MATH 231
Group Work
Quiz Related
1. Compute
T
(
t
),
N
(
t
),
B
(
t
) and
κ
(
t
) for the following curves:
(a)
r
(
t
) =
t
3
j

4
t
k
(b)
r
(
t
) =
1
2
sin
t
i
+
1
2
cos
t
j
+
1
2
t
k
Answers:
(a) There was a terribly cruel typo on this problem which makes doing this extremely ugly. The
problem should read
r
(
t
) =
t
3
j

4
k
. With that, it makes the problem substantially easier. Then
T
(
t
) =
h
0
,
1
,
0
i
,
N
(
t
) =
h
0
,
0
,
0
i
,
B
(
t
) =
h
0
,
0
,
0
i
, and
κ
(
t
) = 0.
(b)
T
(
t
) =
√
2
h
cos
t,

sin
t,
1
i
,
N
(
t
) =
h
sin
t,

cos
t,
0
i
,
B
(
t
) =
±
cos
t,

sin
t,

√
2
²
, and
κ
(
t
) = 4.
2. A gun is ﬁred with angle of elevation 30
◦
. What is the muzzle speed if the maximum height of a shell
is 500 m?
Answer:
We have two equations:
y
=
v
0
sin(
α
)
t

9
.
8
2
t
2
, y
0
=
v
0
sin
α

9
.
8
t
We know that, at the maximum height,
y
= 500 and
y
0
= 0, so we have two equations with two
unknowns (seeing as
α
= 30
◦
). Solving these equations for
v
0
, we ﬁnd that
v
0
≈
198 m/s.
3. At what point does the curve have maximum curvature?