Ball Throw


A small ball is throw up a slope a distance of 30 ft. The slope is 30o from the horizontal and the ball's initial velocity is 40o 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 = yo + voyt - 0.5gt2
     30sin30 = 5 + vo cos40 t - 0.5(32.2) t2
     10 = vo cos40 t - 0.5(32.2) t2

Next, use the x direction equation as second equation since there are two unknowns,
     x = xo + vxot
     25.98 = 0 + vo sin40 t
     vo = 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,
     vo = 40.42 / 1.387

vo = 29.14 ft/s

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