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Chapter 41: Problem 23

A proton has a mass of \(1.6726 \times 10^{-27} \mathrm{~kg}\) and is traveling at \(\mathrm{a}\) speed where \(y=2.294157\); that's \(90 \%\) the speed of light. Determine its total energy. Four significant figures will do. Discuss your answer as it compares with classical energy.

Short Answer

Expert verified
The total relativistic energy of the proton is approximately \(3.453 \times 10^{-10} \, \text{J}\). This is significantly higher than the classical energy prediction due to relativistic effects.

Step by step solution

01

Identify the Required Formula

To find the total relativistic energy of an object, we use the formula: \[ E = rac{mc^2}{ ext{sqrt}(1 - v^2/c^2)} \] where \(E\) is the total energy, \(m\) is the mass of the proton, \(v\) is its velocity, and \(c\) is the speed of light.
02

Substitute Known Quantities

Given that the proton's velocity \(v\) is 90% of the speed of light \(c\), we have \(v = 0.9c\). The mass of the proton \(m\) is \(1.6726 \times 10^{-27} \, \text{kg}\).
03

Calculate the Factor \(1/(\text{sqrt}(1-v^2/c^2))\)

Calculate the Lorentz factor \(y = \frac{1}{\sqrt{1 - (0.9)^2}} = 2.294157\). This was provided in the problem statement.
04

Plug Values into the Energy Formula

Calculate the total energy using \[ E = ym c^2 \]. Substitute the given values: \[ y = 2.294157, \ m = 1.6726 \times 10^{-27} \, \text{kg}, \ c = 3 \times 10^8 \, \text{m/s} \]. Therefore, \[ E = 2.294157 \times 1.6726 \times 10^{-27} \, \text{kg} \times (3 \times 10^8 \, \text{m/s})^2 \].
05

Perform the Calculations

First, calculate \((3 \times 10^8)^2 = 9 \times 10^{16} \, \text{m}^2/\text{s}^2\). Then multiply: \[ E = 2.294157 \times 1.6726 \times 9 \times 10^{-11} \, \text{J} \] which results in \[ E \approx 3.453 \times 10^{-10} \, \text{J} \] (rounded to four significant figures).
06

Compare with Classical Energy

The classical kinetic energy formula is \(KE = \frac{1}{2}mv^2\). Substituting \( m = 1.6726 \times 10^{-27} \, \text{kg} \) and \( v = 0.9 \times 3 \times 10^8 \, \text{m/s} = 2.7 \times 10^8 \, \text{m/s}\), we get \( KE \approx 6.08 \times 10^{-11} \, \text{J} \). The relativistic energy \(3.453 \times 10^{-10} \, \text{J}\) is noticeably greater, illustrating the significant effect of relativistic speeds.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Lorentz Factor
The Lorentz factor is a key concept when discussing relativity, especially when calculating relativistic energy. It's a number that quantifies how much an object's relativistic mass increases as it moves relative to an observer. The formula for the Lorentz factor is \[\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\]where \(v\) is the velocity of the object and \(c\) is the speed of light.
  • In essence, the Lorentz factor tells us how time, length, and relativistic mass are affected by an object's speed.
  • As speed \(v\) nears the speed of light \(c\), the Lorentz factor \(\gamma\) increases significantly.
For a proton traveling at 90% of the speed of light, the Lorentz factor is calculated as \(\gamma = 2.294157\). This means all relativistic effects, including energy, are amplified by this factor.
Classical vs Relativistic Energy Comparison
In physics, it's crucial to differentiate between classical and relativistic energy calculations. Classical physics fails when dealing with objects moving at significant fractions of the speed of light. This discrepancy is evident in our earlier calculations:
  • Classical energy assumes that all massive objects have kinetic energy \(KE = \frac{1}{2}mv^2\).
  • In contrast, relativistic energy uses \[E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}\]which accounts for increases in energy due to relativistic speeds.
In our exercise, the proton's classical energy was calculated to be approximately \(6.08 \times 10^{-11} \, \text{J}\), whereas the relativistic energy was approximately \(3.453 \times 10^{-10} \, \text{J}\). This demonstrates that relativistic effects significantly increase the total energy of fast-moving particles.
Proton Mass
The mass of a proton is a fundamental value in physics, crucial for nuclear and particle physics. Protons have a rest mass of \(1.6726 \times 10^{-27} \, \text{kg}\), which doesn't change regardless of where they are in the universe.
  • Mass plays a critical role in calculating energy, both in classical and relativistic contexts.
  • In relativistic physics, the notion of "relativistic mass" emerges, which combines rest mass and motion effects through the Lorentz factor.
This distinction is essential when calculating energies at high velocities, as the proton's mass is multiplied by the Lorentz factor, augmenting the effective energy of fast-moving protons.
Speed of Light
The speed of light \(c\), approximately \(3 \times 10^8 \, \text{m/s}\), is a fundamental constant in physics. It serves as the ultimate speed limit in the universe, beyond which no material object can travel.
  • The speed of light is central to Einstein's theory of relativity, forming the basis for all relativistic calculations.
  • In our energy calculations, it defines the scale at which relativistic effects become significant.
The constancy of this speed impacts various fundamental principles, such as causality and simultaneity, reinforcing the interconnected nature of space and time in relativity.
Velocity of 90% the Speed of Light
In relativistic scenarios, traveling at velocities close to the speed of light changes the way we perceive time, space, and energy. In the exercise, a proton travels at 90% of the speed of light, which is denoted as \(0.9c\).
  • At such high velocities, relativistic effects are pronounced, necessitating the use of relativistic formulas as opposed to classical ones.
  • A proton traveling at this speed exhibits significantly altered energy as seen by its increased relativistic mass.
Understanding velocities near the speed of light is key in fields such as particle physics and astrophysics, where particles routinely reach such speeds naturally.

