Thomas Calculus 13th [Solutions]

956 Chapter 13 Vector-Valued Functions and Motion in Space
14. By Exercise 14 in Section 13.4, T cost
i sin t j and N sin t i cost
j so that
i j k
B T N cos t sin t 0 k . Also, v
2
3cos t sin t i
2
3sin t cost
j
sin t cost
0
a d 2 2 2 2
3cos sin d d a
t t i 3sin t cost j d d 3cos t sin t i d d 3sin t cost
dt dt dt dt dt dt dt
j
2
3cos t sin t
2
3sin t cost
0
d 2 2
3cos t sin t d 3sin t cost
dt
dt
0 0 0
d d 2 2
3cos t sin t d d 3sin t cost
dt dt dt dt
0
15. By Exercise 15 in Section 13.4, T v sech t tanh t
v a
i a
j and N tanh t sech t
a
i a
j so that
i j k
B T N sech t tanh t 0 k . Also, sinh t 1 cosh t d a
v i j a j 1 sinh t j so
a
a
a a a dt 2
a a
tanh t sech t 0
a a
1 sinh t 0
a
that 0 1 cosh t 0 0 0
a a
0 1 sinh t 0
2
a a
16. By Exercise 16 in Section 13.4, T 1 tanh t i 1 j 1 sech t
2 2 2
k and N sech t i tanh t k so that
i j k
B T N 1 tanh t 1 1 sech t 1 tanh t i 1 j 1 sech t
2 2 2 2 2 2
k . Also, v sinh t i cosh t j k
sech t 0 tanh t
i j k
a cosh t i sinh t j
d a
dt
sinh t i cosh t j and v a sinh t cosh t 1
cosh t sinh t 0
sinh t i cosh t j
2
cosh t
2
sinh t k sinh t i cosh t j k v
2
a
2
sinh t
2
cosh t 1. Thus
sinh t cosh t 1
cosh t sinh t 0
sinh t cosh t 0 1 1 .
2 2 2 2 2
sinh t cosh t 1 sinh t cosh t 1 2cosh t
17. Yes. If the car is moving along a curved path, then
2
0 and aN k | v | 0 a aT T aN
N 0.
18. v constant a d
T v 0 a a
dt
N N is orthogonal to T the acceleration is normal to the path
19. a v a T a 0 d
T v 0 v is constant
dt
Copyright
2014 Pearson Education, Inc.