PICUP Relativistic Dynamics in 1D with a constant force JavaScript Simulation Applet HTML 5

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Relativistic Dynamics in 1D with a constant force

Developed by Larry Engelhardt

In these exercises, you will determine the motion of a proton in a uniform electric field. We will begin by simulating a proton in an electric field using the NON-relativistic version of Newton’s 2nd Law. Then we will modify this simulation to take special relativity into account. In the process, we will observe the transition from non-relativistic to relativistic dynamics. In order to generate results, we will see that we need to be careful when working with non-SI units. In particular, we will need to pay close attention to factors of eV and .

Subject Area	Modern Physics
Level	Beyond the First Year
Available Implementations	IPython/Jupyter Notebook and Python
Learning Objectives	Students will be able to: Execute a working simulation, and explain non-relativistic, constant force motion (Exercise 1) Manipulate and explain the units that appear in the context of relativistic motion (Exercise 2) Interpret plots of energy vs. time for non-relativistic motion (Exercises 2) Observe and explain when the non-relativistic form of Newton’s 2nd Law breaks down (Exercise 3) Derive the relativistic form of Newton’s 2nd Law (Exercise 4) Modify a non-relativistic simulation to incorporate relativity (Exercise 5) Produce and interpret plots for relativistic motion (Exercises 6, 7, 8) Validate numerical results by comparing with an analytical solution (Exercise 7) Apply simulated results to a particle accelerator (Exercises 9 and 10) Rewrite code to store data in arrays using array indices rather than by appending data to arrays (Exercise 11)
Time to Complete	120 min

Exercise 1

Exercise 1:

The file ending with “Version1” contains code to simulate a proton in an electric field using the non-relativistic acceleration derived above. (This code uses the “Euler algorithm” described in the “Theory” section.) Execute this code, and look at the plots of position versus time and speed versus time. Explain why these plots have the shapes that they have.

Exercise 2

Exercise 2:

The file ending with “Version2” contains the same code as the file ending with “Version1” except that a few lines have been added in order to also calculate and plot the non-relativistic energy. Locate the line where the non-relativistic energy is calculated. (The same line appears both inside the loop and before the loop.) Explain why this line is correct. You will need to be very careful with the units and the factors of . Write down the equation for non-relativistic energy (including both rest energy and non-relativistic kinetic energy), and carefully argue why this line of code is correct.

Execute this code, and look at the plot of energy versus time. Why does this plot have the shape that it has? How large is the kinetic energy compared to the rest energy after the proton has been accelerating for 1 second?

Exercise 3

Exercise 3:

Again use the file ending with “Version2” but now increase the value of the maximum time from 1 second to 5 seconds. How large is the kinetic energy compared to the rest energy after the proton has been accelerating for 5 seconds? Look at the plot of speed versus time. Something should bother you! Explain what is wrong

Exercise 4

Exercise 4:

Derive the relativistic form of Newton’s 2nd Law,

a = F γ 3 m, (24)

from the equation . See the “Theory” section for additional background.

Exercise 5

Exercise 5:

Save a copy of the file ending with “Version2” using a file name ending with “Version3”. In this new file, you are going to modify the code in order to take special relativity into account. This will involve the following steps:

Compute the value of the Lorentz factor, .
Use the Lorentz factor to compute the acceleration.
Modify the equation for the energy (both before the loop and within the loop) in order to compute the relativistic energy. Hint: Once you have computed the Lorentz factor, the relativistic energy is actually very simple to compute, but be very careful of the units and factors of .

Execute your modified program, and fix any bugs!

Exercise 6

Exercise 6:

Execute your modified program (Version3) using a maximum time of 5 seconds, and look at the plot of speed versus time. Does the plot look better than the plot that you looked at in Exercise 3? It should! If it doesn’t, there is a bug in your code that needs to be fixed before you continue. Explain why the new plot of speed versus time has the shape that it has.

