# Understanding Motion in Physics Through Numerical Methods

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## Chapter 1: Introduction to Numerical Calculations

Numerical calculations break down complex problems into simpler, manageable tasks. This approach is particularly useful in physics when you have specific information about a particle's position and velocity at a given moment, as well as the net force acting on it, but lack a motion equation.

To predict the particle's future movement, you can segment the problem into small time intervals, allowing for approximations to derive a solution. The plan is to analyze a scenario that permits an analytical solution without numerical methods, followed by applying two distinct numerical techniques for comparison.

### Section 1.1: The Mass on a Spring

The classic example of a simple harmonic oscillator involves a mass attached to a spring on a frictionless horizontal surface. At the equilibrium position (x = 0 meters), the mass experiences no forces and remains stationary. However, when displaced, a restoring force from the spring comes into play.

In this scenario, there is only one force acting in the x-direction, allowing us to express Newton's Second Law as a differential equation. This equation can be simplified into a form that suggests the solution is related to sine and cosine functions. By hypothesizing a solution and substituting initial conditions, we arrive at an analytical formula for the mass's position over time.

Now, let’s utilize this analytical solution to evaluate different numerical methods.

### Section 1.2: The Euler Method

To begin, I'll apply the simplest numerical method, known for its straightforwardness. The initial conditions—position, velocity, and time—are established. For a brief interval (say, 0.1 seconds), I can compute the object's acceleration using the net force. Although acceleration isn't constant, we can use it as an approximation.

Assuming the initial velocity is v1 and the final velocity at the end of the interval is v2, we can express the final velocity as:

v2 = v1 + a * dt

Next, I need to update the position during this short time frame, again assuming constant velocity:

x_final = x_initial + v * dt

This method, referred to as the Euler-Cromer method, leverages the velocity at the end of the interval. After updating the position, the process repeats for subsequent time intervals.

Here's a basic Python code snippet illustrating this approach:

t = 0

dt = 0.1

x = 0.05

k = 10

m = 0.1

v = 0

while t < 3:

a = -k * x / m

v = v + a * dt

x = x + v * dt

fe.plot(t, x)

t = t + dt

It's crucial to understand that the assignment operator in Python functions differently than in algebra. It signifies assigning the computed value to the variable rather than establishing equality.

Let's now compare the Euler method's results with our analytical calculations.

The output shows the Euler method's performance with time steps of 0.1 seconds and 0.01 seconds. Notably, the larger time step yields reasonable results, with oscillation periods closely matching the analytical solution.

## Chapter 2: Exploring Advanced Methods

In this chapter, we delve into the Euler-Richardson method, also known as Runge-Kutta of the second order. This technique improves upon the basic Euler method by considering the slope (acceleration) at the midpoint of the time interval rather than at the start.

### Section 2.1: Understanding the Euler-Richardson Method

When using the Euler method, the velocity after a time interval is calculated based on the acceleration at the beginning of that interval. However, the Euler-Richardson method refines this by evaluating the slope at the midpoint.

To find the midpoint's position, we can initially apply the standard Euler method using a half time interval. This allows us to calculate the midpoint acceleration and subsequently use it to derive the new velocity and position.

Here's how the code would look:

while t < 3:

a = -k * x / m

vm = v + a * dt / 2

xm = x + v * dt / 2

am = -k * xm / m

v = v + am * dt

x = x + vm * dt

fe.plot(t, x)

t = t + dt

The results can be visually compared, although the effectiveness of different methods becomes apparent only with varying time intervals.

In conclusion, while the Euler-Cromer method generally provides satisfactory results, the Euler-Richardson method may yield better accuracy with smaller time intervals. Ultimately, the choice of method depends on the specific requirements of the problem at hand.