4)          A soccer player takes a penalty kick towards the goal. The angle at which the ball leaves
             his/her boot is 53 with respect to the ground. The penalty spot is 30 ft from the goal and the
            cross bar is 8 ft high (g = 32 ft/s², tan(53) = 4/3).

     (a)   If the ball hits the cross bar (i.e. it is exactly 8 ft high after traveling 30 ft), how long does it
            take to reach the bar ? (9)

           
Eliminating v_i between these equations leaves,

     (b)   What must be the initial velocity of the ball, v_i ? (6)

             Given t then,

     (c)    Determine the horizontal and vertical components of the velocity of the ball upon impact
             with the cross bar. (6)

             x component is unchanged,

            
y component is given by,

     (d)    Sketch these velocity components and determine the tangent of the angle the velocity
             vector makes with the vertical at the cross bar. (4)