Exercise 7

Exercise 7:

Now that you have created a new program, you should attempt to validate your program (to test how accurate the numbers actually are). One simple way to do this is using the energy. From the work-energy theorem, the kinetic energy of the proton (starting from rest) is

K E = W = \int F d x, (25)

and since the force is constant, the kinetic energy is simply the product of the (constant) force times the distance traveled. This “analytical” result (which does not involve any approximations) can be compared with the “numerical” kinetic energy from your program,

K E = γ m c 2 - m c 2 . (26)

Add code to your program (after the program’s loop has completed) to compute the kinetic energy both analytically and numerically. Print both results, and make sure that they are similar. (If they aren’t you need to fix something!) Compute the percent error in the numerical kinetic energy. How large is the error? Increase the value of your program’s time step by a factor of 10. How does this affect the error? Why does this happen? (This code uses the “Euler algorithm” described in the “Theory” section.) Before you continue to the next exercise, make sure that the error is a small fraction of a percent.

Exercise 8

Exercise 8:

Increase the maximum time for your simulation until you are able to get the total energy of the proton up to 100 GeV. Discuss the shape of each of the three plots (position, speed, and energy). Why do they have the shapes that they have? (Incorporate the term “ultra-relativistic” into your answer.)

Exercise 9

Exercise 9:

Using an electric field of Volt/meter, how much time does it take to accelerate a proton to an energy of 100 GeV? How far does it travel in that time? (Use your plots from Exercise 8.)

At the Large Hadron Collider (LHC), protons are accelerated to an energy of 8 TeV. Instead of using a longer simulation, extrapolate your results in order to determine how long it would take to accelerate a proton up to TeV using Volt/meter. How far does the proton travel during this time? How many trips around the LHC would the proton make during this time? How does this distance compare to the circumference of the Earth’s orbit around the sun?

Complete_RelativisticDynamics.py

# relativisticDynamicsVersion3.py

from pylab import *

from time import time

start = time()

c = 2.998e8 # Speed of light in m/s

m = 0.938e9 # Mass in eV/c^2

Efield = 5e6 # Electric field in Volts per meter

x = 0 # Position in meters

v = 0 # Velocity in meters/second

t = 0 # Time in seconds

dt = 1e-8 # Time STEP in seconds

lorentz = 1 / sqrt(1 - v**2 / c**2) ## ADDED ##

E = lorentz * m ## MODIFIED ##

# Create arrays using initial values

tArray = array(t)

xArray = array(x)

vArray = array(v)

EArray = array(E)

lorentzArray = array(lorentz) ## ADDED ##

while t < 1e-5:

# The dynamics:

lorentz = 1 / sqrt(1 - v**2 / c**2) ## ADDED ##

a = Efield*c**2/(lorentz**3*m) ## MODIFIED ##

t = t + dt

x = x + v*dt

v = v + a*dt

E = lorentz * m ## MODIFIED ##

# Append the new values onto arrays

tArray = append(tArray, t)

xArray = append(xArray, x)

vArray = append(vArray, v)

EArray = append(EArray, E)

lorentzArray = append(lorentzArray, lorentz) ## ADDED ##

end = time()

print('Elapsed time:', end-start)

# The following lines are added for Exercise 7

KE_numerical = lorentz * m - m

KE_analytical = Efield*x

print('Numerical KE: ', KE_numerical)

print('Analytical KE:', KE_analytical)

print('Percent Difference:', 100*abs(KE_numerical-KE_analytical)/KE_analytical)

# Create plots

figure(1)

plot(tArray, vArray, linewidth=2)

xlabel('Time (sec)')

ylabel('Speed (m/s)')

grid(True)

ylim([0, 3.1e8])

savefig('ultrarelativistic_v_vs_t.png')

show()

figure(2)

plot(tArray, xArray, linewidth=2)

xlabel('Time (sec)')

ylabel('Position (m)')

grid(True)

savefig('ultrarelativistic_x_vs_t.png')

show()

figure(3)

plot(tArray, EArray, linewidth=2)

xlabel('Time (sec)')

ylabel('Energy (eV)')

grid(True)

savefig('ultrarelativistic_E_vs_t.png')

show()

Translations

Code	Language		Translator	Run

Credits

Fremont Teng; Loo Kang Wee; based on code by Larry Engelhardt

Overview:

This document summarizes the main themes and important ideas presented in the provided excerpts from a set of exercises focused on simulating the motion of a proton in a uniform electric field, contrasting non-relativistic and relativistic dynamics. The exercises guide the user through modifying JavaScript code to incorporate special relativity and explore the consequences at high speeds and energies. Key aspects covered include Newton's 2nd Law, energy conservation, the Lorentz factor, relativistic mass increase, and the limitations of non-relativistic physics. The exercises culminate in extrapolating results to understand particle acceleration in high-energy physics facilities like the Large Hadron Collider (LHC).

