**29. Constraints on ***m*_{cloud}
and *v*_{cloud} from *m*_{cloud}
and *v*_{cloud}

The previous section has shown the constraints arising for
*v*_{lurch} and *v*_{cloud} that arise by varying *Θ*_{cl}
and *PE*. But those calculations leave *m*_{cloud} fixed. Here
we take a more general look at *m*_{cloud} and *v*_{cloud}
by varying them both and examining the constraints imposed by *PE*. This
requires us to use SL7A, which solves for *PE* instead of *v*_{cloud}.
It will be seen that *m*_{cloud} and *v*_{cloud} can
roam quite freely but are related roughly inversely when they do. This means
that their product, the momentum of the cloud, tends toward constancy, and is
therefore a more fundamental quantity than *m*_{cloud} or *v*_{cloud}.
It also means that we don't have to worry about their individual ranges, for the
product is the thing that really counts.

This section presents the
constraints separately for *Θ*_{cl} = 30°, 70°, and
105°. For each angle, separate sets of results are presented for masses
of the head ranging from 5 to 10 lb. (As before, however, all solutions for 9
and 10 lb are disallowed by constraints on *PE* and *v*_{frags}.) The last part summarizes the constraints for these
different conditions. There it is seen that the constraints, particularly for
the lurch, are essentially independent of the mass of the head. Most
importantly, the allowed speeds of the lurch are all negative and bracket the
observed value of -0.8 ft s^{-1}.

For *Θ*_{cl}
= 30°

For *Θ*_{cl} = 70°

For *Θ*_{cl} = 105°

Summary of the constraints

Ahead to Grand Summary of Constraints

Back to Constraints on *m*_{cloud}
and *v*_{cloud} from *Θ*_{cl}
and *PE*

Back to Intro to Constraints

Back to Physics of the Head Shot