From - Tue Feb 23 11:14:12 1999
Newsgroups: rec.puzzles
Subject: Re: Jumping off of a swing
From: karr@shore.net (David A Karr)

The exact formula that theoretically maximizes distance, by the way, is:

  Let L = length of the swing's chain;
      h = height above ground at the highest point of your swinging
      R = sqrt((L/2h)^2 - 1/27)
  theta = arc sec((L/2h + R)^(1/3) + (L/2h - R)^(1/3))

Then theta is the angle from vertical at which you should jump to
maximize the distance you land from the base of the swing set.

As long as L/h >= 2*sqrt(3)/9 ~ 0.3849 (which should obtain for most
swing sets), this involves only real arithmetic, not complex numbers.

>I believe your solution is correct. The interesting thing is, I can't
>even get close to that swinging on a real swing & trying to jump.
>[...] increased my distance by bailing out earlier [...]

Remember the h is measured to the highest point of your swing, not
necessarily to the top of the swing set; y is measured from the
ground, not from the lowest point of your swing; and this is not a
horizontal distance.  In reality L is usually signficantly shorter
than the swing set height, and you probably should measure L to your
center of gravity rather than to the seat of the swing, which means
it's even shorter yet.  The smaller L/h is, the sooner you should jump.

Of course you're also not a perfect pendulum when you're swinging:
your body has a moment of inertia, and the chain has substantial
mass too.  I haven't worked out how this would affect the result.


The first thing my nine-year-old pointed out about this problem is
that it's against playground rules to jump from the swing.  This
suggested to me that instead of jumping, you should drop a ball.  You
could even hold the ball between your knees to free your hands, and
you could bring several balls with you so you could run several trials
without having to pump the swing all the way up from a standstill each
time.

Actually this problem begs for someone to build a scale model.

-- 
David A. Karr       "Groups of guitars are on the way out, Mr. Epstein."
karr@shore.net                         --Decca executive Dick Rowe, 1962