Trajectory of an airdropped bomb

[LilFly]

Homework Statement

I want to derive an equation for the place where airdropped bomb is going to hit the ground.

Known data at the moment of drop:
1) position [x,y,z] also and the pitch angle (between the airplane's axis and the ground - we assume that Earth is flat)
2) linear and angular velocities [v_x,v_y,v_z] and [ω_x,ω_y,ω_z]
3) accelerations of the plane [a_x,a_y,a_z]
4) velocity of wind [w_x,w_y,w_z]
5) mass of the bomb m
6) drag coefficient of the bomb Cd which we assume to be constant in all directions
7) g and the air density ρ

Angular velocities cause only additional acceleration of the bomb (as the bomb is attached at some distance from the center of gravity of and aircraft)

Homework Equations

Drag: D=Cd*S/2*ρ*v², S is also given

The Attempt at a Solution

I am pretty sure that what I have to do is to write equations of motion.
First of all I find the net velocity -> sum of the linear velocity and wind velocity.
However what is my problem is how to deal with this additional acceleration caused by the angular velocity? And should I take anything else into consideration?

For example I would start it for z axis as follows
$mz'' = -Q - D_z + ma_z+...$ is this correct and if yes what to do with angular velocities?

Please, do help me :)

 

[mfb]

As long as it is attached to the airplane, the airplane will set its velocity. Afterwards, the airplane does not matter any more. For the point where the bomb gets dropped: yeah, in principle a rotation of the plane would influence the velocity a bit, but I think you can neglect this. Approximation (6) is much worse I guess. Therefore, linear velocities should be fine.

 

[LilFly]

I am also taking care of the 6th approximation, I've given it just not to mess up the picture. Let say bomb is not stiffly attached but has some freedom of movement then how can I include this? (Angular velocities cause only additional acceleration of the bomb (as the bomb is attached at some distance from the center of gravity of and aircraft))

 

[mfb]

Some freedom of movement relative to the airplane, while still attached? Then you'll need a mechanical model of this attachment mechanism.

What is the scope of this problem? Assuming you don't want to drop actual bombs, why do you care about those details?

 

[LilFly]

I am designing an UAV (for competition) which has to drop a package near the given location. I haven't decided yet on how to attach the 'bomb' however I know that it is going to be some kind of a 'hook' so it's not going to be stiff and since at the moment of drop aircraft is going to have big angular velocity I want to include it in my mathematical model. At this point we can introduce some kind of a constant which simulates the hook and the distance from the center of gravity and still I want to somehow include this in equations of motions and I don't know how

 

[mfb]

Ah, well. It all boils down to finding the current position and velocity of the package at release time, nothing else matters. How exactly this is determined will depend on the aircraft and hook design.

 

[LilFly]

Ok. Even if the 'bomb' is stiffly attached to the airplane but the airplane has the angular velocity (and since the bomb is not attached in the centre of gravity), the bomb will gain extra acceleration due to centrifugal force. And I just want to know how to deal with it, given the distance from the bomb's and airplane's centres of gravity.

 

[a_potato]

As long as it is attached to the airplane, the airplane will set its velocity. Afterwards, the airplane does not matter any more. For the point where the bomb gets dropped: yeah, in principle a rotation of the plane would influence the velocity a bit, but I think you can neglect this. Approximation (6) is much worse I guess. Therefore, linear velocities should be fine.

I suspect MFN is right and the assumptions in drag coefficient and it's impact on the final position is probably dominant over issues of initial angular momentum

 

[LilFly]

As I said, I am taking care of drag on my own. And I am not asking about how big the impact is going to be (but as the angular velocity is going to be big the effect will be noticeable), but how to include it in equations of motion if I know it's position in comparison to the airplane's center of gravity.

 

[haruspex]

Ok. Even if the 'bomb' is stiffly attached to the airplane but the airplane has the angular velocity (and since the bomb is not attached in the centre of gravity), the bomb will gain extra acceleration due to centrifugal force

Not so. As has been posted, all that matters are the velocity and position of the bomb at the instant of release. The radial acceleration of the aircraft (i.e. the curved nature of its path) only comes into it as part of calculating that velocity. The 'angular acceleration' of the aircraft doesn't seem to come into it at all.
Going back to the original problem, a rigid tether of any length will seriously affect the movement of the aircraft, and may be hard to achieve. For a cable, you'd need to compute the shape of the cable induced by the drag on that.