model the earth's orbit around the sun using keplers laws in matlab. show 'real-time' calculated values for speed and time in simulation. in matlab

To model the Earth's orbit around the Sun using Kepler's laws in MATLAB and show real-time calculated values for speed and time in the simulation, you can follow these steps:

1. Define the initial conditions:

   • Assume the initial position of the Earth as (x, y) coordinates.
   • Assume the initial velocity of the Earth as (vx, vy) components.

2. Set up the simulation loop:

   • Define the time step for the simulation.
   • Specify the duration of the simulation.

3. Calculate the gravitational force:

   • Use Newton's law of universal gravitation to calculate the gravitational force between the Earth and the Sun.
   • The formula for the gravitational force is: F = G * (M1 * M2) / r^2, where G is the gravitational constant, M1 and M2 are the masses of the Sun and the Earth, and r is the distance between them.

4. Calculate the acceleration:

   • Use Newton's second law of motion to calculate the acceleration of the Earth.
   • The formula for acceleration is: a = F / m, where F is the gravitational force and m is the mass of the Earth.

5. Update the velocity and position:

   • Update the velocity of the Earth using the formula: v = u + (a * dt), where u is the initial velocity, a is the acceleration, and dt is the time step.
   • Update the position of the Earth using the formula: x = x + (v * dt), where x is the initial position and v is the updated velocity.

6. Calculate the speed:

   • Calculate the speed of the Earth using the formula: speed = sqrt(vx^2 + vy^2).

7. Display the real-time values:

   • Use the MATLAB disp function to display the calculated speed and time in the simulation loop.

Here's an example MATLAB code that demonstrates the above steps:

main.m
% Define initial conditions
x = 1.496e+8;  % x-coordinate of Earth's initial position (in km)
y = 0;        % y-coordinate of Earth's initial position (in km)
vx = 0;       % x-component of Earth's initial velocity (in km/s)
vy = 29.29;   % y-component of Earth's initial velocity (in km/s)

% Constants
G = 6.67430e-11;  % Gravitational constant (in m^3/(kg*s^2))
M_sun = 1.989e+30;  % Mass of the Sun (in kg)
M_earth = 5.972e+24;  % Mass of the Earth (in kg)

% Set up simulation parameters
dt = 0.01;  % Time step (in seconds)
duration = 365 * 24 * 3600;  % Duration of simulation (in seconds)

% Simulation loop
for t = 0:dt:duration  
    % Calculate the gravitational force
    r = sqrt(x^2 + y^2);  % Distance between the Earth and the Sun
    F = G * (M_sun * M_earth) / r^2;  % Gravitational force
    
    % Calculate the acceleration
    a = F / M_earth;  % Acceleration
    
    % Update the velocity and position
    vx = vx + a * dt;  % Updated x-component of velocity
    vy = vy;          % Updated y-component of velocity
    x = x + vx * dt;  % Updated x-coordinate of position
    y = y + vy * dt;  % Updated y-coordinate of position
    
    % Calculate the speed
    speed = sqrt(vx^2 + vy^2);  % Speed of the Earth
    
    % Display real-time values
    disp(['Speed: ', num2str(speed), ' km/s']);
    disp(['Time: ', num2str(t/(24*3600)), ' days']);
    
    % Visualization (optional)
    plot(x, y, 'ko');
    xlim([-1.5e+8, 1.5e+8]);
    ylim([-1.5e+8, 1.5e+8]);
    drawnow;
end
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Please note that this model assumes a simplified circular orbit and neglects factors such as the tilt of the Earth's axis and interactions with other celestial bodies.