Minimizing final velocity on a balistic trajectory

[flatmaster]

I thought of the following academic problem while watching a baseball movie.

I ballplayer wishes to throw the game-winning ball to a kid in the stands. He wants to minimize the final velocity of the ball. Not the horizontal component of the velocity, but the the total magnitude of the velocity. Obviously, he controlls the initial angle and initial velocity. A line drive right at the kid is obviously leads to a maximum final velocity, while a high-arching trajectory also leads to a maximum velocity. There is some angle inbetween these two choices that leads to a minimum final velocity.

My intuition says the choice trajectory would be the one where the ball arives at the top of it's parabola with zero vertical velocity.

I don't have it on me know, but I wrote an equation for the final velocity, took it's derivative with respect to initial velocity, and set that equal to zero. Horrible math ensued. Any ideas on other math tricks for this academic problem? Conservation of energy?

 

[tiny-tim]

Conservation of energy?

Hi flatmaster!

Yup … conservation of energy means that KE_f = KE_i - mgh, where m is the mass of the ball, and h is the height of the stand.

So you minimise KE_f by minimising KE_i.

 

[csprof2000]

I would imagine that, ideally, you would want the ball to peak right when he catches it. Assuming conservation of energy, this will be the time when PE is highest... and correspondingly KE will be lowest.

If you threw the ball higher than this, you would have to put more E = PE + KE into it. If you threw it lower, the kid wouldn't be able to catch it.

 

[flatmaster]

I think I see the remainder of the proof. I take tiny tim's suggestion that minimizing KEi is easier mathematically. So I use the kenitic energy equation and use the additional bountry condition of ariving at the point x,y, to eliminate one of my remaining variables.