3. Consider the equation for a mass m > 0 on a spring with spring-constant k > 0, with damping friction b>0: the damped oscillator mx + bi + kx = 0. (a) Define v(t) = x(t) and write this second order equation as a pair of coupled first-order equations for x(t) and v(t). (b) For various values of the parameters, classify all possible solution types, their fixed points and stability. [You should probably divide through by m first..]. Interpret the results physically. (c) Show that if b = 0, then v² + ax² = constant for some a. What is a? Thus show the trajectories are ellipses.

= 1. A particle 1 of mass m₁ is attached to one end of a spring with constant k at position x = = 0. It is initially at rest, with velocity u₁ 0. The other end of the spring is attached to a fixed wall at position x = -l where l is the length of the spring at rest. The whole setup is horizontal along the x-axis. Another particle, 2, of mass m2 travels towards particle 1 with constant velocity u2 (note that u2 < 0 in this setup). The two particles undergo a collision with coefficient of restitution 0 ≤ e ≤ 1. (a) Using the one-dimensional equations of collision (eq (6.44) in the lecture notes) show that immediately after collision the velocity of particle 1 is given by V₁ = (1 + e) m2u2 m1 + m2 Find the velocity v2 of particle 2 immediately after collision. What happens to particle 2 if m₁ = m2 and the collision is elastic (e = 1)? (b) Immediately after collision, particle 1 has velocity v₁ and is at position x = 0, which we take as initial conditions for its subsequent motion under the…

Help with this linear algebra problem, thanks!

(18) Consider a particle with position function 7 (t) = (t, t² – 3t, 2t2). (a) Find the velocity, acceleration and speed of the particle. (b) What can you say about maximum and minimum values of the speed of this particle?