Confusing projectile motion problem?

[bronxbomber91]

Homework Statement

You are designing software for an anti-aircraft system. Your guns have a maximum horizontal range R, and can be fired at any angle (call it "A") above the horizontal. Enemy aircraft are spotted at time t=0, flying at altitude H, distance D, horizontal velocity V toward a point directly above your guns. You are a physicist who understands projectile motion, and what it means to "hit" a target. The following questions explore how you figure out when and in what direction to fire the guns.

a) Make a labeled sketch of the graph (already done)
b) What is the highest H your gun can reach?
c) Write the two equations for the height and horizontal distance of the plane from the guns as a function of time, t, since the time (t=0) it was sighted.
d) Should you program the guns to fire as soon as the plane is sighted? Or should you calculate when to fire and at what angle to the horizontal?
e) In the case the plane is flying low enough that your guns can hit the plane at some angle, at what angle should they be fired to hit it at the highest point of the shell's trajectory?
f) How soon after sighting should they be fired?

Homework Equations

The Attempt at a Solution

a) Our teacher said that our group's sketch was fine so that isn't a problem.

b) My guess was H = -(initial velocity)^2/2g, but I have a feeling that's incorrect.

c) For height, I put H(t) = H, because isn't the plane's altitude constant? For horizontal distance, I put D(t) = D - Vt. This equation is the only one i have any confidence in.

d,e,f) Honestly I'm completely stumped, as was the rest of my class. Any help would be greatly appreciated.

 

[redargon]

ok, for b) you're almost correct, but does a negative answer make sense? what would cancel out that negative?

I think you answer for c) makes sense.

d) if we think about it, if the plane altitude is the same as the guns maximum height, then you could only hit the plane when it was directly overhead. but you'd have to fire at a time slightly before the plane got to that position, to take into account the time the bullet takes to get to that position. There hasn't been any mention of bullet speed, but we could just call it Vinitial, as you did for b).

that's about as much as i got for you for the moment.

 

[tomcue]

I would recommend taking a step back and breaking down the problem into smaller parts. First, let's consider the maximum horizontal range R of your guns and the altitude H of the enemy aircraft. If the altitude of the aircraft is greater than the maximum range of your guns, then it is not possible to hit the aircraft. Therefore, the highest H that your gun can reach is equal to the maximum range R of your guns.

Next, let's consider the equations for height and horizontal distance of the plane from the guns. For the height, you are correct in saying that it is constant since the altitude of the plane is not changing. However, for the horizontal distance, the equation D(t) = D - Vt assumes that the plane is flying in a straight line at a constant velocity. This may not always be the case, so it would be more accurate to use the equation D(t) = D + Vxt, where Vx is the horizontal component of the plane's velocity.

Now, should you program the guns to fire as soon as the plane is sighted? It may seem like the logical choice, but it may not always be the most effective. If the plane is flying at a constant altitude and velocity, then yes, the guns should be fired as soon as the plane is sighted. However, if the plane is flying in a curved path or changing altitude, then it would be more effective to calculate when and at what angle to fire the guns in order to hit the plane at its highest point. This would require more complex calculations, but it would increase the chances of hitting the target.

In the case that the plane is flying low enough for your guns to hit it at some angle, the angle at which the guns should be fired to hit the highest point of the shell's trajectory would depend on the initial velocity and angle of the plane. This would require further calculations and may not always be possible to determine without more information.

As for how soon after sighting the plane should the guns be fired, it would depend on the speed of the guns and the distance to the target. The faster the guns and the closer the target, the sooner they should be fired. However, this would also require taking into account the reaction time of the gun operator and any delays in the firing mechanism.

In summary, while the problem may seem confusing at first, it can be broken down into smaller parts and solved using equations and calculations. As a