When baseball players throw the ball in from the outfield, they usually allow it

Question

When baseball players throw the ball in from the outfield, they usually allow it to take one bounce before it infield, on the theory the ball arrives sooner that way. Suppose the angle at which a bounced ball leaves the same as the angle at which the outfield threw it, as in the figure, but that the ball's speed after the ball half of what it was before the bounce. Assume the balls is always thrown with the same initial speed, at what angle theta should the fielder throw the ball to make if go the same distance D with one bounce ( blue path ) as a ball thrown upward at 50..0 degree with no bounce ( green path )? Determine the ratio of the times for the one-bounce and no-bounce throws. (one-bounce time/ no-bounce time)

Explanation / Answer

a)D=vcos50*t = vcos*t'+v/2*cos*t''

t=vsin50/g   t'=vsin/g t''=v/2*sin/g

vcos50*t = vcos*t'+v/2*cos*t''

=>cos50*sin50= cossin+cossin/4 = 5/8*sin2

=>=26o

b)(t'+t")/t=1.5sin/sin50 = 0.85

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