Main Themes and Important Ideas:

Transition from Non-relativistic to Relativistic Dynamics: The core objective of these exercises is to illustrate the shift in the behavior of particles as their speed approaches the speed of light. The simulation starts with the non-relativistic version of Newton's 2nd Law and progressively incorporates relativistic corrections.

The introduction states: "In these exercises, you will determine the motion of a proton in a uniform electric field. We will begin by simulating a proton in an electric field using the NON-relativistic version of Newton’s 2nd Law. Then we will modify this simulation to take special relativity into account. In the process, we will observe the transition from non-relativistic to relativistic dynamics."

Importance of Units in Relativistic Physics: The exercises emphasize the necessity of careful unit handling, especially when dealing with non-SI units like electron volts (eV) and the speed of light (c).

The introduction notes: "In order to generate results, we will see that we need to be careful when working with non-SI units. In particular, we will need to pay close attention to factors of eV and $c$ ."

Non-relativistic Simulation and its Limitations: Exercises 1-3 focus on using the non-relativistic Newton's 2nd Law ( $F = ma$ ) to simulate the proton's motion. These exercises highlight that in this model, speed increases linearly with time under a constant force, and kinetic energy increases quadratically. However, Exercise 3 points out a critical flaw at longer timescales: the non-relativistic model allows the speed to exceed the speed of light, which is physically impossible.
Relativistic Modification of Newton's 2nd Law: Exercise 4 requires the derivation of the relativistic form of Newton's 2nd Law, which introduces the Lorentz factor ( $\gamma$ ).

The derived equation is: " $a = \frac{F}{\gamma^3 m}$ ". This shows that as the speed increases and $\gamma$ becomes larger, the acceleration for a constant force decreases significantly.

Incorporating Special Relativity into the Simulation: Exercise 5 guides the user to modify the simulation code to include relativistic effects by:

Calculating the Lorentz factor: "$lorentz = 1 / sqrt(1 - v2 / c2)$".
Using the relativistic acceleration formula.
Calculating the relativistic energy: " $E = \gamma m$ ". The hint emphasizes careful handling of units and factors of $c^2$ .

Relativistic Energy and Kinetic Energy: The exercises introduce the concept of relativistic energy, including rest energy ( $mc^2$ ) and relativistic kinetic energy ( $KE = \gamma mc^2 - mc^2$ ). Exercise 7 involves comparing the numerical kinetic energy from the simulation with the analytical work done by the constant force over the distance traveled ( $KE = W = \int F dx = Fx$ ).
The Euler Algorithm and Numerical Accuracy: The simulation utilizes the Euler algorithm, a first-order numerical method for solving differential equations. Exercise 7 prompts the user to investigate the impact of the time step ( $dt$ ) on the accuracy of the simulation, demonstrating that smaller time steps lead to more accurate results but require longer computation times.
Ultra-relativistic Regime: Exercise 8 explores the "ultra-relativistic" regime where the proton's kinetic energy is much larger than its rest energy. In this regime, the speed approaches the speed of light, and further energy input primarily increases the relativistic mass and energy, with only a marginal increase in speed.
Application to High-Energy Physics: Exercise 9 connects the simulation to real-world applications by asking the user to extrapolate the results to the energies achieved at the LHC (8 TeV) and to consider the time and distance required for such acceleration under a given electric field. This exercise highlights the immense scale and energy involved in modern particle accelerators.

