Mathematics
Beauty and the Beast [2]
Beauty and the Beast [1]

A Gift That Keeps On Giving

We see from “Seek-Lock-Strike!” Again that given the missile’s position
where and
are themselves functions of time
It means
That is, let
We also have (see “Seek-Lock-Strike!”)
Since
Substitute (2) into (1) yields
It follows that , the position of the missile satisfies the initial-value problem
To obtain the missile’s trajectory, we solve (4) numerically using the Runge-Kutta algorithm. It integrates (4) from to
(see “Seek-Lock-Strike!”).
Fig. 1
The missile strike is illustrated in Fig. 1 and 2.

Fig. 2

Fig. 3
The trajectories shown are much smoother than those in “Seek-Lock-Strike!” Animated.
“Seek-Lock-Strike!” Animated

In “Seek-Lock-Strike!” Again, we obtained the missile’s trajectory. Namely,
where
Since the fighter jet maintains its altitude (), the missile must strike it at
. Setting
in
gives
Hence, we can plot
Fig. 1
We can also illustrate “Seek-Lock-Strike” in an animation:
Fig. 2
Fig. 3
“Seek-Lock-Strike!” Again

We can derive a different governing equation for the missile in “Seek-Lock-Strike!“.

Fig. 1
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
where

Fig. 2
Using Omega CAS Explorer, we compute the missile’s striking time (see Fig. 3). It agrees with the result obtained previously.

Fig. 3
Exercise-1 Obtain the missile’s trajectory from (*).
“Seek-Lock-Strike!” Simplified

There is an easier way to derive the governing equation ((5), “Seek-Lock-Strike!“) for the missile.
Solving
for we have
From
we also have
Equate (1) and (2) gives
Differentiate (3) with repect to we obtain
(see Fig. 1)

Fig. 1
Let (4) bcomes
Since , dividing
through yields
the governing equation for the missile.
Newton’s Pi Simplified

We know from “arcsin” :
Integrate from to
gives
i.e.,
Rewrite the integrand as
so that by the extended binomial theorem (see “A Gem from Issac Newton“),
Hence,
And,
It follows that by ,
Let we have
And so,
Fig. 1
See also “Newton’s Pi“.
Given prove:
proof
Since
Exercise-1 Compute by applying the extended binomial theorem to
Exercise-2 Can we compute by applying the extended binomial theorem to
Explain.
Newton’s Pi
Fig. 1
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)
And,
It means
Area (triangle ABC)
Moreover,
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“)
simplifies beautifully:
Expressing (*) by (3), (2) and (4), we have
Therefore,
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.
Fig. 2
A Gem from Isaac Newton
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,
where
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:
Let
Differentiate (1) with respect to yields
We have
That is,
From
we see that is a solution of initial-value problem
By we also have
Express (3) as
and then differentiate it with respect to
gives us
i.e.,
Since
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),
And so,
Consequently, for
Multiply throughout, we obtain
i.e.,
Prove is convergent:
Proof
Prove
Proof
that is,
Therefore,
Prove
Proof