Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

CHAPTER 13 PROBLEMS PLUS 0 181
The projectile hits the ground when (v sina)t- ~gt 2 = 0 => t = 2 ; sin a, so the distance traveled by the projectile is
(2v/g)sin o 1 (211/g)sin
L(a) = lv(t)l dt = .9
1 0 0
v ) 2 v2
( t - - sin a + 2 cos 2 a dt
.9 9
t - (v /g) sino
=g 2
[
+ [(vl g) cosa]2 ln (t-~sino +
2 g
[using Formula 2 I in the Table of lntegrals]
g
=
[v.
2 (~.rna)' + (~ -gsma oo.o )' + (~'"'")' m ( ~ •ina+ (~ •mo)' + (~=a Y)
+~sin o
g (~ •ina )' + (~=o )'- (~oooo )' m( -~•ina+ (~•ina )' + (~=o Y)]
2
= g_ [~sino · ~+ v
cos 2 2
a ln(~ sino+~) +~ sino·~ - v cos 2 a ln(- ~ sino+~)]
2 g g g2 g g g g g2 g g
v 2 . + v 2 2 1n( (vl g)sina+v/g) v 2 . + v 2 2
1 (1+sino)
= - stno -cos a = -sma -cos an
g 2f1 - (vI g) sin a +vI g g 2g 1 - sin a
We want to maximize L( a) for 0 s; a s; 1r / 2.
v
2
v
2
[ 2 1 - sin a 2 cos a . ( 1 + sin a)]
L'(a) =-coso+ - cos a · . · . 2 1
2 - 2cosa smo ln .
g g +sma (1-smo) . 1 -sm o
v 2 • v 2 [
= - cos a + - cos 2 . ( 1 + sin
2
a)]
2 a ·-- - 2 cos a sm a ln .
g g coso , 1 -sma
v 2 v 2 [ • (l+sina)] v 2 . = - COSO! [ 2- SIDQ . • ln (1+ sina . ) ]
9 g 1 -SlDO! 9 1 -SJDO! ,
= - COSO!+- COSO! 1-SlDO! ln
L( a ) has critical points for 0 < a < 1r 12 when L' (a) = 0 => 2 - sin a ln ( ~ + s~ a) = 0 [since cos a =1= 0].
- SIDQ
Solving by graphing (or using a CAS) gives a~ 0.9855. Compare values at the critical point and the endpoints:
L(O) = 0, L( 1r 12) = v 2 I g, and £(0.9855) ~ 1.20v 2 I g. Thus the distance traveled by the projectile is maximized
for a ~ 0.9855 or ~ 56°.
7. We can write the vector equation as r (t) = ·at 2 + bt + c where a= (a1, a2, a a}, b = (b1 , b2 , ba }, and c = (c1, c2, c3).
Then r ' (t) = 2t a+ b which says that each tangent vector is the sum of a scalar multiple of a and the vector b . Thus the
tangent vectors are all parallel to the plane determined by a and b so the curve must be parallel to this plane. [Here we assume
that a and bare nonparallel. Otherwise the tangent vectors are all parallel and the curve lies along a single line.] A normal
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