Key Equations and Definitions:

Non-relativistic Newton's 2nd Law: $F = ma$
Lorentz Factor: $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$
Relativistic Newton's 2nd Law (for acceleration parallel to force): $a = \frac{F}{\gamma^3 m}$
Relativistic Energy: $E = \gamma mc^2$
Rest Energy: $E_0 = mc^2$
Relativistic Kinetic Energy: $KE = E - E_0 = \gamma mc^2 - mc^2$
Work Done by a Constant Force: $W = Fx$

Code Snippet Analysis:

The provided Python code snippet (Complete_RelativisticDynamics.py) represents "Version3" of the simulation, incorporating relativistic dynamics. Key modifications compared to earlier versions include:

Calculation of Lorentz Factor: "lorentz = 1 / sqrt(1 - v**2 / c**2) ## ADDED ##"
Modification of Acceleration: "a = Efield*c**2/(lorentz**3*m) ## MODIFIED ##" This implements the relativistic form of Newton's 2nd Law. The inclusion of $c^2$ and the use of mass in eV/ $c^2$ suggest careful unit handling.
Modification of Energy: "E = lorentz * m ## MODIFIED ##" This calculates the total relativistic energy (excluding the $c^2$ factor, implying energy is in eV).
Storage of Lorentz Factor: lorentzArray = append(lorentzArray, lorentz) ## ADDED ##
Calculation of Numerical and Analytical Kinetic Energy:KE_numerical = lorentz * m - m
KE_analytical = Efield*x
print('Numerical KE: ', KE_numerical)
print('Analytical KE:', KE_analytical)
print('Percent Difference:', 100*abs(KE_numerical-KE_analytical)/KE_analytical)
These lines calculate the kinetic energy using the relativistic formula and the work-energy theorem, respectively, and compare them to assess the accuracy of the numerical simulation.

Conclusion:

The provided excerpts outline a comprehensive set of exercises designed to teach relativistic dynamics through simulation. By starting with non-relativistic physics and progressively incorporating relativistic corrections, the exercises aim to provide a practical understanding of how the behavior of particles changes at high energies. The emphasis on units, numerical methods, and comparison with analytical results reinforces key concepts and highlights the importance of relativistic physics in understanding phenomena at the forefront of scientific research, such as particle acceleration in colliders like the LHC. The inclusion of the Python code snippet illustrates the practical implementation of these relativistic principles in a computational environment.

Study Guide: Relativistic Dynamics in 1D

Key Concepts

Non-relativistic Dynamics: The study of motion using classical Newtonian mechanics, where mass is considered constant and velocities are much smaller than the speed of light.
Relativistic Dynamics: The study of motion incorporating the principles of special relativity, where mass increases with velocity and the speed of light is a fundamental limit.
Newton's Second Law: In its non-relativistic form, states that the force acting on an object is equal to the mass of the object times its acceleration (F = ma).
Relativistic Momentum: The momentum of an object moving at relativistic speeds, given by p = γmv, where γ is the Lorentz factor.
Relativistic Energy: The total energy of an object moving at relativistic speeds, given by E = γmc², which includes both its rest energy (mc²) and kinetic energy.
Rest Energy: The energy an object possesses due to its mass, given by E₀ = mc².
Relativistic Kinetic Energy: The difference between the total relativistic energy and the rest energy: KE = γmc² - mc².
Lorentz Factor (γ): A factor that describes how space and time are affected by relative motion, given by γ = 1 / √(1 - v²/c²), where v is the velocity of the object and c is the speed of light.
Uniform Electric Field: An electric field where the force on a charged particle is the same in magnitude and direction at all points.
eV (electronvolt): A unit of energy equal to the amount of energy gained (or lost) by the charge of a single electron moving across an electric potential difference of one volt. Often used in particle physics.
c: The speed of light in a vacuum, approximately 2.998 x 10⁸ m/s.
Euler Algorithm: A first-order numerical procedure for solving ordinary differential equations with a given initial value. It's a straightforward method but can accumulate errors, especially with larger time steps.
Ultra-relativistic: Refers to particles moving at speeds very close to the speed of light (v ≈ c), where relativistic effects are dominant.
GeV (gigaelectronvolt) and TeV (teraelectronvolt): Units of energy equal to 10⁹ eV and 10¹² eV, respectively. Commonly used to describe the energies of particles in high-energy physics.
Extrapolation: Estimating values beyond the range of the given data.

