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Q5. 1) A rocket has an initial mass M and fuel fraction. The speed at the exhaust is C. a constant. The rocket is launched vertically upwards from rest. The fuel is burnt at a variable rate A=aße-t (a) Find the mass function m(t). (b) Show that the time at burnout is T--In (1-²). [8 marks] 2) The graph below shows the evolution of a potential U(z) as a function of the distance (red curve) corresponding to the a particle moving along the z-axis and the evolution of the kinetic energy (K) and potential energy (U) of the system. The energy is expressed in Joule (J). The points 1₁, 12, 13, 14, and z5 are particular points along the motion. (1) This is a plot of the potential -) energy U versus position x. w (a) The total energy of the system is 5 J. Where the kinetic energy is maximum? and what is its value? Describe the motion of a particle starting from a location between 2 and 3 (but not including those points). The velocity can be positive or negative (going right or left on the graph respectively). (b) If the total energy of the system is 2.5 J. describe the motion of a particle starting between 13 and 14. [12 marks] [Total: 20 marks]