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Thank you again very much! You are very generous with your time sir. This got me 90% of the way there. For me, I had to make a bit of a modification to the final quartic equation to get it working properly:

$$\begin{align} \\ t^4 \left(\frac {g^2} 4 \right) + t^2 \left(r_v g - \left( \tan \theta \right)^2 v_h^2 \right) + t \left( 2 \left( \tan \theta \right)^2 \vec r_h \cdot \vec v_h \right) + \left( \tan \theta \right)^2 r_h^2 - \vec r_h^2 &= 0 \end{align}$$

Thank you again very much! You are very generous with your time sir. This got me 90% of the way there. For me, I had to make a bit of a modification to the final quartic equation to get it working properly:

$$\begin{align} \\ t^4 \left(\frac {g^2} 4 \right) + t^2 \left(r_v g - \left( \tan \theta \right)^2 v_h^2 \right) + t \left( 2 \left( \tan \theta \right)^2 \vec r_h \cdot \vec v_h \right) + \left( \tan \theta \right)^2 r_h^2 - \vec r^2 &= 0 \end{align}$$

Thank you again very much! You are very generous with your time sir. This got me 90% of the way there. For me, I had to make a bit of a modification to the final quartic equation to get it working properly:

$$\begin{align} \\ t^4 \left(\frac {g^2} 4 \right) + t^2 \left(r_v g - \left( \tan \theta \right)^2 v_h^2 \right) + t \left( 2 \left( \tan \theta \right)^2 \vec r_h \cdot \vec v_h \right) + \left( \tan \theta \right)^2 r_h^2 - \vec r^2 &= 0 \end{align}$$

Thank you again very much! You are very generous with your time sir. This got me 90% of the way there. For me, I had to make a bit of a modification to the final quartic equation to get it working properly:

$$\begin{align} \\ t^4 \left(\frac {g^2} 4 \right) + t^2 \left(r_v g - \left( \tan \theta \right)^2 v_h^2 \right) + t \left( 2 \left( \tan \theta \right)^2 \vec r_h \cdot \vec v_h \right) + \left( \tan \theta \right)^2 r_h^2 - \vec r_h^2 &= 0 \end{align}$$

Thank you again very much! This got me 90% of the way there. For me, I had to make a bit of a modification to the final Quatric equation:

$$\begin{align} \\ t^4 \left(\frac {g^2} 4 \right) + t^2 \left(r_v g - \left( \tan \theta \right)^2 v_h^2 \right) + t \left( 2 \left( \tan \theta \right)^2 \vec r_h \cdot \vec v_h \right) + \left( \tan \theta \right)^2 r_h^2 - \vec r^2 &= 0 \end{align}$$

Thank you again very much! You are very generous with your time sir. This got me 90% of the way there. For me, I had to make a bit of a modification to the final quartic equation to get it working properly:

$$\begin{align} \\ t^4 \left(\frac {g^2} 4 \right) + t^2 \left(r_v g - \left( \tan \theta \right)^2 v_h^2 \right) + t \left( 2 \left( \tan \theta \right)^2 \vec r_h \cdot \vec v_h \right) + \left( \tan \theta \right)^2 r_h^2 - \vec r^2 &= 0 \end{align}$$