(a) Vessel A is pursuing vessel B, so at time t, vessel A must be heading right at vessel B. That is, the tangent line to the curve of pursuit at P must pass through the point Q (see Figure 3.18). For this to be true, show that (4) (b) We know the speed at which vessel A is traveling, so we know that the distance it travels in time t is at. This distance is also the length of the pursuit curve from (0, 0) to (x, y). Using the arc length formula from calculus, show that dy dx = (5) at = (6) y - Bt x-1 = 5₁² V1 + [y' (u)]² du. Solving for t in equations (4) and (5), conclude that y-(x-1) (dy/dx) = == S V₁ + [y' (u)]³² du . В (c) Differentiating both sides of (6) with respect to x, derive the first-order equation dw В (x-1)- VI+w², dx α where w = dy/dx. P = (x₂) -BQ = (1, 3t) (1, 0) X
An interesting geometric model arises when one tries to determine the path of a pursuer chasing
its prey. This path is called a curve of pursuit. These problems were analyzed using methods of
calculus circa 1730 (more than two centuries after Leonardo da Vinci had considered them). The
simplest problem is to find the curve along which a vessel moves in pursuing another vessel that
flees along a straight line, assuming the speeds of the two vessels are constant.
Let’s assume that vessel A, traveling at a speed a, is pursuing vessel B, which is traveling at a
speed b. In addition, assume that vessel A begins (at time t = 0) at the origin and pursues vessel
B, which begins at the point (1, 0) and travels up the line x = 1. After t hours, vessel A is located
at the point P = 1x, y2, and vessel B is located at the point Q = 11, bt2 (see Figure 3.18). The
goal is to describe the locus of points P; that is, to find y as a function of x.
Please look at images for questions. The formatting isn't correct if i type it out.
![(a) Vessel A is pursuing vessel B, so at time t, vessel A must be heading right at vessel B.
That is, the tangent line to the curve of pursuit at P must pass through the point Q
(see Figure 3.18). For this to be true, show that
(4)
(b) We know the speed at which vessel A is traveling, so we know that the distance it
travels in time t is at. This distance is also the length of the pursuit curve from (0, 0) to
(x, y). Using the arc length formula from calculus, show that
(5)
dy
dx
(6)
=
at =
= f* V₁ + [y'(u) ³² du.
y - Bt
x-1
Solving for t in equations (4) and (5), conclude that
y-(x-1) (dy/dx) 1
α
=
В
(c) Differentiating both sides of (6) with respect to x, derive the first-order equation
dw B
(x-1)-
V1+w²,
dx
where w := dy/dx.
——
α
0
=
P = (x, y)
V₁ + [y' (u)]² du.
BQ = (1, 1)
(1, 0)
X](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf1a092e-bde2-4aac-9f0c-dc6ade47f653%2F0d25c068-6de3-48ec-ad5d-89a5a586e709%2F5ee8yw_processed.png&w=3840&q=75)
![(d) Using separation of variables and the initial conditions x = 0 and w= dy/dx = 0 when
t = 0, show that
(7)
1/[(1 − x)-B/α – (1 − x)²/a].
(e) For a > B-that is, the pursuing vessel A travels faster than the pursued vessel B-use
equation (7) and the initial conditions x = 0 and y = 0 when t = 0, to derive the curve
of pursuit
y =
dy
dx
y =
=W=
(1-x) 1+B/a
1 + ß/a
(1-x)¹-B/a7 αβ
1-B/a
(f) Find the location where vessel intercepts vessel B if a > B.
(g) Show that if a = B, then the curve of pursuit is given by
+
- [ (1 − x)² – 1 ] – In ( 1 − x)
-
-
212
Will vessel A ever reach vessel B?
-x)}.
q² B²](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf1a092e-bde2-4aac-9f0c-dc6ade47f653%2F0d25c068-6de3-48ec-ad5d-89a5a586e709%2Flrm5n5_processed.png&w=3840&q=75)
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