A small ball is throw up a slope a distance of 30 ft. The slope is 30^o from the horizontal and the ball's initial velocity is 40^o from the vertical as shown. The ball leaves the persons hand 5 ft from the ground. What is the smallest initial velocity?

Vertical and Horizontal Distances

The ball travels in both the x and y directions. Two equations will be needed. First, examine the y direction,
     y = y_o + v_oyt - 0.5gt²
     30sin30 = 5 + v_o cos40 t - 0.5(32.2) t²
     10 = v_o cos40 t - 0.5(32.2) t²

Next, use the x direction equation as second equation since there are two unknowns,
     x = x_o + v_xot
     25.98 = 0 + v_o sin40 t
     v_o = 40.42/t

Combining the two equations
     10 = (40.42/t) cos40 t - 16.1 t²
     20.96 = 16.1 t²
     t = 1.387 s

Substitute back into either equation to get initial velocity,
     v_o = 40.42 / 1.387

v_o = 29.14 ft/s