1. The problem statement, all variables and given/known data A particle undergoes simple harmonic motion in both the x and y directions simultaneously. Its x and y coordinates are given by x=asin(wt) y=bcos(wt) a. Eliminate t from these equations to show that the path of the particle in the x-y plane is an ellipse. b. Calculate the kinetic and potential energy at a point on the particle’s trajectory. Show that the ellipse is a path of constant total energy, and show that the total energy is given by the sum of the separate energies of the x and y oscillations. 2. Relevant equations Equation of an ellipse: (x/a)^2+(y/b)^2=1 KE = 1/2=mv^2 PE = ?? 3. The attempt at a solution a) I rearranged the parametric eqns. like so: x/a=sinwt and y/b=coswt, squared both sides and using trig identities eliminated the term (sin^2wt+cos^2wt) leaving 1 on the right side and therefore getting the equation of an ellipse: (x/a)^2+(y/b)^2=1 b)This is where I'm having some troubles... I am pretty sure that I can calculate the KE of the particle by looking at: KE(x)= 1/2*mv(t)^2 = (1/2)mw^2(acoswt)^2 KE(y)= -(1/2)mw^2(bsinwt)^2 but I don't know how to come up with the equations for the potential energy. Regrading the fact that the ellipse is a path of constant total energy, I am thinking of showing that when the particle is on the x axis it means that y=0 thus eliminating the y component of the energy, same goes for when its on the y axis. which essentially can be modelled with a cosine function where energy is always constant but changes from component to component. I'd appreciate any help or some input on my attempt at the solution. Thanks!