A shuttle is to rendezvous with a space station which is in a circular orbit at an altitude of 250 mi above the surface of the earth. The shuttle has reached an altitude of 40 mi when its engine is turned off at Point B. Knowing that at that time the velocity v0 of the shuttle forms an angle

Question

A shuttle is to rendezvous with a space station which is in a circular orbit at an altitude of 250 mi above the surface of the earth. The shuttle has reached an altitude of 40 mi when its engine is turned off at Point B. Knowing that at that time the velocity [latex]\mathbf{ v }_{ 0 }[/latex] of the shuttle forms an angle [latex]{ \phi }_{ 0 }=55°[/latex] with the vertical, determine the required magnitude of [latex]\mathbf{ v }_{ 0 }[/latex] if the trajectory of the shuttle is to be tangent at A to the orbit of the space station.

Accepted Answer

Conservation of energy:

[latex]\begin{aligned}T_{B} & =\frac{1}{2} m u_{0}^{2} & V_{B} & =-\frac{G M m}{r_{B}} \T_{A} & =\frac{1}{2} m u_{A}^{2} & V_{A} & =-\frac{G M m}{r_{A}}\end{aligned}[/latex]

[latex]G M=g R^{2} \quad ext { (Eq.12.30) }[/latex]

[latex]T_{A}+V_{A}=T_{B}+V_{B}[/latex]

[latex]\frac{1}{2} \cancel m u_{0}^{2}-\frac{g R^{2}}{r_{B}} \cancel m=\frac{1}{2} \cancel m u_{A}^{2} \frac{g R^{2}}{r_{A}} \cancel m[/latex]

[latex]\begin{aligned}& r_{A}=3960+250=4210 \mathrm{\ mi} \& u_{A}^{2}= u_{0}^{2}-\frac{2 g R^{2}}{r_{B}}\left\lgroup1-\frac{r_{B}}{r_{A}} ight group \& r_{B}=3960+40=4000 \mathrm{\ mi} \& u_{A}^{2}= u_{0}^{2}-\frac{2(32.2)(3960 imes 528)^{3}}{(4000 imes 5280)}\left\lgroup 1-\frac{4000}{4210} ight group \& u_{A}^{2}= u_{0}^{2}-66.495 imes 10^{6}\quad\quad ext{(1)}\end{aligned}[/latex]

Conservation of angular momomentum: [latex]\quad r_{A} u_{A}=r_{B} u_{B} \sin \phi_{0}:[/latex]

[latex] u_{A}=(4000 / 4210) u_{0} \sin 55^{\circ}=0.77829 u_{0}[/latex]

Eqs. (2) and (1) [latex]\quad\left[1-(0.77829)^{2} ight] u_{0}^{2}=66.495 imes 10^{6}[/latex]

[latex] u_{0}=12,990 \mathrm{\ ft} / \mathrm{s\blacktriangleleft}[/latex]

Question 13.194: A shuttle is to rendezvous with a space station which is in ...

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