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## Homework Statement

Let's suppose that we wanted to launch a spacecraft of mass m out of the Solar System.

a) If we want to launch the spacecraft directly from Earth, what boost Δv would be required

and what direction relative to the Earth's velocity about the Sun would this boost be pointed

in?

b) Now let us assume that we wanted to save energy (reduce the magnitude of Δv) by using a

gravitational slingshot off of a single planet. Assume that all the planets are on circular

orbits with the same orbital plane as the Earth, and for now assume that the planets are point

masses. Which planet would you choose and why? The necessary information about

planets is available on Wikipedia. What would be the magnitude and direction of Δv?

c) Real planets have surfaces, and bad things happen to spacecraft (and astronauts) that crash

into the surfaces of planets. Does this change your answer to part b of this problem? Why

might it?

## Homework Equations

[itex] E=\frac{1}{2}μv^2-\frac{GMμ}{r}=\frac{GMμ}{2a}; a=r_1+r_2 [/itex]

## The Attempt at a Solution

Part A)

Let [itex] E_p = \frac{1}{2}μ(v_p)^2-\frac{GMμ}{r_1}=\frac{GMμ}{r_1+r_2} [/itex] be the energy at the periapsis.

Then the corresponding magnitude of the velocity of the spacecraft at periapsis is: [itex] v_p = \sqrt{2GM(\frac{1}{r_1}-\frac{1}{r_1+r_2})} [/itex]

Also the magnitude of the velocity around the 1st circular orbit is : [itex] v_1=\sqrt{\frac{GM}{r_1}}[/itex]

So in order to boost from the circular orbit around Earth to the transitional elliptical transfer orbit requires [itex] v_1+Δv=v_p [/itex] or [itex] Δv=v_p-v_1 [/itex]

Thus [itex] Δv = \sqrt{\frac{GM}{r_1}}(\sqrt{\frac{2r_2}{r_1+r_2}}-1) [/itex]

As [itex] r_2 →∞ , Δv→\sqrt{\frac{GM}{r_1}}(\sqrt{2}-1)[/itex]

Now for the direction: [itex] (v_p)^2 = \vec{v_p}\cdot\vec{v_p} = \vec{v_1}\cdot\vec{v_1}+\vec{Δv}\cdot\vec{Δv}+2\vec{v_1}\cdot\vec{Δv} [/itex] which is clearly maximized for [itex] Δv [/itex] in the same direction as [itex] v_1 [/itex]

(everything in part A makes sense to me, but I want to be sure I didn't make any careless errors)

Part B) I know I need to minimize the first expression for [itex] Δv [/itex] with respect to [itex] r_2 [/itex] i.e [itex] \frac{d}{dr_2}Δv=0 [/itex] and then verify that that second derivative of this expression evaluated at the extrema is positive. Then I need to check which planet fits the radial orbit I find from minimizing [itex] Δv [/itex] wrt [itex] r_2 [/itex]. However, before I do this Is my expression for [itex] Δv [/itex] correct?

Part C) I personally don't see any reason why the planets having surface areas affects anything in part B, but just because I don't see a reason it doens't mean there isn't one. Any clues on this one?

Thank you in advance!

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