Problem 1.16
[Difficulty: 3]
Given:
Long bow at range, R = 100 m. Maximum height of arrow is h = 10 m. Neglect air resistance.
Find:
Estimate of (a) speed, and (b) angle, of arrow leaving the bow.
Plot:
(a) release speed, and (b) angle, as a function of h
Solution:
Let
V
u i
v j
V
i
j)
0
0
0
=
+
=
+
0
0
0
±
±
(cos
±
sin
±
θ
θ
Σ
F
m
mg
y
dv
dt
=
= −
, so v = v
0
– gt, and
t
f
=
2t
v=0
= 2v
0
/g
Also,
mv
dv
dy
mg, v dv
g dy, 0
v
2
gh
0
2
= −
= −
−
= −
Thus
h
v
2g
0
2
=
(1)
Σ
F
m
du
dt
0, so u
u
const, and R
u t
2u v
g
0
0
f
0
0
x
=
=
=
=
=
=
(2)
From Eq. 1:
v
2gh
0
2
=
(3)
From Eq. 2:
u
gR
2v
gR
2
2gh
u
gR
8h
0
0
0
2
2
=
=
∴
=
Then
2
1
2
0
2
2
0
2
0
2
0
8
2
and
2
8
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
=
+
=
+
=
h
gR
gh
V
gh
h
gR
v
u
V
(4)
s
m
7
.
37
m
10
1
m
100
s
m
8
81
.
9
m
10
s
m
81
.
9
2
2
1
2
2
2
2
0
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
×
×
+
×
×
=
V
From Eq. 3:

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