shooting yourself down - more physics?

In article ,
"Rich S." wrote:

Smitty Two asked:

If you guys aren't tired of arithmetic, I'm wondering if any fighter
pilots have shot themselves down. Probably not, but imagine you're
flying straight and level at 1000 mph and fire a 500 mph projectile at
an enemy that's directly ahead but at a higher altitude. You miss him;
the bullet follows a parabolic path, returning to your altitude just as
you overtake it. Discounting air friction, at what angle was the bullet
fired?

Before you find yourself in the same trap as me, let me ask for some
clarification:

1. Since you stipulate a vacuum, are we flying above the Earth?
2. How high?
3. Are we flying at an absolute altitude, e.g. following an orbital path?
4. Is the plane flying level at the moment of firing with only the gun
pointed up at the target airplane?

Rich "Without a clue" S.

I think I'm already in that trap, but it wasn't my intention (or yours,
I know) for this to be a trick question based on ambiguity in the
phrasing.

But, sure, I'll address your concerns.

1. I didn't stipulate a vacuum. My engine and wing don't work very well
without air. I said, to simplify calculations, ignore the effects of
friction on the projectile. We are flying above the earth.

2. At an altitude at which acceleration due to gravity is 32 feet per
second per second. Assume this remains constant throughout.

3. With the exception of the airplanes and the guns, this is a
pre-Columbian question. The earth is flat. Or, if you prefer, we're
flying tangentially to its curved surface.

4. The plane is flying level, as I believe I stated.

"Smitty Two" wrote in message
news
I think I'm already in that trap, but it wasn't my intention (or yours,
I know) for this to be a trick question based on ambiguity in the
phrasing.

But, sure, I'll address your concerns.

1. I didn't stipulate a vacuum. My engine and wing don't work very well
without air. I said, to simplify calculations, ignore the effects of
friction on the projectile. We are flying above the earth.

2. At an altitude at which acceleration due to gravity is 32 feet per
second per second. Assume this remains constant throughout.

3. With the exception of the airplanes and the guns, this is a
pre-Columbian question. The earth is flat. Or, if you prefer, we're
flying tangentially to its curved surface.

4. The plane is flying level, as I believe I stated.

Kewl! That makes it easy!

I don't know.

Rich S.

Smitty Two wrote:

1. I didn't stipulate a vacuum. My engine and wing don't work very well
without air. I said, to simplify calculations, ignore the effects of
friction on the projectile. We are flying above the earth.

You said discounting air friction. This is key to the problem. No air
friction would be the physical equivalent of being in a vacuum.

If there were no air friction, the projectile would continue ahead with
a forward speed of 1500 mph - its own 500 mph from its spent propellant
charge, and the 1000 mph imparted to it from the airframe.

Air friction, however, acts on the projectile from the moment it departs
the muzzle, and acts to slow the projectile in a continuous manner. The
aircraft continues to maintain its 1000 mph since it has a propulsive
device that is assumed to continue to operate. Air friction (along with
gravity) are the forces that conspire to ALLOW the aircraft to overtake
its projectile and contribute to shooting itself down.

So... are we or are we not including air friction? If we are, I am not
going to hazard a guess.. my math gland atrophied long ago after I
passed calculus. If we are not, then the problem is unsolvable in level
flight on earth.

Dave

In article t,
Dave S wrote:

Smitty Two wrote:

1. I didn't stipulate a vacuum. My engine and wing don't work very well
without air. I said, to simplify calculations, ignore the effects of
friction on the projectile. We are flying above the earth.

You said discounting air friction. This is key to the problem. No air
friction would be the physical equivalent of being in a vacuum.

If there were no air friction, the projectile would continue ahead with
a forward speed of 1500 mph - its own 500 mph from its spent propellant
charge, and the 1000 mph imparted to it from the airframe.

Air friction, however, acts on the projectile from the moment it departs
the muzzle, and acts to slow the projectile in a continuous manner. The
aircraft continues to maintain its 1000 mph since it has a propulsive
device that is assumed to continue to operate. Air friction (along with
gravity) are the forces that conspire to ALLOW the aircraft to overtake
its projectile and contribute to shooting itself down.

So... are we or are we not including air friction? If we are, I am not
going to hazard a guess.. my math gland atrophied long ago after I
passed calculus. If we are not, then the problem is unsolvable in level
flight on earth.

Dave

It's a hypothetical question, of course. I think I stated it fairly
clearly. Bill has already answered it correctly, with a second from
Alex. My calculus skills have also eroded over the years, which is one
reason I wanted to discount friction. But, I'd be curious to learn what
the real world answer would be, including friction, which of course is
dependent on a whole host of factors. (Of course, by the time the plane
caught up with the bullet in the real world, the friction would likely
have reduced its speed to something *relatively* harmless.)

Now, for some practical math, I hope one of these WW1 guys around here
can help me not shoot the prop off my RV when I put the .50 on the front.

Smitty Two wrote:
In article t,
Dave S wrote:

Smitty Two wrote:

1. I didn't stipulate a vacuum. My engine and wing don't work very well
without air. I said, to simplify calculations, ignore the effects of
friction on the projectile. We are flying above the earth.

You said discounting air friction. This is key to the problem. No air
friction would be the physical equivalent of being in a vacuum.

If there were no air friction, the projectile would continue ahead with
a forward speed of 1500 mph - its own 500 mph from its spent propellant
charge, and the 1000 mph imparted to it from the airframe.

...

So... are we or are we not including air friction? If we are, I am not
going to hazard a guess.. my math gland atrophied long ago after I
passed calculus. If we are not, then the problem is unsolvable in level
flight on earth.



Neglecting airfriction, the trajectory relative to the aircraft,
of the bullets fired from that aricraft will be the same
as the trajectory of bullets fired from a fixed position,
relative to that fixed position. As OP noted, the gunner hits
his own aircraft by firing staight up.

No Calculus required to reach that conclusion.


It's a hypothetical question, of course. I think I stated it fairly
clearly. Bill has already answered it correctly, with a second from
Alex. My calculus skills have also eroded over the years, which is one
reason I wanted to discount friction. But, I'd be curious to learn what
the real world answer would be, including friction, which of course is
dependent on a whole host of factors. (Of course, by the time the plane
caught up with the bullet in the real world, the friction would likely
have reduced its speed to something *relatively* harmless.)

That all depends on the relative velocity. If he fires upward and
forward and then does a low yo-yo to intercept I expect the bullets
could come down hard enough to hurt.


--

FF

(snip)
... my math gland atrophied long ago after I passed calculus.
(snip)

That calculus was *nasty* stuff ...
and matrix algebra was *worse*

I haven't recovered either, after 35+ years!

Peter

Peter Dohm wrote:
(snip)
... my math gland atrophied long ago after I passed calculus.
(snip)

That calculus was *nasty* stuff ...
and matrix algebra was *worse*

I haven't recovered either, after 35+ years!

Peter


Geeze, a buncha light weights. The calculus series and linear
algebra you guys took were rough? Differential equations? Hah! Take a
course in imaginary variables.

Dan, U.S. Air Force, retired