Quiz

Explain the fundamental difference in how non-relativistic and relativistic dynamics treat the mass of an object as its velocity changes. Why is this difference significant at very high speeds?
In the context of a proton moving through a uniform electric field, describe how the force on the proton is determined. How does this force relate to the proton's acceleration in the non-relativistic case?
What is the Lorentz factor, and how does it depend on the velocity of an object? Explain its significance in the equations for relativistic momentum and energy.
Write down the equation for the total non-relativistic energy of a particle, including both rest energy and kinetic energy. How does the calculation of kinetic energy differ in the relativistic case?
According to the provided text, what is "wrong" with the plot of speed versus time when a non-relativistic simulation of a proton in an electric field is run for a sufficiently long time? Explain the underlying physical reason for this issue.
State the relativistic form of Newton's Second Law as presented in the text. How does the inclusion of the Lorentz factor modify the relationship between force and acceleration at high velocities?
In Exercise 5, what are the three main steps involved in modifying the simulation code to account for special relativity? Briefly describe the purpose of each of these modifications.
Explain the terms "numerical kinetic energy" and "analytical kinetic energy" as they relate to the simulation described in the text. Why might there be a difference between these two values, and how can the accuracy of the numerical result be improved?
Describe the general shapes of the position versus time, speed versus time, and energy versus time plots for a proton accelerated to ultra-relativistic energies in a constant electric field. Explain the physical reasons behind these shapes.
How does the energy gained by a charged particle moving through an electric field relate to the work done on the particle? In the case of a constant electric field, how can the kinetic energy gained be calculated analytically?

Quiz Answer Key

Non-relativistic dynamics assumes that the mass of an object remains constant regardless of its velocity. In contrast, relativistic dynamics states that the mass of an object increases as its velocity approaches the speed of light. This difference becomes significant at very high speeds because the inertia of the object increases substantially, making further acceleration much more difficult.
The force on the proton in a uniform electric field is given by the product of its charge and the electric field strength (F = qE). In the non-relativistic case, this constant force results in a constant acceleration according to Newton's Second Law (a = F/m).
The Lorentz factor (γ) is given by γ = 1 / √(1 - v²/c²), where v is the object's velocity and c is the speed of light. It increases as the velocity approaches c. In relativistic equations, γ accounts for the effects of time dilation and length contraction, leading to increases in momentum and energy beyond what classical physics predicts.
The total non-relativistic energy is E = mc² + (1/2)mv², where mc² is the rest energy and (1/2)mv² is the non-relativistic kinetic energy. Relativistically, kinetic energy is the difference between the total energy E = γmc² and the rest energy mc², so KE = γmc² - mc².
The speed versus time plot in the non-relativistic simulation increases linearly without bound. This is wrong because it violates the principle of special relativity, which states that no object with mass can reach or exceed the speed of light in a vacuum.
The relativistic form of Newton's Second Law presented is a = F / (γ³m). The inclusion of the γ³ term in the denominator indicates that as the velocity (and thus γ) increases, the acceleration for a given force decreases significantly compared to the non-relativistic case.
The three main steps are: 1) Compute the Lorentz factor (γ) based on the current velocity. 2) Use the Lorentz factor to calculate the relativistic acceleration (a = F / (γ³m)). 3) Modify the energy equation to calculate the relativistic energy (E = γmc²). These modifications ensure that the simulation adheres to the principles of special relativity.
Numerical kinetic energy is the kinetic energy calculated at each time step within the simulation using the computed velocity and the relativistic energy formula. Analytical kinetic energy, in this context, refers to the kinetic energy calculated using a theoretical, exact formula derived from the constant force and the distance traveled. Differences can arise due to the approximations inherent in the numerical method (Euler algorithm), and the accuracy can be improved by decreasing the time step (dt).
The position versus time plot will initially show a parabolic increase but will flatten out at later times as the speed approaches c. The speed versus time plot will show an initial linear increase but will then curve and asymptotically approach the speed of light, never exceeding it. The energy versus time plot will show a continuously increasing curve, becoming steeper as the velocity approaches c, reflecting the increasing kinetic energy due to the work done by the constant force.
The energy gained by a charged particle in an electric field is equal to the work done on it by the electric force. For a constant electric field, the work done (and hence the kinetic energy gained) is given by W = Fd = qEd, where q is the charge, E is the electric field strength, and d is the distance traveled.

