Missile Defense System - Collide Missiles

[kchurchi]

Homework Statement

You are designing a missile defense system that will shoot down incoming missiles that pass over a perimeter defense post. The set-up is shown below. An incoming missile passes directly above the defense base. Radar at the base can measure the height, h, and speed, v1, of the incoming missile. Your Patriot Rocket is set to fire at an angle of θ = 46.0 degrees from vertical. You design the Patriot Rocket so the magnitude of its acceleration is given by:

a2 = A e−bt

where A can be set on your Patriot Rocket as it is fired, and b = 0.10 s-1. The direction of your Patriot Rocket's vector acceleration stays at the same angle, θ, for the entire trip. If an incoming missile passes over the defense base at a height of 4.60 km and at a constant speed of 740.0 m/s (this means that v1 is constant), solve for the value of A your Patriot Rocket must have in order to hit the incoming missile. You will also need to enter results from intermediate steps of your calculation, including the time ∆t in between launch and impact, and the horizontal distance ∆x from the launch station to the impact position.

http://lon-capa.mines.edu/res/csm/csmphyslib/Mechanics/Kinematics/2D_Projectiles/MissileDefenseSystem.jpg

Part A) Find Δt
Part B) Find Δx
Part C) Find A

Homework Equations

a1x(t) = 0 m/s^2
v1x(t) = v1x(0) = v1
x1(t) = v1*t

a2x(t) = a2*sin$\theta$ = A*e^(-b*t)*sin$\theta$
v2x(t) = v2x(0) -(A/b)*e^(-b*t)*sin$\theta$
x2(t) = v2x(0)*t +(A/b^2)*e^(-b*t)*sin$\theta$

a2y(t) = a2*cos$\theta$ = A*e^(-b*t)*cos$\theta$
v2y(t) = v2y(0) -(A/b)*e^(-b*t)*cos$\theta$
y2(t) = v2y(0)*t +(A/b^2)*e^(-b*t)*cos$\theta$

The Attempt at a Solution

Part A) I know that when the missiles collide x2(t) = x1(t). I tried to use that relationship to find the change in time, but I am not sure what v2x(0) is (the initial velocity of missile 2 in the x-direction). I think as soon as I figure out part A I will be able to do the other parts. I am just a bit stuck.

 

[voko]

The picture is inaccessible.

 

[kchurchi]

Does this link work?

 

[azizlwl]

1. It is not a constant velocity. You have to find the displacement equation.
2. You have the vertical distance to cover before interception.
3. Assuming level flight of the missille, the distance covered by the missile is equal to distance traveled by the patriot, that is patriot horizontal component.

 

[Nickie]

I would like to provide some suggestions to approach this problem:

1. Start by drawing a diagram of the situation, labeling the relevant quantities such as height, speed, and angle.

2. Use the given equations for the motion of the missiles and the acceleration of the Patriot Rocket to set up a system of equations.

3. Since the missiles collide when x2(t) = x1(t), you can use this relationship to eliminate the time variable and solve for A.

4. To find v2x(0), you can use the fact that the missile is moving at a constant speed of 740.0 m/s and the angle θ = 46.0 degrees. This will give you the initial velocity in the x-direction.

5. Once you have solved for A, you can use the equations for the motion of the missiles to find the values of Δt and Δx.

6. Make sure to pay attention to units and use the appropriate units for each quantity in your calculations.

7. Finally, check your answer to see if it makes sense. Does the acceleration of the Patriot Rocket seem reasonable? Is the time and distance traveled by the missile and the Patriot Rocket realistic?

I hope these suggestions help you in solving this problem. Good luck!