Tossing a ball

Once upon a time I was playing catch with a small child. The child was just learning to catch a ball, and so I drew closer to her because she dropped the last one I threw. However,  the dear child was habitually teased by an older sibling for being a “baby” and she was therefore keen to appear grown up. When I took a step towards her, she took a step back to avoid ridicule from her sibling, keeping the distance between us constant.

What to do?

And thus was born a calculus/physics problem. For a fixed range, determine the initial velocity of a projectile such that the target is hit and yet the final speed is minimized. (You want to minimize the final speed to make the ball as easy as possible to catch.)

Note that there are two adjustable parameters in the projectile’s initial velocity; you can take them to be the two components of the velocity, or you can take them to be the initial speed and the initial angle of inclination.

Have fun!


About Santo D'Agostino

I have taught mathematics and physics since the mid 1980s. I have also been a textbook writer/editor since then. Currently I am working independently on a number of writing and education projects while teaching physics at my local university. I love math and physics, and love teaching and writing about them. My blog also discusses education, science, environment, etc. Further resources, and online tutoring, can be found at my other site
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2 Responses to Tossing a ball

  1. Joey says:

    There is also the mass and material of the object being thrown. So perhaps a softer ball so any misses will not be painful. There is always Nerf. Hehe.

  2. Santo says:

    Good point!

    Another thing I take away from your humorous comment is the disconnect between the typically artificial textbook problems (the one I suggested in this post being an example) and reality. We (meaning I) have to find a way to do a better job at injecting more reality into our university courses … although it’s hard to do because of the constant pressure to “cover” the material.

    Thanks for your comment, Joey!

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