Instance Models

This section transforms the problem defined in the problem description into one which is expressed in mathematical terms. It uses concrete symbols defined in the data definitions to replace the abstract symbols in the models identified in theoretical models and general definitions.

Calculation of landing time

Refname	IM:calOfLandingTime
Label	Calculation of landing time
Input	\({v_{\text{launch}}}\), \(θ\)
Output	\({t_{\text{flight}}}\)
Input Constraints	\[{v_{\text{launch}}}\gt{}0\]\[0\lt{}θ\lt{}\frac{π}{2}\]
Output Constraints	\[{t_{\text{flight}}}\gt{}0\]
Equation	\[{t_{\text{flight}}}=\frac{2\,{v_{\text{launch}}}\,\sin\left(θ\right)}{g}\]
Description	\({t_{\text{flight}}}\) is the flight duration (\({\text{s}}\)) \({v_{\text{launch}}}\) is the launch speed (\(\frac{\text{m}}{\text{s}}\)) \(θ\) is the launch angle (\({\text{rad}}\)) \(g\) is the magnitude of gravitational acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
Notes	The constraint \(0\lt{}θ\lt{}\frac{π}{2}\) is from A:posXDirection and A:yAxisGravity, and is shown in Fig:Launch. \(g\) is defined in A:gravAccelValue. The constraint \({t_{\text{flight}}}\gt{}0\) is from A:timeStartZero.
Source	–
RefBy	IM:calOfLandingDist, FR:Output-Values, and FR:Calculate-Values

Detailed derivation of flight duration:

We know that \({{p_{\text{y}}}^{\text{i}}}=0\) (A:launchOrigin) and \({{a_{\text{y}}}^{\text{c}}}=-g\) (A:accelYGravity). Substituting these values into the y-direction of GD:posVec gives us:

\[{p_{\text{y}}}={{v_{\text{y}}}^{\text{i}}}\,t-\frac{g\,t^{2}}{2}\]

To find the time that the projectile lands, we want to find the \(t\) value (\({t_{\text{flight}}}\)) where \({p_{\text{y}}}=0\) (since the target is on the \(x\)-axis from A:targetXAxis). From the equation above we get:

\[{{v_{\text{y}}}^{\text{i}}}\,{t_{\text{flight}}}-\frac{g\,{t_{\text{flight}}}^{2}}{2}=0\]

Dividing by \({t_{\text{flight}}}\) (with the constraint \({t_{\text{flight}}}\gt{}0\)) gives us:

\[{{v_{\text{y}}}^{\text{i}}}-\frac{g\,{t_{\text{flight}}}}{2}=0\]

Solving for \({t_{\text{flight}}}\) gives us:

\[{t_{\text{flight}}}=\frac{2\,{{v_{\text{y}}}^{\text{i}}}}{g}\]

From DD:speedIY (with \({v^{\text{i}}}={v_{\text{launch}}}\)) we can replace \({{v_{\text{y}}}^{\text{i}}}\):

\[{t_{\text{flight}}}=\frac{2\,{v_{\text{launch}}}\,\sin\left(θ\right)}{g}\]

Calculation of landing position

Refname	IM:calOfLandingDist
Label	Calculation of landing position
Input	\({v_{\text{launch}}}\), \(θ\)
Output	\({p_{\text{land}}}\)
Input Constraints	\[{v_{\text{launch}}}\gt{}0\]\[0\lt{}θ\lt{}\frac{π}{2}\]
Output Constraints	\[{p_{\text{land}}}\gt{}0\]
Equation	\[{p_{\text{land}}}=\frac{2\,{v_{\text{launch}}}^{2}\,\sin\left(θ\right)\,\cos\left(θ\right)}{g}\]
Description	\({p_{\text{land}}}\) is the landing position (\({\text{m}}\)) \({v_{\text{launch}}}\) is the launch speed (\(\frac{\text{m}}{\text{s}}\)) \(θ\) is the launch angle (\({\text{rad}}\)) \(g\) is the magnitude of gravitational acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
Notes	The constraint \(0\lt{}θ\lt{}\frac{π}{2}\) is from A:posXDirection and A:yAxisGravity, and is shown in Fig:Launch. \(g\) is defined in A:gravAccelValue. The constraint \({p_{\text{land}}}\gt{}0\) is from A:posXDirection.
Source	–
RefBy	IM:offsetIM and FR:Calculate-Values

