Trajectory Equations, Threshold Conditions, and Value Validation

Geometric and Trajectory Limitations

• Trajectory Range and Angle Sensitivity:
  • The parameter $\theta$ dictates the overall distance and clearance of the motion.
  • If the angle $\theta$ is insufficiently small, the object does not go over the target distance $z$.
  • Instead of passing over $z$, the moving object prematurely stops by the ball.
  • Under these specific geometrical conditions, the governing equation is no longer valid or applicable.

Mathematical Domain and Error Analysis

• Mathematical Error Condition:

  • An error state occurs when the initial velocity and angle product falls below the threshold determined by time and target position:     $v_0 \theta < 2 t z$
  • When this inequality holds ($v_0 \theta < 2 t z$), the expression located inside the square root becomes negative.
  • A negative term inside the square root produces an evaluation error, causing the mathematical model to return an error.

• Domain Restrictions:

  • When $\theta$ (or $\text{zeta}$) drops below a defined threshold value, the trajectory equation cannot be legally evaluated.
  • Demonstrating that the function returns an error below this threshold confirms the boundary limit of the mathematical model.

Equation Verification and Value Validation

• Structural Validity:

  • The algebraic structure of the derived trajectory equation is accurate within its valid domain.

• Numerical Parameter Check:

  • After verifying the underlying equation structure, all numerical parameter values must be verified against the domain constraints to ensure physical applicability.