Finding the Minimum Velocity for UFO Intercepting Missile (question)

[marcosvalle]

Hi!

I've just created this question for my undergraduated students that are going to apply for an engineering institute. I'm not quite sure if it's a nice question, so I'd like to know your opinions or suggestions about it (and yours solutions, of course!)

A flying saucer with mass 5m and radius R is at rest and preparing to destroy our beloved Earth. An aircraft is therefore sent to prevent it and shoots a missile with mass m and velocity v₁ at the same plane of the flying saucer. The UFO reacts and launches an intercepting missile of mass m and velocity v₂ that makes a 30^o angle with the direction of the first projectile. If both projectiles meet after a distance h from the launching point of the second missile and then keep moving together, find the minimal value of v₂ for which the aircraft will not hit the target.

PS.: Ignore friction and gravitational forces

Please check the attached picture.

Sorry for the bad english, hope you guys understand it! =]

Ty!

 

[russ_watters]

Welcome to PF!

How are you hoping your students will answer it? Can you show us?

 

[marcosvalle]

What I think it's interesting about this question is the fact that the UFO also recoils after shooting the missile. So, what I expect them to answer is something like:

1-) Adopt a reference frame and calculate how much time the UFO's missile takes to hit the other projectile.
2-) Using momentum conservation principle, find what distance does the flying saucer travel before the missiles meet.
3-) With a little bit more of cinematics, find the necessary angle that the system composed by the two missiles together will make with corrdinates axis after collision. This might be done after realizing that the minimum value for v2 occurs when the direction of the system tangencies the UFO's circunference.
4-) Using MCP once again, find v2.

What you think?

 

[CWatters]

You had better explain in the question that the UFO is subject to recoil. I imagined it "at rest" supported on three legs in a field :-)

 

[nasu]

Which direction is the gravity field in that picture? Are these objects in space, above the Earth?

 

[marcosvalle]

You had better explain in the question that the UFO is subject to recoil. I imagined it "at rest" supported on three legs in a field :-)

Ohh, that is a good point! I intended to say it was initially at rest, but still flying, not supported. May be that problem is due to bad translation, since I first wrote the statement of the question in my native language.

Thank you very much, I will try to correct it.

 

[marcosvalle]

Which direction is the gravity field in that picture? Are these objects in space, above the Earth?

The picture shows a top view of the situation, so the gravitational acceleration vector is pointing down (into the page). I do not think it would be necessary to use gravity in this question, since it is being ingored, though.

Still, it is also a good point and I will check to make it clear.

Thank you very much!

 

[marcosvalle]

When I tried solving this question, following the steps I put above, I got stuck in a system of two equations, involving trigonometric equations like:

cosθ = a + b*v₂

sinθ = c - d*sin(2θ),

where a,b are positive constants and c, d are functions of v₂. I am not really sure if these are exactly the equations I foud before, but they were pretty similar.

So, I tought about trying to indicate an approximation in the statement of the question, in order to simplify the math, but found nothing. Does anybody see an easy solution?

If I can't find any, I will modify the problem and ask for the expression that gives the minimal value of v₂, instead of the value itself.

Any other suggestion?

 

[nasu]

The picture shows a top view of the situation, so the gravitational acceleration vector is pointing down (into the page). I do not think it would be necessary to use gravity in this question, since it is being ingored, though.

Still, it is also a good point and I will check to make it clear.

Thank you very much!

Sorry, I did not see that part. (ignore gravity)

So I understand you mean a purely kinematic problem.
Why do you give this info about masses and the fact that they move together? Is this relevant to the problem?

And is the airplane's position determined?
Is there some missing information regarding the position of the airplane relative to the UFO, at the moment the projectile is launched?

 

[marcosvalle]

Sorry, I did not see that part. (ignore gravity)

So I understand you mean a purely kinematic problem.
Why do you give this info about masses and the fact that they move together? Is this relevant to the problem?

And is the airplane's position determined?
Is there some missing information regarding the position of the airplane relative to the UFO, at the moment the projectile is launched?

Yes, It is a purely kinematic problem. Masses are given so you can use momentum conservation when attempting to find the recoil velocity of the UFO and relation the angle of deviation of the system of projectiles and their velocity (note the missiles do not explode when they meet).

I do not believe the airplane's position is relevant, since you could set t=0 as the instant when second missile is launched. What matters is that the first missile moves in a given direction with a given velocity, from what we should be able to conclude when the UFO would launch it's own, except for the velocity of this new missile, which we do not know.

If I am still not clear, please tell me, ok? Thanks!