Essay Format Questions

Discuss the transition from non-relativistic to relativistic dynamics for a proton moving in a uniform electric field. How do the predictions of classical mechanics diverge from those of special relativity as the proton's velocity increases, particularly with respect to acceleration and the attainment of the speed of light?
Explain the importance of using relativistic equations when simulating the motion of particles at high energies, such as those encountered in particle accelerators. Using the example of a proton in an electric field, describe the consequences of using non-relativistic approximations in such scenarios.
The Lorentz factor plays a central role in special relativity. Discuss how the Lorentz factor affects the concepts of time, length, mass, momentum, and energy for an object moving at a significant fraction of the speed of light.
Describe the Euler algorithm as a numerical method for solving equations of motion. What are its advantages and limitations, particularly in the context of simulating relativistic dynamics? How does the choice of time step affect the accuracy of the simulation?
Consider the implications of the relationship E=mc² in the context of the energy gained by a proton accelerating in an electric field. How does this fundamental equation relate to the concepts of rest energy and relativistic kinetic energy, and what does it tell us about the nature of mass and energy?

Glossary of Key Terms

Non-relativistic: Relating to speeds much less than the speed of light, where classical Newtonian mechanics is a good approximation.
Relativistic: Relating to speeds approaching the speed of light, where the effects of special relativity become significant.
Newton's 2nd Law: The principle that the acceleration of an object is directly proportional to the net force acting on the object and inversely proportional to its mass (F=ma in non-relativistic form).
Relativistic Momentum (p): The momentum of a particle moving at relativistic speeds, given by p = γmv.
Relativistic Energy (E): The total energy of a particle moving at relativistic speeds, given by E = γmc².
Rest Energy (E₀): The energy equivalent of the mass of a particle at rest, given by E₀ = mc².
Relativistic Kinetic Energy (KE): The energy of a particle due to its motion at relativistic speeds, KE = E - E₀ = γmc² - mc².
Lorentz Factor (γ): The factor γ = 1 / √(1 - v²/c²), which accounts for relativistic effects.
Uniform Electric Field: An electric field with constant magnitude and direction.
eV (electronvolt): A unit of energy equal to the energy gained by an electron moving through a potential difference of one volt.
c: The speed of light in a vacuum (approximately 2.998 x 10⁸ m/s).
Euler Algorithm: A first-order numerical method for approximating the solution of a differential equation.
Ultra-relativistic: Moving at a speed very close to the speed of light.
GeV (gigaelectronvolt): 10⁹ electronvolts, a unit of energy.
TeV (teraelectronvolt): 10¹² electronvolts, a unit of energy.
Extrapolation: Estimating values based on extending a known sequence of values or trend beyond the observed data.

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Frequently Asked Questions: Relativistic Dynamics in 1D with a Constant Force

1. What is the purpose of the simulation described in these exercises?

The primary goal of these exercises is to understand the motion of a proton in a uniform electric field, comparing and contrasting non-relativistic and relativistic dynamics. By using JavaScript simulations, the exercises aim to illustrate how special relativity affects the motion, energy, and acceleration of a charged particle as its speed approaches the speed of light. The exercises also emphasize the importance of unit consistency, particularly when working with non-SI units like electron volts (eV) and the speed of light (c).

2. How does the non-relativistic simulation of a proton in an electric field work, and what do the resulting plots of position and speed versus time show?

The non-relativistic simulation is based on Newton's second law of motion ( $F=ma$ ), where the constant electric field exerts a constant force on the proton, resulting in a constant acceleration. The code (in "Version1") uses the Euler algorithm to iteratively update the proton's velocity and position over small time steps. The plot of position versus time shows a parabolic shape, indicating motion with constant acceleration. The plot of speed versus time shows a linear increase, also consistent with constant acceleration.

3. What is non-relativistic energy, and why is it important to consider units and factors of $c^2$ when calculating it in the simulation?

Non-relativistic energy, in this context, refers to the sum of the rest energy ( $mc^2$ ) and the non-relativistic kinetic energy ( $\frac{1}{2}mv^2$ ). When implementing this in code (as seen in "Version2"), careful attention to units and factors of $c^2$ is crucial because the mass ( $m$ ) is given in eV/ $c^2$ , the energy is often tracked in eV, and the velocity is in m/s. Incorrect handling of these factors would lead to erroneous energy calculations and plots. The energy versus time plot in the non-relativistic case will show an increasing trend due to the increasing kinetic energy as the proton accelerates.

