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Computation Projectile Motion Atmospheric Effects Consider simple ofai resistance proportional tothe projectiles velocity: 1--bi Furthermore, consider the proportionality depends asfollows Compare the trajectories air and where exponentially decreases Charged Particles Cons themation of charged particle under Force the magnetic fields. Coriolis Effect Conside the effects the motion of particles dueto the Earth's rotation abouti axis. Using north-south east west, up-down choice faxis where the position, velocity, and acceleration vectors canbe written as: - Suppose Earth rotates about axis with angular velocity vector n terms of the angle of latitude, 6: The Coriolis term for theacceleration observed from the non-inertial reference frame of the form: Include projectile motion and analyze the results. Page 1/4 Gravity Usethe Verlet -Stormer algorithm tomodel satellite motion Arethe orbits stable? Consider different kinds of orbits energy and angular momentum conserved over time? Coupled Oscillators Three Mass System . Analyze: linearly -coupled systemofthree " masses shown inthefigure. Suppose alithe masses are equal and k 4, & k3-K Writea system equations o motion - - DO and usethe semi-implicit method to 00 numerically solve. Describe normal modes of thesystem Consider the weakly-coupled case where x " Consid Create simulation that shows (with lines connecting themas springs) function ol 2 Dimensional Heat Equation Create ulation the 2D Heat Equation Boundan Condit Neumann BC This should bedoneon asquare initial temperature profiles fsimple shapes. . You shoul make video evolution similar what shown here. Wave Equation Consider thefollowing initial boundan value Wave Equation With fixed boundary conditions w(t,0)- u(t,L)-0 Arbitrary initi al conditions for both position velocity. u(0,x)-f(x) u,(0,x)-g(x) Page Usethe Central- (CTCS) finite method approximation wherethe following notation willbe followed: x-jAx Wherei [0,1,2, such thatin Tand xx -L The Wave Equation becomes: At2 Which renders Where levels. Meaning knowing as well. Toget this information, Resulting in 2Atg(z) Substitutin recurrence +2(1 Atg(x) Page The numerical solution However itis worth noting the choice of - 1 is known as the magic time- step and the Consider simple initial and start from rest i.e. g(x) -0 Create simulation shows the following BC's o Dirichlet BC Both ends Neumann BC- -Both ends ar free: o Absorbing Boundary Condition (ABC). Both end totally absort energy. User - 1 for this Repeat for auniform initial velocity profile function: g(x)- The simulation should be similar that shown here (for fixed ends)

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function [T,X]=SolvesemiImplctEuler(f,h,X0,n)
X=X0;
T=0;

%Loop to solve by Semi-implicit Euler
for i=1:n
    dX=f(T(end),X(end,:));
    Xn=X(end,:)+h*dX';
    X=cat(1,X,Xn);
    T=cat(1,T,T(end)+h);
end...
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