---
title: Lecture 5 Homework
author: Aidan Sharpe
date: February 24th, 2025
geometry: margin=1in
---

### 1.1 What are the individual principles used to create a balanced weapon system?
The weapon system should be robust, simple, and predictable.

### 1.2 Name three of the measures of weapon effectiveness.
Weapon systems may be measured in terms of circular error probability ($\text{CEP}_\text{XX}$), probability of mission kill ($P_\text{MK}$), probability of guidance ($P_G$), and probability of hit, damage, or kill ($P_H$, $P_D$, $P_K$).

### 1.3 Is the ENU coordinate system in invariant?
No, because all three directions (East, North, and up) are dependent on where on the surface you are.

### 1.4 Assume only one illuminator and one launcher in the combat system, what two limitations of the combat system determine the depth of fire?
With only one illuminator, only one semi-active missle can be supported at a time. With only one launcher, the time between firings is limited by the the launchers minimum time between missile launches ($\Delta T_{L_\text{min}}$).

### 1.5 What are the two main philosophies in weapon scheduling? How are they different?
#### Quickdraw
Shoot early and often to maximize the number of shot opportunities.

#### Sharpshooter
Maximize the performance of every shot by trading firing rate for shot quality.


### 2.1 Delay between missile launches ($\Delta T_L$)
$$\Delta T_L = T_H \left(1 + \frac{v_t}{v_m}\right)$$

```python
import numpy as np
import matplotlib.pyplot as plt


MIN_DELAY = 5
T_HOMING = 8
V_MISSILE = 700


def launch_delay(t_homing, v_missile, v_target):
    return t_homing*(1 + v_target/v_missile)


def main():
    # Range of target speeds from 0 to 1200 (inclusive)
    v_target = np.arange(1200+1)

    # Delta T_L for each target speed
    delays = launch_delay(T_HOMING, V_MISSILE, v_target)

    plt.plot(v_target, delays, label=r"$\Delta T_L$")
    plt.hlines(0, v_target[-1], MIN_DELAY,
	      label=r"$\Delta T_{L_\text{min}}$", color='r')
    plt.legend()
    plt.savefig("launch-delay.png")
    plt.show()
```

![](./launch-delay.png)

Since the homing time is longer than the minimum time between missile launches, there is no such target speed where $\Delta T_{L_\text{Min}} = \Delta T_L$.

### 2.2 Depth of Fire

$$\text{TET} = \frac{\text{ROF} - R_\text{min}}{V_T}$$
$$\text{TOF}_1 = \frac{\text{ROF}}{V_M + V_T}$$
$$\text{DOF} = \frac{\text{TET} + \text{TOF}_1}{T_H} + 1$$


```python
import numpy as np
import matplotlib.pyplot as plt

TARGET_RANGE = 20E3
MIN_RANGE = 2E3
MIN_DELAY = 5
T_HOMING = 8
V_MISSILE = 700

def depth_of_fire(target_range, min_range, t_homing, v_target, v_missile):
    t_engagement = (target_range - min_range) / v_target
    t_flight = target_range / (v_missile + v_target)
    return (t_engagement + t_flight)/t_homing

def main():
    # Range of target speeds from 0 to 1200 (inclusive)
    v_target = np.arange(1200+1)
    
    dof = depth_of_fire(TARGET_RANGE, MIN_RANGE, T_HOMING, v_target, V_MISSILE)
    print("DOF for V_T=200:", depth_of_fire(TARGET_RANGE, \
    	 MIN_RANGE, T_HOMING, 200, V_MISSILE))

    plt.plot(v_target, dof, label="Depth of Fire")
    plt.savefig("depth-of-fire.png")
    plt.show()
```

![](./depth-of-fire.png)

DOF, where $V_T = 200$[m/s] is 14.0278 shots.