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A particle moves in a vertical plane under gravity and an air resistance whose magnitude per unit mass is k times the speed of the particle, where k is a positive constant. The coordinates of the particle at time t, with respect to a horizontal axis Ox and a vertically upward axis Oy, are (x,y).

Show that

d^2 x /dt^2 = -k dx/dt

d^2y/dt^2 = - k dy/dt - g ...... (*)

Given that a projectile with a terminal speed of 30.5 m/s is fired when t=0 from O with an initial speed of 30.5 m/s at an inclination of 45 degree.

(a)Solve (*) for x and y in term of t.
(b) Plot the trajectory of the projectile.
(c) Compare the path trajectory to that when there is no air resistance.

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