MATLAB: Writing a script for the Equation Of Motion

accelerationdistanceMATLABmatlab functionmotion

Hi,

I want to write a MATLAB script for the second equation of motion s=vo+1/2at^2. How would I approach this? Any help would be appreciated.

I am trying to simulate the trajectory of a bullet and I am finding it quite difficult as no values are assigned for conditions and therefore struggling as to where begin. Thank you

Best Answer

If you neglect friction from the air, then acceleration is g, 9.81 m/s, directed downwards.

V0 is the muzzle velocity; there is a whole Wikipedia article about muzzle velocities of various kinds of bullets. Pick one.

t is simulated time.

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It depends how your RK4 code is written. Remember that you have to convert the higher order equation to a system of equations of order 1 at first. In your case these are 2 equations, one for the velocity and one for the acceleration. All you "simulate" is the time, the position and the velocity. To get velocity^2 and acceleration you can simply evaluate the function with the formerly calculated position and velocity.

To do this, your RK4 must accept a vector as input. Because you did not post any code, it is hard to suggest any real code. But simply try to implement this advice and show us the code.

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If you want to use the odeNN functions you need to rewrite your 2 2nd-order ODEs into 4 1st-order ODEs, something like this:

function dxdydvxdvydt = your_eqs_of_motion(t,xyvxvy,other,parameters)
x = xyvxvy(1);  % If my ode-equations are complicated 
y = xyvxvy(2);  % I prefer to explicitly extract the
vx = xyvxvy(3); % variables into something that's
vy = xyvxvy(4); % easily human-readable
m = other;
Fx = force_functionx(t,x,y,vx,vy,other,parameters); % Adjust inputs as necessary
Fy = force_functiony(t,x,y,vx,vy,other,parameters);
% or possibly preferable:
% [Fx,Fy] = force_function_xny(t,x,y,vx,vy,other,parameters);
dxdydvxdvydt = [vx;vy;Fx/m;Fy/m];
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Then you can integrate the equations of motion something like this:

m = 12; % you know what to set for this one
x0y0vx0vy0 = [1e-10,5.08e-5,0,0]; % Your initial conditions
t_span = [0,2*pi];                % a time-span of interest
[t_out,xyvxvy] = ode45(@(t,xyvxvy) your_eqs_of_motion(t,xyvxvy,m,[]),t_span,x0y0vx0vy0);

When integrating equations of motion you typically also have to carefully check that the constants of motion we know should be conserved (total energy, angular momentum, etc.) are reasonably conserved - the general ODE-integrating functions are not guaranteed to do this.

HTH

Best Answer

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