This project models low-level bombing with differential equations. This involves determining where a bomb lands when you drop it from a specific height and velocity, or when and where to release a bomb when you are given a specific, stationary target. The two ideas are Galileo's law of gravity and air friction due to the high speed of the bomb. They are combined with Newton's law, F=mA. Vectors play a role, because the air friction is an increasing function of speed, not velocity.

Figure 45.1: A Hypothetical Bomb Trajectory

We place a coordinate system with its origin at the target with the horizontal x-axis aligned so the plane has positive x-vleocity. We will use mks units, so the gravitational constant g=9.8.

We will model air friction by assuming that the force due to air friction is proportional to a power of the speed and acts in the direction opposite the velocity vector, |F_air|=ks^p, with the same direction as -V.

Figure 45.2: A Vector Force for Gravity and a Vector Force for Air Resistance

One of the two forces acting on the bomb as it falls is air resistance, F_air. This force acts to slow the bomb opposite to the direction in which the bomb tries to travel. It acts in an opposite direction to the velocity vector at any given time, as Figure 45.2 illustrates.

The Air's Force
1) What is the unit vector in the direction of V, the velocity vector? Express your answer in terms of u, v, and s.

2) What is the unit vector in the direction opposite V, or in the direction of F_air?

3) Using the results from the two previous exercises, show that

Verify that |F_air|=ks^p and that the direction of F_air is opposite the direction of V.

The other force acting on the bomb is due to the weight of the bomb due to gravity. The force due to gravity varies according to mass and has magnitude mg, where g is the acceleration due to gravity, approximately equal to 9.8 meters/second².

The Total Force
Now that we have the two forces acting on the bomb in vector form, compute the total force, F_total, as a vector sum, F_total=F_air+W. Express your answer in terms of u, v, m and g,

F=mA
You should be familiar with Newton's law F=mA from Chapter 10 of the text or from your physics course. This can be applied to our model, where F is the force vector, F_total the total force acting on the bomb, and A is the acceleration vector,

Now rewrite this equation in terms of u, v, m and g solving for A, ,

In order to use the AccDEsoln program to solve the equations, you must use the phase variable trick. This transforms the two second-order differential equations that you just derived into a system of four first-order differential equations.

A First-Order 4-D System
Write in terms of first derivatives and the phase variables.

These four equations can be plugged into the AccDEsol program; however you also need the initial conditions. A plane flying horizontally at height h, a horizontal distance j before the target at speed s_i gives:

You will also need a specific value of the nonlinear power p in the air resistance term. In the Bungee project we used p=7/5. Linear air resistance would be p=1. Another simple nonlinear choice is p=2. We would like you to try several choices of p. Each choice will require an associated constant of proportionality, k. Fortunately, we know the terminal velocity of the bombs from data at the testing center. When we drop these kinds of bombs vertically from a test tower, they speed up to approximately 89.4 meters per second and then continue to fall at that speed. Apparently, the air friction balances gravity at that speed.

Measuring k from Terminal Velocity
You can use simple algebra to compute the value of k. Since in a vertical drop, we have the equations

What does "terminal velocity" say about ? Solve this equation for k in terms of p and compute the specific values for p=1,7/5,2.

45.1 Significance of Vector Air Resistance

The vector nature of the air resistance means that a bomb released at high speed will have a big vector force opposite to its direction of motion. If the bomb is inclined downward, the resistance will therefore have an upward component. The bomb won't fall if this component of resistance equals its weight. Of course, the resistance slows the bomb down horizontally, so it does not fly along with the bomber for long. Once it slows down, weight exceeds the upward component of resistance and it falls faster and faster until the vertical resistance again builds up.

Problem 45.1 Low-Flying Bombs

Run some computer experiments with bombers flying 350 mph (be careful with units), dropping bombs from various initial heights. Use several values of the resistance power p and its associated k. Show that the flying effect of bombs is most important in hitting a target if you wish to bomb from a low level.

Figure 45.3: Force Components

Now that you have a good understanding of how this model works and what the equations mean both graphically and mathematically, it is time to put the computer to work.

Problem 45.2 Hitting the Target

You are flying at altitude h at velocity s_i and decide to drop a bomb. Where do you release the bomb so that it hits the target at (0,0)? Use several values of p. How much room for error do you have in releasing the bomb so that you will still hit within a meter of your target? How does varying your altitude and initial velocity affect the room for error that you have?

It is also interesting to follow the path of the bomb. This can be graphed with the computer's help. In addition, you can follow the descent by following the angle the tangent line to the bomb's path forms with the ground. This can tell you the angle at which the bomb hits, or if the bomb nearly "floats" when first released.

Problem 45.3

Determine the angle at impact of the bomb. How do changes in h and s_i affect this?

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