity

In the context of special relativity, the total energy of a particle is the sum of its rest mass energy and its kinetic energy. For a proton, this is given by the equation E = mc² + K.E., where E is the total energy, m is the rest mass, and c is the speed of light. Understanding this relationship is crucial for calculating the speed of a proton when its total energy is known.

Relativistic Momentum

Relativistic momentum is defined as p = γmv, where γ (gamma) is the Lorentz factor, m is the rest mass, and v is the velocity of the particle. As the speed of a particle approaches the speed of light, its momentum increases significantly, which is essential for understanding how energy and speed relate in high-energy physics scenarios, such as with a proton at 1000 GeV.

Lorentz Factor

The Lorentz factor, denoted as γ, is a crucial component in relativity that accounts for time dilation and length contraction at high speeds. It is defined as γ = 1 / √(1 - v²/c²). This factor becomes significant when calculating the speed of particles moving close to the speed of light, as it affects both energy and momentum, making it essential for solving the given problem.

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