All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Science
Not enough information! (Posted on 2018-03-23) Difficulty: 2 of 5
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.

No Solution Yet Submitted by Danish Ahmed Khan    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution proposed solution | Comment 1 of 7
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


  Posted by Charlie on 2018-03-23 11:51:25
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (2)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (2)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2019 by Animus Pactum Consulting. All rights reserved. Privacy Information