A proton (rest mass 1.67 ×10^ ×27 kg ) has total energy that is 4.00 times its rest energy. What

A proton (rest mass 1.67 ×10^ ×27 kg ) has total energy that...

Key Concepts

Relativistic Energy

In special relativity, a particle's total energy is expressed as E = ?mc², where m is the rest mass, c is the speed of light, and ? (the Lorentz factor) accounts for the effects of motion at relativistic speeds. The rest energy (mc²) is the energy inherent to the particle when it is at rest, and any additional energy contributes to its kinetic energy.

Relativistic Kinetic Energy

The kinetic energy of a relativistic particle is the difference between its total energy and its rest energy, given by KE = E_total ? mc². This formulation highlights that kinetic energy becomes significant as the velocity approaches the speed of light, where classical kinetic energy formulas no longer apply.

Energy–Momentum Relation

The relationship E² = (pc)² + (mc²)² connects a particle’s total energy with its momentum and mass. This relation is fundamental in relativistic mechanics as it allows the determination of momentum when the total energy and rest mass are known, proving essential for analyzing particles moving at high speeds.

Lorentz Factor

The Lorentz factor, ? = 1/?(1 ? v²/c²), quantifies how much time, length, and relativistic mass (or energy) change for a particle moving at a significant fraction of the speed of light. It is crucial for connecting the observed quantities in different inertial frames and is central to all calculations involving relativistic speeds.

Relativistic Momentum

Relativistic momentum is given by p = ?mv, indicating that as a particle’s speed increases, its momentum increases by the Lorentz factor. This concept is critical since, near the speed of light, momentum does not simply scale linearly with velocity as it does in non-relativistic mechanics.

Transcript

00:01 Given that the total energy is four times the rest energy, and we also know the mass of the proton, we can go ahead and get the kinetic energy.

00:11 We know that the total energy is going to be the kinetic energy plus the rest energy.

00:16 So we can get the kinetic energy by subtracting the rest energy from the total energy.

00:22 The kinetic energy in this case would then be three times the rest energy.

00:27 So then this gives us three times the mass of the proton times the square of the speed of light.

00:39 Because the rest energy, again, is mc squared, and we know that the total energy was four times mc squared.

00:46 So we're simply subtracting the rest energy off from the total energy, and the rest energy is mc squared itself.

00:55 So if we say 4 mc squared minus mc squared, then we're left with 3 mc squared for the kinetic energy.

01:04 So then when we calculate this, we have the following numbers.

01:08 We have 3 times 1 .67 times 10 to the minus 27 times the square of the speed of light, which is 3 .00 times 10 to the 8 squared.

01:18 So then this comes out to 4 .50 times 10 to the minus 10 joules.

01:27 So next, we can use the equation e squared equals mc squared squared plus pc squared to get the momentum.

01:42 So we know the total energy once again is four times the rest energy.

01:46 So we substitute that in for the total energy, which is e.

01:50 And then on the right hand side we have the rest energy squared so that's mc squared squared squared and then we have plus p c squared and we want to know p which is the momentum so squaring the left side gives us 16 .00 m squared c to the fourth and then on the right side we have m squared c to the fourth and plus p squared c squared so when we continue here we have we can actually subtract the m squared c of the fourth that's on the right side from both sides of the equation.

02:34 So then on the left hand side we'll have 16 .00 m squared c of the fourth minus m square c of the fourth.

02:42 And this will now equal p squared c squared.

02:47 This gives us 15 .00 m squared c of the fourth equaling p squared c squared.

02:54 So now we can divide both sides by c squared, and this gives us p squared equaling 15 .00 m squared c squared.

03:04 And then we just take the square root of both sides, and we have the square root of 15 .00 times mc...

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