Bisection Method (More accurate).

M03_Kinetic_Energy_v3.m

%% Newtonian vs. Relativistic Energy Accuracy
% Determines the maximum speed a particle can have such that its 
% kinetic energy described as 0.5*m*v^2 with an error of no greater 
% than <tolerance>.
close all; clear all; clc;

% Input - The tolerance of the accuracy of the Newtonian kinetic energy 
% compared to the relativistic energy.
tolerance = .005;  

% Speed of light  [m/s]
c = 299792458;            

% Bisection parameters
maxiters = 1000;
hi = c;
lo = 0;
v = -1;

% How accurate we are in our calculation.                       
% Not to be confused with tolerance of energy calculation (aka error).
TOL = 1e-4; 

% Use bisection to determine max velocity to keep energy within
% the specified tolerance.
for i=1:maxiters  
    
    % "Guess" the velocity
    v_prev = v;
    v = (hi - lo) / 2 + lo;    
    
    % Calculate our energies
    g = (1 - v^2/c^2)^(-1/2);   % Gamma
    K = 0.5*v^2;            	% Newtonian kinetic energy per unit mass
    E = g*c^2 - c^2;            % Relativistic kinetic energy per unit mass
    
    % Calculate energy error
    err = abs(K - E) / E;
    %fprintf('%4d: %14.4f  %6.4f  %10.4e  %10.4e  %10.8f\n', i, v, g, E, K, err)
    fprintf('%4d:  %14.4f  %8.6fc  %10.8f\n', i, v, v/c, err)
    
    % Velocity is "close enough"
    if abs(v - v_prev) < TOL
        break
    % Not close enough, adjust velocity
    else
        % Error is too big, try a lower velocity
        if err > tolerance
            hi = v;
        % Not close enough to target value, increase velocity
        else
            lo = v;
        end
    end
end