There is a vertical pole perpendicular to the horizontal plane. From point P on the plane, 2 projectiles are fired simultaneously at different velocities. The first projectile is fired at an angle of 30o and it hits the foot of the pole. The second projectile is fired at an angle of 60o and it hits the top of the pole
It is further known that the projectiles hit the pole at the same time.
Find the angle subtended by the pole from P.
Let the 30° projectile (P1) be fired at velocity v1, and the 60° projectile (P2) at velocity v2. The distance to the pole is d.
The x component of P1's velocity is v1*cos(30°) so it reaches the pole in d/(v1*cos(30°)) units of time. We're going to use seconds as the unit of time with velocities in feet per second and distances in feet.
The y component of P1's velocity is v1*sin(30°) Its height after t seconds is v1*sin(30°)*t-(g/2)*t^2.
This is desired to be zero for the second time, as it reaches the bottom of the pole:
v1*sin(30°)*t = (g/2)*t^2
v1*sin(30°) = (g/2)*t
t = v1*sin(30°) / (g/2)
t = 2*v1*sin(30°) / g
so d/(v1*cos(30°)) = 2*v1*sin(30°) / g
and d = 2*v1*sin(30°)*(v1*cos(30°)) / g
d = v1^2*sin(60°)/ g
The x component of P2's velocity is v2*cos(60°) so it reaches the top of the pole in d/(v2*cos(60°)) seconds.
The y component of P2's velocity is v2*sin(60°). Its height after t seconds is v2*sin(60°)*t-(g/2)*t^2.
So after d/(v2*cos(60°)) units of time its vertical location is v2*sin(60°)*d/(v2*cos(60°))-(g/2)*(d/(v2*cos(60°)))^2.
The ratio of the height of the top of the pole to the pole's distance is therefore
v2*sin(60°)*d/(v2*cos(60°))-(g/2)*(d/(v2*cos(60°)))^2
-----------------------------------------------------
d
= v2*sin(60°)/(v2*cos(60°))-(g/2)*d/(v2*cos(60°))^2
= v2*sin(60°)/(v2*cos(60°))-(g/2)*(v1^2*sin(60°)/g)/(v2*cos(60°))^2
= v2*sin(60°)/(v2*cos(60°))-(v1^2*sin(60°)/2)/(v2*cos(60°))^2
= v2*sin(60°)/(v2*cos(60°))-v1^2*sin(60°)/(2*(v2*cos(60°))^2)
The subtended angle is the arctan of this quotient.
A table for the angle of elevation in degrees of the top of the pole using the formula:
v1 \ v2 25 50 75 100 125 150 175 200 225 250 275 300
10 55.5 59.0 59.6 59.8 59.8 59.9 59.9 59.9 60.0 60.0 60.0 60.0
20 31.9 55.5 58.1 59.0 59.4 59.6 59.7 59.8 59.8 59.8 59.9 59.9
30 -37.3 47.9 55.5 57.6 58.5 59.0 59.3 59.4 59.6 59.6 59.7 59.8
40 -69.7 31.9 51.1 55.5 57.3 58.1 58.7 59.0 59.2 59.4 59.5 59.6
50 -79.1 0.0 43.9 52.4 55.5 57.0 57.8 58.4 58.7 59.0 59.2 59.3
60 -83.1 -37.3 31.9 47.9 53.1 55.5 56.8 57.6 58.1 58.5 58.8 59.0
70 -85.2 -59.0 12.6 41.5 49.9 53.6 55.5 56.7 57.4 57.9 58.3 58.6
80 -86.4 -69.7 -13.4 31.9 45.6 51.1 53.9 55.5 56.5 57.3 57.8 58.1
90 -87.2 -75.5 -37.3 18.2 39.8 47.9 51.9 54.1 55.5 56.4 57.1 57.6
100 -87.8 -79.1 -53.4 0.0 31.9 43.9 49.4 52.4 54.3 55.5 56.4 57.0
110 -88.2 -81.4 -63.4 -20.0 21.3 38.7 46.3 50.4 52.8 54.4 55.5 56.3
120 -88.5 -83.1 -69.7 -37.3 7.7 31.9 42.5 47.9 51.1 53.1 54.5 55.5
130 -88.7 -84.3 -73.9 -50.1 -8.0 23.3 37.8 45.0 49.1 51.6 53.4 54.6
140 -88.9 -85.2 -76.9 -59.0 -23.8 12.6 31.9 41.5 46.7 49.9 52.1 53.6
150 -89.1 -85.9 -79.1 -65.2 -37.3 0.0 24.7 37.2 43.9 47.9 50.6 52.4
When the velocities are equal, the two projectiles both fall at the bottom of the pole. If the 60° projectile has lower velocity than the 30° projectile it falls short of the pole. As the 60° projectile gets faster and faster relative to the 30° projectile, the subtended angle seems to approach 60°. Since what we are talking about is the angle that the projectile makes in the line of sight (it's assumed the pole extends that high), this makes sense, as a really fast projectile will get to look more and more like a straight line.
