基于割线法的Matlab 3D抛体：求解落至x轴的抛射角b

Alright, let's break down how to use the secant method to find that launch angle ( b ) (the angle between the positive x-axis and negative y-axis) that gets your projectile to land right on the x-axis, starting from an initial height ( z=1.4,\text{m} ). I'll walk you through this step by step, tailored to your given parameters:

1. Define Your Target Function First #

Our end goal is to find a ( b ) where the projectile's final z-coordinate hits 0. So we'll define a target function ( f(b) ):

( f(b) = z_{\text{final}}(b) )

Here, ( z_{\text{final}}(b) ) is the z-coordinate of the projectile when it finishes its trajectory (we need this value to equal 0).

To recap your fixed parameters:

• Horizontal launch angle ( a=30^\circ )
• Initial velocity ( v_0=25,\text{m/s} )
• Air drag coefficient ( c=0.070 )
• Height-dependent wind speed: ( a(z)=7+0.35z )
• Initial height ( z_0=1.4,\text{m} )

2. How the Secant Method Works (In Plain Terms) #

The secant method is perfect here because it doesn't require calculating derivatives (which would be a huge pain with your coupled, wind-dependent motion equations). It works by:

1. Starting with two initial guesses for ( b )
2. Drawing a straight line (secant) between the two points on the ( f(b) ) curve
3. Using where that line crosses the ( f(b)=0 ) axis as your next guess
4. Repeating until your guess is accurate enough

The formal iteration formula you'll use is:

b_{n+1} = b_n - f(b_n) * (b_n - b_{n-1}) / (f(b_n) - f(b_{n-1}))

3. Step-by-Step Implementation #

Step 1: Pick Initial Guesses #

Start with two reasonable initial angles ( b_0 ) and ( b_1 ). For example:

• First, calculate a rough angle ignoring drag and wind (using basic projectile motion) to get ( b_0 )
• Pick a second angle ( b_1 ) slightly larger or smaller (e.g., if ( b_0=10^\circ ), try ( b_1=15^\circ ))
• Critical note: Make sure ( f(b_0) ) and ( f(b_1) ) have opposite signs—this guarantees there's a root between them.

Step 2: Calculate Initial Function Values #

For each initial angle, solve the projectile's motion equations (you'll need numerical integration like Runge-Kutta here, since the wind and drag make it non-analytical) to find the final z-coordinate. That gives you ( f(b_0) ) and ( f(b_1) ).

Step 3: Iterate Until Convergence #

Plug your values into the secant formula to get ( b_2 ), then calculate ( f(b_2) ). Repeat this process until one of these conditions is met:

• The difference between consecutive angles is tiny: ( |b_{n+1} - b_n| < \epsilon ) (use something like ( 10^{-6},\text{rad} ) or ( 0.0001^\circ ) for precision)
• The final z-coordinate is practically 0: ( |f(b_{n+1})| < \delta ) (e.g., ( 0.001,\text{m} ))

Step 4: Verify Your Result #

Once you have a converged ( b ), plug it back into the motion equations to simulate the trajectory. Double-check that the projectile lands on the x-axis (z ≈ 0) to make sure you didn't hit a false root.

4. Key Tips to Avoid Headaches #

• Stick to consistent units: Decide whether you're using radians or degrees for angles, and keep all length/speed units matched (meters and seconds here).
• Numerical integration accuracy: Use a reliable method like 4th-order Runge-Kutta for solving the motion equations—this ensures your ( f(b) ) calculations are precise enough for the secant method to converge.
• If it doesn't converge: Try adjusting your initial guesses. If your first two angles are too close or both give ( f(b) ) with the same sign, the secant method might wander off instead of converging.

内容的提问来源于stack exchange，提问作者michelle