We can derive a different governing equation for the missile in “Seek-Lock-Strike!“.
Looking from a different viewpoint (Fig. 1), we see
Solving (1) for ,
We also have
Equate (1) and (2) gives
The governing eqaution emerges after differentiate (4) with respect to
We let so and express (5) as
Using Omega CAS Explorer, we compute the missile’s striking time (see Fig. 3). It agrees with the result obtained previously.
Exercise-1 Obtain the missile’s trajectory from (*).
There is an easier way to derive the governing equation ((5), “Seek-Lock-Strike!“) for the missile.
for we have
we also have
Equate (1) and (2) gives
Differentiate (3) with repect to we obtain
(see Fig. 1)
Let (4) bcomes
Since , dividing through yields
the governing equation for the missile.
We know from “arcsin” :
Integrate from to
Rewrite the integrand as
so that by the extended binomial theorem (see “A Gem from Issac Newton“),
It follows that by ,
Let we have
See also “Newton’s Pi“.
Exercise-1 Compute by applying the extended binomial theorem to
Exercise-2 Can we compute by applying the extended binomial theorem to Explain.
Shown in Fig. 1 is a semicircle centered at C with radius = . Its equation is
Simplifying and solving for gives
We see that
Area (sector OAC) = Area (sector OAB) + Area (triangle ABC)
Area (triangle ABC)
Since is one-third of the angle forming the semicircle, the sector is likewise a third of the semicircle. Namely,
Area (sector OAC) Area (semicircle) =
Area (sector OAB) is the area under the curve from its starting point to the point i.e.,
Area (sector OAB)
By the extended binomial theorem: (see “A Gem from Isaac Newton“)
Expressing (*) by (3), (2) and (4), we have
Observe first that
and so we replace by its binomial expansion. As a result,
Substituting (6) into (5) then yields
Fig. 2 shows that with just ten terms (0 to 9) of the binomial expression, we have found correct to seven decmal places.
The Binomial Theorem (see “Double Feature on Christmas Day“, “Prelude to Taylor’s theorem“) states:
Provide and are suitably restricted, there is an Extended Binomial Theorem. Namely,
Although Issac Newton is generally credited with the Extended Binomial Theorem, he only derived the germane formula for any rational exponent (i.e., ).
We offer a complete proof as follows:
Differentiate (1) with respect to yields
we see that is a solution of initial-value problem
By we also have
Express (3) as
and then differentiate it with respect to
we see that is also a solution of initial-value problem (2).
Hence, by the Uniqueness theorem (see Coddington: An Introduction to Ordinary Differential Equations, p. 105),
Multiply throughout, we obtain
Prove is convergent: