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[Physltest] [Phys-L] Re: projectiles



Regarding Jack's comments:

David-
I don't recognizem my solution as identical to yours. Either
one of us has made an error, or I gave up to soon on the algebra.
Jack

and

David, one of us has made an error. The solutions are identical in
the special case of q=0, which explains why we got nearly identical
solutions for the case of small q. We can communicate privately
until we agree on a solution.
Regards,
Jack

Jack, we *already* agree on a solution. Your implicit equation for
the angle a is clearly not *formally* identical to my explicit
formula for it, but your equation *is* equivalent to mine in the
sense that they both have the same value of a as the solution for a
given value of q. In fact, it is fairly straightforward to derive
my formula from your equation. Neither of us made an error. You
just gave up on the algebra a bit to soon.

Below are a few more details of the steps in the derivation that I
outlined in my previous response to Ken Fox (except I am now using
Jack's notation for the variables rather than my/Anthony's notation).

First let's define s == 1 - cos(2*a). Substitution of 1 - s for
cos(2*a) and s/2 for sin^2(a) into Jack's equation 2.) gives (after
dividing both the numerator and denominator on the RHS by sin(a):

1 - s = q/(1 + sqrt(1 + 2*q/s))

Multiplication on both sides by the denominator gives:

(1 - s)*(1 + sqrt(1 + 2*q/s)) = q

Isolating the sqrt() term gives:

(1 - s)*sqrt(1 + 2*q/s) = q - 1 + s

Squaring out both sides and canceling the common terms on both sides
gives:

1 + 2*q/s - 4*q = (q - 1)^2

Solving for the only appearance of s in the above equation gives:

s = 2*q/((q - 1)^2 + 4*q - 1)

Simplifying the denominator and canceling the common factor of q in
the fraction on the right gives:

s = 2/(q + 2)

This means that 1 - s = 1 - 2/(q + 2) = q/(q + 2)

Now since we had defined s as 1 - cos(2*a) this means that we can
write a as:

a = (1/2)*arccos(1 - s) = (1/2)*arccos(q/(q + 2)) QED

Also, if we are willing to write this in terms of 1/q (to Ken Fox's
chagrin about what happens when q --> 0) we can also write:

a = (1/2)*arccos(1/(1 + 2/q))

David Bowman
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