Table 1 Derivatives of F(t)

F⁽⁴⁾(t)

\(6\left( {{\varvec{a}_1}^2 + {\varvec{a}_2}^2 + 2{\varvec{a}_1}{\varvec{a}_2}\cos \beta } \right)\)

F⁽⁴⁾(t) is a constant and greater than zero

F⁽³⁾(t)

\(F^{{(4)}} (t)t - 6\varvec{v}_{trans} \left( {\varvec{a}_{1} + \varvec{a}_{2} \cos\beta } \right)\)

F⁽³⁾(t) has only one solution named t₃₁

\(F^{\prime\prime}\left( t \right)\)

\(- 0.5F^{{(4)}} (t)t^{2} + F^{{(3)}} (t)t + 2\left( {\varvec{v}_{trans}^{2} - \varvec{a}_{2} R\sin\beta } \right)\)

\(F^{\prime\prime}\left( t \right)\) has two solutions named t₂₁ and t₂₂

\(F^{\prime}\left( t \right)\)

\(\frac{1}{3}F^{{(4)}} (t)t^{3} + \frac{1}{6}F^{{(3)}} (t)t^{2} - \frac{2}{3}F^{\prime\prime}(t)t + \frac{{10}}{3}\left( {\varvec{v}_{trans}^{2} - \varvec{a}_{2} R\sin\beta } \right)t\)

\(F^{\prime}\left( t \right)\) has three solutions named t₁₁, t₁₂ and t₁₃