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4 members have voted

Wizard
Joined: Oct 14, 2009
• Posts: 22244
November 15th, 2020 at 7:42:47 PM permalink
This one is definitely worth of it's own thread. It pushes against the limits of my math ability.

An airplane is directly 8 miles overhead from you, traveling at a speed of 600 mph. The plane maintains the same altitude and same direction. At that moment, you fire surface-to-air missile at it. The missile always travels at a speed of 2,000 mph and always points directly at the airplane.

How far away will the wreckage fall, assuming it will fall directly below the point of impact?

For full credit, I want to see an expression for the answer and solution.
Last edited by: Wizard on Nov 15, 2020
It's not whether you win or lose; it's whether or not you had a good bet.
DogHand
Joined: Sep 24, 2011
• Posts: 397
November 15th, 2020 at 8:24:42 PM permalink
Quote: Wizard

This one is definitely worth of it's own thread. It pushes against the limits of my math ability.

An airplane is directly 8 miles overhead from you, traveling at a speed of 600 mph. At that moment, you fire surface-to-air missile at it. The missile always travels at a speed of 2,000 mph and always points directly at the airplane.

How far away will the wreckage fall, assuming it will fall directly below the point of impact?

For full credit, I want to see an expression for the answer and solution.

Since you didn't specify the airplane's flight vector, I'll assume it is flying vertically away from you. Thus, the missile will strike the plane at a time of t = 8 miles/ (2000 - 600) mph = 1/175 hrs = 20.57... seconds, and the wreckage will fall right on top of you, killing you.

Dog Hand
Wizard
Joined: Oct 14, 2009
• Posts: 22244
November 15th, 2020 at 9:01:03 PM permalink
Quote: DogHand

Since you didn't specify the airplane's flight vector, I'll assume it is flying vertically away from you. Thus, the missile will strike the plane at a time of t = 8 miles/ (2000 - 600) mph = 1/175 hrs = 20.57... seconds, and the wreckage will fall right on top of you, killing you.

Dog Hand

That is pretty funny. I added to the wording that the plane maintains the same altitude and direction.
It's not whether you win or lose; it's whether or not you had a good bet.
ThatDonGuy
Joined: Jun 22, 2011
• Posts: 4390
November 16th, 2020 at 10:00:26 AM permalink
Quote: Wizard

How far away will the wreckage fall, assuming it will fall directly below the point of impact?

It isn't that hard to determine how far in terms of horizintal distance the airplane would travel.

8 miles = about 12.875 km
Assume the acceleration due to gravity is a constant 9.81 m/sec2
The time it takes to fall is t = sqrt(2 x 12,875 / 9.81) = about 51.23 seconds
The horizontal distance = 51.23 / 3600 hours x 600 miles/hour = about 8.54 miles.

It's a little bit above my normal math pay grade; I haven't worked with DEs in decades. Here's what I have so far:

Let xt and yt be the missile's location at time t in seconds; all distances are in miles
x0 = y0 = 0
At time t, the airplane's location is (600 t/3600, 8), or (t/6, 8)

The angle at time t, measured from the positive-x direction toward the positive-y direction, is Anglet = tan-1((8 - yt) / (t/6 - xt)
The missile's velocity = 2000/3600 miles/second = 5/9
δyt/dt = 5/9 sin(Anglet) = 5/9 ((8 - yt) / sqrt((8 - yt)^2 + (t/6 - xt)^2)))
δxt/dt = 5/9 cos(Anglet) = 5/9 ((t/6 - xt) / sqrt((8 - yt)^2 + (t/6 - xt)^2)))

Wizard
Joined: Oct 14, 2009
• Posts: 22244
November 16th, 2020 at 10:40:26 AM permalink
I just want to make it perfectly clear I'm not asking for the distance from the missile launcher to the point of impact in the sky. I am asking for the horizontal distance between the missile launcher and the point on the ground directly below the point of impact.

It would be easy to get a very good estimate of the answer in a spreadsheet. However, I'm looking for a mathematical solution.

Here is an integral you may need, which you could also easily look up.

1/sqrt(1+x2) dx = ln|x + sqrt(1+x2)|
It's not whether you win or lose; it's whether or not you had a good bet.
Wizard
Joined: Oct 14, 2009
• Posts: 22244
November 16th, 2020 at 1:19:47 PM permalink
Could it be I've stumped the math geniuses of the forum?
It's not whether you win or lose; it's whether or not you had a good bet.
unJon
Joined: Jul 1, 2018
• Posts: 1971
November 16th, 2020 at 1:26:37 PM permalink
Quote: Wizard

Could it be I've stumped the math geniuses of the forum?

This appears to be the same question as the greyhound and the hare, which I remember working out in Calc 2 in college. Have not had time to sit down with it.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
Wizard
Joined: Oct 14, 2009
• Posts: 22244
November 16th, 2020 at 1:32:51 PM permalink
Quote: unJon

This appears to be the same question as the greyhound and the hare, which I remember working out in Calc 2 in college. Have not had time to sit down with it.

I will allow for extra time with this one. I spent a good part of the weekend on it, to be honest.

The spoiler will reveal the path of the missile and will give away the approximate answer. Don't click it if you don't want to know.

It's not whether you win or lose; it's whether or not you had a good bet.
FleaStiff
Joined: Oct 19, 2009
• Posts: 14484
November 16th, 2020 at 8:53:18 PM permalink
i doubt any of the scatter pattern would be diretly below impact point, else airshows would be different.

Predictable? One of the first pilots killed when Japanese fighter pilots dove into parachuting Americans was a pilot who died from a .38 calibre pistol wound!
FleaStiff
Joined: Oct 19, 2009