uations independently to compute the components of the velocity vector. v(t)=v(0)+ F:()l,n()=n0)+1/F()d,n2()=n20)+/F(出 and the position ()=0+mb2(),y(t)=01 m)dt,2()=20)+m/n:()d There are also one-dimensional cases in which the equations of motion can be integrated analytically(see review on rectilinear motion). In some other situations, it is possible to partially solve these equations nalytically. For instance, if the forces can be derived from a potential, we might be able to write expressions e.g. conservation of total energy, that are satisfied by the motion even though we may not know exactly know what the motion is Numerical Integration The most general way of solving the equations of motion is by numerical integration. In this case we do not compute the solution of the problem but an approximation to it. It is typically useful to work only with first order equations, so we writ 1 -Fr(a, y, z, vx, Vy, U2, t) . Fy(a, y, 2,t) 1 F2(x,3,2,Ux,Uy,U2,t) This is a set of 6 first order odes, with initial conditions x(0)=r0,y(0)=30,z(0)=20,v(0)=tx0,vy(O)=vy0,v2(0) The above set of equations is sometimes written in shorthand notation as X=F(X, t), with X(O=Xo where X=(r, g, 2, Ux, Uy, U2), F=(ux, Uy, U2, F=/m, Fy/m, F:/m), and Xo =(ro, y0, 20, 020, Uy0, vxo) The array X is usually referred to as the vector of state variables, even though it is not truly a vector. It hould not be confused with at physical vector such as the velocity or acceleration. Note that the components of x do not even have the same units ortunately, the theory for numerically solving systems of ordinary differential equations, is very mature and good algorithms are available. Here we use a simple"Forward Euler Method" for illustration. Consider a time increment At(this At should be"small enough"so that the computed results are accurate, but on theequations independently to compute the components of the velocity vector, vx(t) = vx(0) + 1 m Z t 0 Fx(t) dt , vy(t) = vy(0) + 1 m Z t 0 Fy(t) dt , vz(t) = vz(0) + 1 m Z t 0 Fz(t) dt , and the position, x(t) = x(0) + 1 m Z t 0 vx(t) dt , y(t) = y(0) + 1 m Z t 0 vy(t) dt , z(t) = z(0) + 1 m Z t 0 vz(t) dt . There are also one-dimensional cases in which the equations of motion can be integrated analytically (see review on rectilinear motion). In some other situations, it is possible to partially solve these equations analytically. For instance, if the forces can be derived from a potential, we might be able to write expressions, e.g. conservation of total energy, that are satisfied by the motion even though we may not know exactly know what the motion is. Numerical Integration The most general way of solving the equations of motion is by numerical integration. In this case we do not compute the solution of the problem but an approximation to it. It is typically useful to work only with first order equations, so we write, x˙ = vx y˙ = vy z˙ = vz v˙x = 1 m Fx(x, y, z, vx, vy, vz, t) v˙y = 1 m Fy(x, y, z, vx, vy, vz, t) v˙z = 1 m Fz(x, y, z, vx, vy, vz, t) This is a set of 6 first order ODE’s, with initial conditions x(0) = x0, y(0) = y0, z(0) = z0, vx(0) = vx0, vy(0) = vy0, vz(0) = vz0 . The above set of equations is sometimes written in shorthand notation as X˙ = F(X, t), with X(0) = X0 (2) where X = (x, y, z, vx, vy, vz) T , F = (vx, vy, vz, Fx/m, Fy/m, Fz/m) T , and X0 = (x0, y0, z0, vx0, vy0, vz0) T . The array X is usually referred to as the “vector” of state variables, even though it is not truly a vector. It should not be confused with at physical vector such as the velocity or acceleration. Note that the components of X do not even have the same units. Fortunately, the theory for numerically solving systems of ordinary differential equations, is very mature and good algorithms are available. Here we use a simple “Forward Euler Method” for illustration. Consider a time increment ∆t (this ∆t should be “small enough” so that the computed results are accurate, but on the 2