Matter & Interactions 2nd ed. Practice Problems
Aaron Titus | High Point University
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15a0001     Initial speed of a space probe traveling from Earth to Jupiter along a ``least energy" orbit     15a0001
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Question

 

To send a probe from earth to another outer planet, it is most efficient (i.e. conserves the most fuel) to put the probe into an orbit about Sun so that when it is nearest Sun (perihelion), its path is tangent to the earth's orbit, and when it is furthest from Sun (aphelion), its path is tangent to the outer planet's orbit. Besides the fuel necessary to put the probe into orbit (i.e. escape from Earth) and to make it orbit the outer planet once it gets there, no fuel is necessary during travel from Earth to the outer planet. For this reason, the probe's orbit in this case is called a "least-energy" orbit.

Suppose the probe travels from Earth to Jupiter as shown below. The radius of the Earth's nearly circular orbit is , and the radius of Jupiter's nearly circular orbit is .


Figure: The path of a probe leaving Earth at perihelion and arriving at Jupiter at aphelion.

If the probe's speed at aphelion should be the same as the speed of Jupiter ( ), what should be the probe's speed at perihelion, when it leaves Earth with its thrusters turned off? Neglect interactions of the probe with Earth and Jupiter, and assume that the probe's motion is completely determined by its interaction with Sun.