A particle has position function r(t) and velocity function v(t) = { 2e“, sin(−t), ln(t+1) ) (a) Find its acceleration function a(t). (b) At time t=0, its position is r(0)=i+k. Find r(t).

C (a) Write down the equation of motion for the point particle of mass m moving in the Kepler potential U(x) = -A/x+B/x² where x is the particle displacement in m. (b) Introduce a dissipative term in the equation of motion assuming that the dissipative force acting on a particle is proportional to the partical velocity with the coefficient of proportionality v. Write down the modified equation of motion. From the dimensionless equation of motion modified by dissipation Compare to question (2 b) derive a linearised equation of motion around a stable equilibrium state. Solve it and determine what is the range of values of the free parameter B for which the particle motion is oscillatory or non-oscillatory? What is the critical value of that determines a transition from the subcritical case with the oscillatory motion to the super-critical case with the non-oscillatory motion?