4. Periodic impulses. A mass weighing four pounds stretches a spring 24 in. Suppose the mass is at equilibrium and is set into motion with a velocity of 6 inches per second. The mass is in a medium where friction is small, and we assume it is zero. Note that the acceleration due to gravity near the earth's surface where the experiment takes place can be taken as 32 ft/s². (a) Suppose that in addition to the conditions described above, a hammer strikes the mass every π/2 seconds starting at t = 0. A delta function with impulse of 1 pound-second models the force of each blow to the mass. Find the function x(t) for the displacement from equilibrium in feet of the mass at time t in seconds. Express x(t) as a piecewise defined function without the use of unit step or Heaviside functions and such that in each time window it is a different multiple of the same sinusoidal function. (b) Now assume the same spring-mass system is hit with a hammer strike more frequently, with a time delay of π/4 seconds between consecutive blows. Find x(t). Express x(t) following the guidelines in part (a). (c) Explain how the frequency of the blows affects the long term behavior of the system in parts (a) and (b).

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