where's the plane?

[BMAD]

Michelle, john, and Kiru are all outside at the time a Jet is observed in the sky. Kiru hears the Jet's sonic boom two seconds before Michelle and seven seconds before John. if you were to draw a line to connect their locations they would form a triangle. Michelle is 10000 feet from Kiru. Kiru is 15000 ft from John. And John is 6000 ft from Michelle. Find a formula to derive the possible locations of where the sonic boom originated.

sound travels 1100 ft per second.

[k-man]

I'm making a couple of simplifying assumptions here:

1) we're ignoring the curvature of the earth and the triangle KJM with sides 10000ft, 15000ft, 6000ft is planar.

2) the sonic boom is a momentary event that occurred and the sound from this event was moving with a constant speed of 1100ft/sec in all directions.

Let's define our coordinates in the following way (1/100 ft scale):

Kiru (K) is in the origin (0,0,0)

John (J) is 15,000 ft along the X axis (150,0,0)

Michelle (M) is on the XY plane. We can calculate her coordinates from the following 2 equations:

1) x²+y²=100²

2) (x-150)²+y²=60²

Solving these equations gives us the coordinates for M (289/3, sqrt(6479)/3, 0) or approximately (96.33, 26.33, 0)

Let r be the distance from the boom to the origin (K), then the distance to M is r+22 and the distance to J is r+77. We can now construct 3 equations describing 3 spheres centered in K, M and J and corresponding radii of r, r+22 and r+77.

1) x²+y²+z²=r²

2) (x-150)²+y²+z²=(r+77)²

3) (x-289/3)²+(y-sqrt(6479)/3)²+z²=(r+22)²

Solving these equations we get

x = (16571 - 154 r)/300

y = (24706 r - 506819) / (300 sqrt(6479))

z = +/- sqrt(-50259 r²+ 16141917 r - 565551724) / (5 sqrt(6479))

Now we need to find the range of r for which there are real solutions for z: (-50259 r² + 16141917 r - 565551724) > 0

This gives us a valid range for r:

(16141917-5 sqrt(5874609121953))/100518 < r < (16141917+5 sqrt(5874609121953))/100518, or approximately 40.0239 < r < 281.151

This range corresponds to a range of solutions for where a jet can be. If you know how high the jet was flying (z) then you can figure out the rest using the formulas above.

[bonanova]

Are you taking into account that a sonic boom sweeps along the ground at the speed of the plane that generates it? Or are we to assume the sound originates from a fixed point and propagates at its native speed?

[Rob_Gandy]

No clue about how to come up with a formula, but I did draw it out.

there are two answers.

1) I constructed the triangle KJB with the scale 1cm=1000ft.

2) I constructed two circles circle M and circle J with radii of 2.2cm and 7.7cm respectively.

3) The intersections of these two circles (points A and B) indicate where the sound 'is' when Kiru hears it or the locations that are 2.2 secs from Michelle and 7.7 secs from John.

4) Now the sound must originate from a point equidistant from K and A or K and B in other words on the perpendicular bisector of KA or KB.

5) I constructed KA, KB and their perpendicular bisectors (CD and EF) and marked the midpoints of KA and KB (M and N).

Now lets look at the point of origin associated with KA and its perpendicular bisector. (The other went off the page)

6) I constructed ray MA and ray JA by extending the radii of the two circles from 2).

7) Ray MA intersects CD at G and ray JA intersects CD at H.

8) I constructed the angle bisector of angle GAH (ray AL).

9) Ray AL intersects CD at O which should be one possible point of origin for the sonic boom.

If I had a larger sheet of paper steps 6-9 could be repeated to find the other point.

I didn't really think about the height of the plane though....oops

Edit:bad tags

Edited February 21, 2013 by Rob_Gandy

[BMAD]

Are you taking into account that a sonic boom sweeps along the ground at the speed of the plane that generates it? Or are we to assume the sound originates from a fixed point and propagates at its native speed?

for the sake of simplicity and hopefully so i can actually declare a right answer, let's assume it originates from a fixed point.