# Missile Defense System - Collide Missiles

• kchurchi
In summary, the Patriot Rocket needs to have an initial velocity of v2x(0) in order to hit the incoming missile.
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.

Last edited by a moderator:
The picture is inaccessible.

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.

Last edited:

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!

## 1. What is a missile defense system?

A missile defense system is a defensive technology designed to intercept and destroy incoming missiles before they reach their intended target. These systems use various methods such as missiles, lasers, and projectiles to neutralize the threat of incoming missiles.

## 2. How does a missile defense system work?

Missile defense systems use a combination of detection, tracking, and interception technologies to identify and eliminate incoming missiles. These systems rely on sophisticated radar and other sensors to detect and track the missiles, and then fire interceptor missiles to collide with and destroy the incoming threat.

## 3. What is the purpose of "colliding" missiles in a missile defense system?

The purpose of colliding missiles in a missile defense system is to physically destroy the incoming threat. By intercepting and colliding with the missile, the system can neutralize the threat and prevent it from reaching its intended target.

## 4. Can a missile defense system guarantee 100% success?

No, a missile defense system cannot guarantee 100% success. While these systems are highly advanced and effective, there is always a small chance that a missile can slip through the defenses. However, having a missile defense system greatly increases the chances of intercepting and destroying incoming missiles.

## 5. How effective is a missile defense system against different types of missiles?

The effectiveness of a missile defense system can vary depending on the type of missile being intercepted. These systems are generally more effective against short-range and medium-range ballistic missiles, while long-range missiles may pose a greater challenge. However, advancements in technology and continuous testing and development of these systems aim to improve their effectiveness against all types of missiles.

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