- 01-math Derivation of Kinematic Velocity Vector - 02-math Hitting a Moving Target: The Math of Projectile Motion - 03-math Hitting Moving Targets in Games: Kinematic Projectile Interception Guide - 04-math Accelerated Target Kinematics: Projectile Aiming with Motion Equations - 05-physics Kinematic Analysis of Projectile Target Interception - 06-acceleration Determining Initial Velocity and Coincidence Time for Particles with Constant Acceleration - 07-continuation Extended Kinematic Analysis of Projectile Interception: Constant Velocity and Beyond - 08-equations Advanced Kinematic Formulations for Projectile Interception - 09-numerical-methods Numerical Methods for Projectile Interception: Solving the Quartic Equation
Full Link: 01-math.md
This document comprehensively derives the components of a kinematic vector ✈️ using detailed position and velocity differences 📏. It introduces vector notations to streamline complex calculations 🔄, ultimately simplifying to expressions involving square roots and position vectors ⭕. The derivation concludes with a final boxed vector result 📦.
Full Link: 02-math.md
This document delves into the mathematics of projectile motion in 3D environments 🎯. It outlines a method to determine the precise velocity components for a projectile to intercept a moving target 🚀. The setup involves initial positions and velocities of both entities 📍. Key equations are presented, including those for position matching and speed constraints ⏳. Solving these equations yields two possible solutions for velocity 💡. Practical steps for game implementation are included 🎮.
Full Link: 03-math.md
🎯 This comprehensive guide demystifies the calculations needed to make projectiles hit moving targets in your game. 🚀 It introduces key variables like positions, velocities, and speeds that are essential for accurate targeting. 📊 The core math involves aligning coordinates at the time of impact using three primary equations. 💡 Solving these equations yields two potential velocity solutions for the projectile, giving flexibility in trajectory choices. 🛠️ Step-by-step implementation advice helps integrate this into game engines effectively. ⚠️ Assumptions about constant target motion are highlighted, emphasizing the need for real-time recalculations in varied scenarios.
Full Link: 04-math.md
🤔 This document adapts projectile aiming calculations for targets that accelerate, using constant acceleration equations to match positions over time. 📐 It formulates three position-matching equations for coordinates x, y, z plus a speed constraint, incorporating initial positions, velocities, and accelerations. 🔢 Solving results in a complex quartic equation in time t, resolved numerically via iterative methods like Newton-Raphson or bisection. ⚡ Practical considerations include multiple solutions, computational intensity, real-time recalculation, and the possibility of no feasible hit. 🎯 Overall, it enables precise kinematic calculations in dynamic environments for accurate projectile targeting despite increased complexity.
Full Link: 05-physics.md
This white paper explores kinematic principles for intercepting moving targets with projectiles 🚀. It details mathematical frameworks for constant velocity scenarios, providing analytical solutions 🧮. The analysis extends to constant acceleration cases, necessitating numerical methods 🔢. Practical considerations and limitations in real-world applications are thoroughly discussed ⚠️. Appendices include explicit formulas for projectile velocities 📈.
Full Link: 06-acceleration.md
The problem explores two particles A and B moving in 3D space with constant acceleration vectors. 🔭 It formulates their position vectors using kinematic equations with initial conditions. 📐 The goal is to find particle B's initial velocity components and the coincidence time t*, subject to a velocity magnitude constraint. ⏱️ This requires solving a system of equations for the coincidence condition across three dimensions. 🧮 Component-wise derivations are provided for x, y, and z coordinates. ◰ Assumptions include known accelerations and initial states for both particles. 📋 If analytical solutions prove challenging, numerical approximation methods are suggested. 🔢
Full Link: 07-continuation.md
This paper extends kinematic interception analysis from constant velocity targets to constant acceleration scenarios 🚀. It reinterprets geometric interpretations and introduces time-centric approaches with abundant mathematical equations 📐. Derivations cover vector formulations, quadratic solutions, and quartic equations for advanced cases 🔄. Numerical methods like Newton-Raphson are recommended for solving complex cases 🧮. The conclusion emphasizes the comprehensive toolkit provided and hints at future extensions 🌌.
Full Link: 08-equations.md
📐 This document expands on kinematic frameworks for projectile interception by introducing new equations through coordinate transformations, parametric approaches, and energy principles. 🚀 For constant velocity scenarios, it uses rotated coordinate systems and parametric elimination to solve for interception times and velocities. 🔢 The constant acceleration case is analyzed with matrix formulations and detailed polynomial expansions for a comprehensive solution. ⚡ Energy-based constraints add a physical interpretation to the velocity calculations, enriching the mathematical exploration.
Full Link: 09-numerical-methods.md
This extended paper delves into numerical techniques for resolving the quartic equation arising from projectile interception under constant acceleration. 🚀 It comprehensively covers the Bisection Method with iterative midpoint calculations and error bounds. 📏 Next, the Newton-Raphson Method is explained with quadratic convergence formulas and derivative computations. 📈 The Secant Method is presented as an efficient alternative approximating derivatives via two-point slopes. 🔄 Emphasis is placed on computing interception time t* and projectile velocity v_β using detailed mathematical equations. ⏱️ Practical aspects include initial guess selection, convergence criteria, and handling multiple roots for stable solutions. ⚖️ This equation-rich approach enhances the kinematic framework for real-world applications. 🌍