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Most popular questions from this chapter

A person in a spaceship holds a meterstick as the ship shoots past the Earth with a speed \(u\) parallel to the Earth's surface. What does the person in the ship notice as the stick is rotated from parallel to perpendicular to the ship's motion? The stick behaves normally; it does not change its length, because it has no translational motion relative to the observer in the spaceship. However, an observer on Earth would measure the stick to be \((1 \mathrm{~m}) \sqrt{1-(v / \mathrm{c})^{2}}\) long when it is parallel to the ship's motion, and \(1 \mathrm{~m}\) long when it is perpendicular to the ship's motion.

As a rocket ship sweeps past the Earth with speed \(u\), it sends out a pulse of light ahead of it. How fast does the light pulse move according to people on the Earth? Method 1 With speed \(c\) (by the second postulate of Special Relativity). Method 2 Here \(v_{O^{\prime} O}=u\) and \(v_{P O^{\prime}}=c\). According to the velocity addition formula, the observed speed will be (since \(u=\mathrm{c}\) in this case ) $$ v_{P O}=\frac{v_{P O}+v_{o^{\prime} o}}{1+\frac{v_{P O} \cdot v_{o^{\prime}} O}{c^{2}}}=\frac{v+c}{1+(v / c)}=\frac{(v+c) c}{c+v}=c $$

How fast must an object be moving if its corresponding value of \(y\) is to be \(1.0\) percent larger than \(y\) is when the object is at rest? Give your answer to two significant figures. Use the definition \(\gamma=1 / \sqrt{1-(v / \mathrm{c})^{2}}\) to find that at \(v=0, y=1.0\). Hence, the new value of \(y=1.01(1.0)\), and so $$ 1-\left(\frac{v}{\mathrm{c}}\right)^{2}=\left(\frac{1}{1.01}\right)^{2}=0.980 $$ Solving yields \(v=0.14 \mathrm{c}=4.2 \times 10^{7} \mathrm{~m} / \mathrm{s}\)

Two twins are \(25.0\) years old when one of them sets out on a journey through space at nearly constant speed. The twin in the spaceship measures time with an accurate watch. When he returns to Earth, he claims to be \(31.0\) years old, while the twin left on Earth knows that she is \(43.0\) years old. What was the speed of the spaceship? The spaceship clock as seen by the space-twin reads the trip time to be \(\Delta t_{\mathrm{S}}\), which is \(6.0\) years long. The Earth-bound twin sees her brother age \(6.0\) years, but her clocks tell her that a time \(\Delta t_{\mathrm{M}}=18.0\) years has actually passed. Hence, \(\Delta t_{\mathrm{M}}=y \Delta t_{\mathrm{S}}\) becomes \(\Delta t_{\mathrm{S}}=\Delta t_{\mathrm{M}} \sqrt{1-(v / \mathrm{c})^{2}}\) and so $$ 6=18 \sqrt{1-(v / \mathrm{c})^{2}} $$ from which \((\mathrm{u} / \mathrm{c})^{2}=1-0.111\) or \(v=0.943 \mathrm{c}=2.83 \times 10^{8} \mathrm{~m} / \mathrm{s}\)

The insignia painted on the side of a spaceship is a circle with a line across it at \(45^{\circ}\) to the vertical. As the ship shoots past another ship in space, with a relative speed of \(0.95 \mathrm{c}\), the second ship observes the insignia. What angle does the observed line make to the vertical?
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