Detailed derivation of landing position:

We know that \({{p_{\text{x}}}^{\text{i}}}=0\) (A:launchOrigin) and \({{a_{\text{x}}}^{\text{c}}}=0\) (A:accelXZero). Substituting these values into the x-direction of GD:posVec gives us:

\[{p_{\text{x}}}={{v_{\text{x}}}^{\text{i}}}\,t\]

To find the landing position, we want to find the \({p_{\text{x}}}\) value (\({p_{\text{land}}}\)) at flight duration (from IM:calOfLandingTime):

\[{p_{\text{land}}}=\frac{{{v_{\text{x}}}^{\text{i}}}\cdot{}2\,{v_{\text{launch}}}\,\sin\left(θ\right)}{g}\]

From DD:speedIX (with \({v^{\text{i}}}={v_{\text{launch}}}\)) we can replace \({{v_{\text{x}}}^{\text{i}}}\):

\[{p_{\text{land}}}=\frac{{v_{\text{launch}}}\,\cos\left(θ\right)\cdot{}2\,{v_{\text{launch}}}\,\sin\left(θ\right)}{g}\]

Rearranging this gives us the required equation:

\[{p_{\text{land}}}=\frac{2\,{v_{\text{launch}}}^{2}\,\sin\left(θ\right)\,\cos\left(θ\right)}{g}\]

Offset

Refname	IM:offsetIM
Label	Offset
Input	\({p_{\text{land}}}\), \({p_{\text{target}}}\)
Output	\({d_{\text{offset}}}\)
Input Constraints	\[{p_{\text{land}}}\gt{}0\]\[{p_{\text{target}}}\gt{}0\]
Output Constraints
Equation	\[{d_{\text{offset}}}={p_{\text{land}}}-{p_{\text{target}}}\]
Description	\({d_{\text{offset}}}\) is the distance between the target position and the landing position (\({\text{m}}\)) \({p_{\text{land}}}\) is the landing position (\({\text{m}}\)) \({p_{\text{target}}}\) is the target position (\({\text{m}}\))
Notes	\({p_{\text{land}}}\) is from IM:calOfLandingDist. The constraints \({p_{\text{land}}}\gt{}0\) and \({p_{\text{target}}}\gt{}0\) are from A:posXDirection.
Source	–
RefBy	IM:messageIM, FR:Output-Values, and FR:Calculate-Values

Output message

Refname	IM:messageIM
Label	Output message
Input	\({d_{\text{offset}}}\), \({p_{\text{target}}}\)
Output	\(s\)
Input Constraints	\[{d_{\text{offset}}}\gt{}-{p_{\text{target}}}\]\[{p_{\text{target}}}\gt{}0\]
Output Constraints
Equation	\[s=\begin{cases}\text{\(``\)The target was hit.‘’}, & \|\frac{{d_{\text{offset}}}}{{p_{\text{target}}}}\|\lt{}ε\\\text{\(``\)The projectile fell short.‘’}, & {d_{\text{offset}}}\lt{}0\\\text{\(``\)The projectile went long.‘’}, & {d_{\text{offset}}}\gt{}0\end{cases}\]
Description	\(s\) is the output message as a string (Unitless) \({d_{\text{offset}}}\) is the distance between the target position and the landing position (\({\text{m}}\)) \({p_{\text{target}}}\) is the target position (\({\text{m}}\)) \(ε\) is the hit tolerance (Unitless)
Notes	\({d_{\text{offset}}}\) is from IM:offsetIM. The constraint \({p_{\text{target}}}\gt{}0\) is from A:posXDirection. The constraint \({d_{\text{offset}}}\gt{}-{p_{\text{target}}}\) is from the fact that \({p_{\text{land}}}\gt{}0\), from A:posXDirection. \(ε\) is defined in Sec:Values of Auxiliary Constants.
Source	–
RefBy	FR:Output-Values and FR:Calculate-Values