4. What limitation of the non-relativistic simulation becomes apparent when the simulation time is increased, and why does this occur?

When the simulation time in the non-relativistic model (Version 2) is increased to 5 seconds, the speed versus time plot will show the proton's speed exceeding the speed of light ( $c$ ). This is a fundamental flaw of the non-relativistic approach, which does not account for the speed limit imposed by special relativity. In reality, as an object's speed increases and approaches $c$ , its acceleration under a constant force decreases, preventing it from exceeding $c$ .

5. How is the relativistic form of Newton's second law different from the non-relativistic form, and how does the Lorentz factor play a role?

The relativistic form of Newton's second law in 1D is given by $a = \frac{F}{\gamma^3 m}$ , where $a$ is the acceleration, $F$ is the force, $m$ is the rest mass, and $\gamma$ is the Lorentz factor ( $\gamma = 1/\sqrt{1 - v^2/c^2}$ ). The Lorentz factor, which depends on the object's velocity ( $v$ ) relative to the speed of light ( $c$ ), accounts for relativistic effects. As the velocity increases, $\gamma$ increases, and $\gamma^3$ increases even more rapidly, causing the acceleration for a given force to decrease significantly compared to the non-relativistic case ( $a = F/m$ ).

6. What modifications are necessary to update the simulation to incorporate special relativity (as done in "Version3")?

To modify the simulation for relativistic dynamics, the following steps are crucial:

Calculate the Lorentz factor ( $\gamma$ ) at each time step based on the current velocity.
Use the relativistic form of Newton's second law ( $a = F/(\gamma^3 m)$ ) to calculate the acceleration. This ensures that as the velocity approaches $c$ , the acceleration decreases, preventing the speed from exceeding $c$ . The force $F$ due to the electric field is $qE$ , where $q$ is the charge of the proton and $E$ is the electric field strength.
Compute the relativistic energy using the formula $E = \gamma mc^2$ (or $E = \gamma m$ if mass is in eV/ $c^2$ ). This replaces the non-relativistic energy calculation and correctly accounts for the increase in mass-energy with velocity.

7. How can the accuracy of the numerical simulation (using the Euler algorithm) be assessed, and how does the time step affect this accuracy?

The accuracy of the numerical simulation can be assessed by comparing the numerical kinetic energy ( $KE_{numerical} = \gamma mc^2 - mc^2$ ) with the analytical kinetic energy ( $KE_{analytical} = Fx$ ), where $F$ is the constant force and $x$ is the distance traveled. The percent difference between these two values indicates the error. The Euler algorithm is a first-order method, and its accuracy depends on the size of the time step ( $dt$ ). Decreasing the time step generally increases the accuracy by reducing the discretization error, but it also increases the computational time. Increasing the time step will typically increase the error.

8. What happens to the plots of position, speed, and energy versus time for a proton accelerated to ultra-relativistic energies (close to the speed of light)?

When a proton is accelerated to ultra-relativistic energies, the plots exhibit the following characteristics:

Position versus time: The plot will initially show a curve with increasing slope (like the non-relativistic case), but as the speed approaches $c$ , the slope (which represents velocity) will become nearly constant, resulting in a nearly linear position versus time graph.
Speed versus time: The speed will increase rapidly at first but will then level off, asymptotically approaching the speed of light ( $c$ ). It will never exceed $c$ .
Energy versus time: The total energy ( $E = \gamma mc^2$ ) will continue to increase, even as the speed approaches $c$ . This is because the Lorentz factor $\gamma$ keeps increasing as the kinetic energy increases, even if the velocity增量 becomes very small. In the ultra-relativistic regime, the total energy becomes much larger than the rest energy ( $mc^2$ ).

When the total energy reaches 100 GeV, the proton will be in the ultra-relativistic regime, and these trends will be evident in the simulation plots.

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Details: Written by Loo Kang Wee; Parent Category: Physics; Category: 06 Modern Physics; Published: 30 April 2018; Created: 30 April 2018; Last Updated: 22 March 2025; Hits: 4567

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