Answer the following problem: Conservation of Energy A particle moves with position vector (t) = x(t)+ y(t)+ z(t). Let (t) and (t) be its velocity and acceleration vectors. Show that 2 = (t) (t) We...

Answer the following problem: Conservation of Energy A particle moves with position vector r(t) = x(t)i + y(t)j + z(t)k . Let v(t) and a(t) be its velocity and acceleration vectors. Show that 12 ddt v(t)2 = a(t) ·v(t) We now derive the principle of Conservation of Energy. The kinetic energy of a particle of mass m moving with speed ? is (1/2)m?2. Suppose the particle has potential energy f(r) at the position r due to a force filed F = -?f. If the particle moves with position vector r(t) and velocity v(t). Then the Conservation of Energy principle says that Total energy = Kinetic energy + Potential energy = 12mv(t)2 + f(r(t)) = Constant. Let P and Q be two points in space and let C be a path from P to Q parameterized by r(t) for t0 = t = t1 , where r(t0) = P and r(t1) = Q. Using part (a) and Newton’s law F =ma, show Work done by F = Kinetic energy at Q – Kinetic energy at P. as particle moves along C Use the Fundamental Theorem of Calculus for Line Integrals to show that Work done by F = Potential energy at P – Potential energy at Q. as particle moves along C Use parts (a) and (b) to show that the total energy at P is the same as at Q. This problem explains why force vector fields which are path-independent are usually called conservative (force) vector fields.