Thomas Calculus 13th [Solutions]

Section 13.2 Integrals of Vector Functions; Projectile Motion 935
dr
16. i j k dt ti t j tk C
dt
1 ; dr
(0) 0 0i 0j 0k C
dt
1 0 C1
0
dr
2 2 2
( ti t j tk);
r ti t j tk dt t
i
t
j
t
k C
dt
2 2 2 2;
(0) 10 10 10
2 2 2
0 0 0
2 2 2
i j k C i j k C i j k
2 2 2
2 10 10 10 2 10 10 10
t 10 t 10 t 10
2 2 2
r i j k
r i j k
dv
17. a 3 i j k v( t) 3 ti t j tk C 1;
the particle travels in the direction of the vector
dt
(4 1) i (1 2) j (4 3) k 3i j k (since it travels in a straight line), and at time t 0 it has speed 2
2 dr
6
2 2
9 1 1
1 t t t t
dt
11 11 11
v (0) 3 i j k C v ( ) 3 i j k
3 2 6 1 2 2 1 2
r ( t ) t t i t t j t 2 t k C
2 11 2 11 2 11
2 ; r(0) i 2j 3k C2
3 2 6 1 2 2 1 2 2 1 2
t t t t t t t t 2 t
2 11 2 11 2 11 2 11
r ( ) 1 i 2 j 3 k 3 i j k i 2 j 3 k
dv
dt
18. a 2 i j k v( t) 2 ti t j tk C 1;
the particle travels in the direction of the vector
(3 1) i 0 ( 1) j (3 2) k 2i j k (since it travels in a straight line), and at time t 0 it has speed 2
2 dr
4 2 2
4 1 1
1 t t t t
dt
6 6 6
v (0) 2 i j k C v ( ) 2 i j k
2 6 1 2 2 1 2
r ( t ) t t i t t j t 2 t k C
6 2 6 2 6
2 ; r(0) i j 2k C2
2 4 1 2 2 1 2 2 1 2
t t t t t t t t 2 t
6 2 6 2 6 2 6
r ( ) 1 i 1 j 2 k 2 i j k i j 2 k
19.
1000 m 21,000 m
x v0 t t t
1 km (840m/s)(cos60 )
( cos ) (21km) (840 m/s)(cos60 ) 50 seconds
20.
2
v
R 0
sin 2 and maximum R occurs when
g
2 2
v 0 (9.8)(24,500) m /s 490 m/s
2
v0
2
45 24.5 km (sin 90 )
9.8m/s
0
21. (a) t
(b)
(c)
2v
sin 2(500 m/s)(sin 45 )
g
2
9.8 m/s
72.2 seconds; R
2 2
0 (500 m/s)
2
v
g
sin 2 (sin 90 ) 25,510.2 m
9.8 m/s
5000 m
x ( v0 cos ) t 5000 m (500 m/s)(cos 45 ) t t 14.14 s; thus,
(500 m/s)(cos 45 )
1 2 1 2 2
y v0 t gt y
2 2
y
( sin ) (500 m/s)(sin 45 )(14.14s) (9.8 m/s ) (14.14s) 4020 m
2
( v0
sin )
max 2g
2
2(9.8m/s )
(500m/s)(sin 45 )
2
6378m
22.
1 2 1 2 2 2
y y0 v0 t gt y t t y t t
2 2
( sin ) 32ft (32ft/sec)(sin 30 ) (32ft/sec ) 32 16 16 ;
the ball hits the ground when
3
x v0 t x t
2
2
y 0 0 32 16t 16t t 1or t 2 t 2sec since t 0; thus,
( cos ) (32ft/sec)(cos30 ) 32 (2) 55.4ft
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