In fact the angle seems to depend solely on the ratio of the velocities.
velocity elevation
ratio angle
1.00 0.0
1.50 43.9
2.00 52.4
2.50 55.5
3.00 57.0
3.50 57.8
4.00 58.4
4.50 58.7
5.00 59.0
5.50 59.2
6.00 59.3
6.50 59.4
7.00 59.5
7.50 59.6
8.00 59.6
8.50 59.7
9.00 59.7
9.50 59.7
10.00 59.8
10.50 59.8
11.00 59.8
11.50 59.8
12.00 59.8
12.50 59.8
13.00 59.9
13.50 59.9
14.00 59.9
14.50 59.9
15.00 59.9
15.50 59.9
16.00 59.9
16.50 59.9
17.00 59.9
17.50 59.9
18.00 59.9
18.50 59.9
19.00 59.9
19.50 59.9
20.00 59.9
20.50 59.9
21.00 59.9
21.50 59.9
22.00 59.9
22.50 60.0
23.00 60.0
23.50 60.0
24.00 60.0
24.50 60.0
25.00 60.0
25.50 60.0
26.00 60.0
26.50 60.0
27.00 60.0
27.50 60.0
28.00 60.0
28.50 60.0
29.00 60.0
29.50 60.0
30.00 60.0
30.50 60.0
31.00 60.0
31.50 60.0
32.00 60.0
32.50 60.0
33.00 60.0
33.50 60.0
34.00 60.0
34.50 60.0
35.00 60.0
35.50 60.0
36.00 60.0
36.50 60.0
37.00 60.0
37.50 60.0
38.00 60.0
38.50 60.0
39.00 60.0
39.50 60.0
40.00 60.0
40.50 60.0
41.00 60.0
41.50 60.0
42.00 60.0
42.50 60.0
43.00 60.0
43.50 60.0
44.00 60.0
44.50 60.0
45.00 60.0
45.50 60.0
46.00 60.0
46.50 60.0
47.00 60.0
47.50 60.0
48.00 60.0
48.50 60.0
49.00 60.0
49.50 60.0
50.00 60.0
50.50 60.0
51.00 60.0
51.50 60.0
52.00 60.0
52.50 60.0
53.00 60.0
53.50 60.0
54.00 60.0
54.50 60.0
55.00 60.0
55.50 60.0
56.00 60.0
56.50 60.0
57.00 60.0
57.50 60.0
58.00 60.0
58.50 60.0
59.00 60.0
59.50 60.0
60.00 60.0
60.50 60.0
61.00 60.0
61.50 60.0
62.00 60.0
62.50 60.0
63.00 60.0
63.50 60.0
64.00 60.0
64.50 60.0
65.00 60.0
65.50 60.0
66.00 60.0
66.50 60.0
67.00 60.0
67.50 60.0
68.00 60.0
68.50 60.0
69.00 60.0
69.50 60.0
70.00 60.0
70.50 60.0
71.00 60.0
71.50 60.0
72.00 60.0
72.50 60.0
73.00 60.0
73.50 60.0
74.00 60.0
74.50 60.0
75.00 60.0
75.50 60.0
76.00 60.0
76.50 60.0
77.00 60.0
77.50 60.0
78.00 60.0
78.50 60.0
79.00 60.0
79.50 60.0
80.00 60.0
80.50 60.0
81.00 60.0
81.50 60.0
82.00 60.0
82.50 60.0
83.00 60.0
83.50 60.0
84.00 60.0
84.50 60.0
85.00 60.0
85.50 60.0
86.00 60.0
86.50 60.0
87.00 60.0
87.50 60.0
88.00 60.0
88.50 60.0
89.00 60.0
89.50 60.0
90.00 60.0
90.50 60.0
91.00 60.0
91.50 60.0
92.00 60.0
92.50 60.0
93.00 60.0
93.50 60.0
94.00 60.0
94.50 60.0
95.00 60.0
95.50 60.0
96.00 60.0
96.50 60.0
97.00 60.0
97.50 60.0
98.00 60.0
98.50 60.0
99.00 60.0
99.50 60.0
100.00 60.0
DefDbl A-Z
Dim crlf$, x(2), dr
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr(13) + Chr(10)
pi = Atn(1) * 4
dr = pi / 180
dist = 2
x(1) = dist / 2
k = Tan(30 * dr) / (2 * x(1))
a = k * x(1) ^ 2
Text1.Text = Text1.Text & " "
For v2 = 25 To 300 Step 25
Text1.Text = Text1.Text & mform(v2, "#####0")
Next v2
Text1.Text = Text1.Text & crlf
For v1 = 10 To 150 Step 10
Text1.Text = Text1.Text & mform(v1, "##0") & " "
For v2 = 25 To 300 Step 25
ratio = v2 * Sin(60 * dr) / (v2 * Cos(60 * dr)) - v1 ^ 2 * Sin(60 * dr) / (2 * (v2 * Cos(60 * dr)) ^ 2)
angle = Atn(ratio) / dr
Text1.Text = Text1.Text & mform(angle, "###0.0")
Next
Text1.Text = Text1.Text & crlf
Next
Text1.Text = Text1.Text & crlf
Text1.Text = Text1.Text & crlf
v1 = 1
For v2 = 1 To 100 Step 0.5
Text1.Text = Text1.Text & mform(v2, "#0.00") & " "
ratio = v2 * Sin(60 * dr) / (v2 * Cos(60 * dr)) - v1 ^ 2 * Sin(60 * dr) / (2 * (v2 * Cos(60 * dr)) ^ 2)
angle = Atn(ratio) / dr
Text1.Text = Text1.Text & mform(angle, "###0.0") & crlf
Next
Text1.Text = Text1.Text & crlf & " done"
End Sub
Function mform$(x, t$)
a$ = Format$(x, t$)
If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
mform$ = a$
End Function
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Posted by Charlie
on 2018-03-23 11:51:25 |
proposed solution
