Often in designing orbits for satellites, people use what is termed a \"gravitat

Question

Often in designing orbits for satellites, people use what is termed a "gravitational slingshot effect." The idea is as follows: A satellite of mass ms and speed vs,i circles around a planet of mass mp that is moving with speed vp,i in the opposite direction. See the diagram below:

Although the satellite never touches the planet, this interaction can still be treated as a collision because of the gravitational interaction between the planet and satellite during the slingshot. Since gravity is a conservative force, the collision is elastic.

Use an x-axis with positive pointing to the right.
Solve for the unknowns below algebraically first, then use the following values for the parameters.

mp = 8.60E+24 kg
ms = 960 kg
vs,i  x = 2.405E+4 m/s
vp,i  x = -4.81E+4 m/s

Solve for the final velocity of the satellite after the collision.
vs, f   x =  
1pts

Explanation / Answer

mp = 8.60 x 1024 kg
ms = 960 kg
vs,ix = 2.405 x 104 m/s
vp,ix = - 4.81 x 104 m/s

vs,fx = ?
vp,fx = ?

using conservation of momentum

mpvp,ix + msvs,ix = mpvp,fx + msvs,fx

(8.60 x 1024 ) (- 4.81 x 104) + (960) (2.405 x 104) = (8.60 x 1024 ) vp,fx + (960) vs,fx

(- 4.13 x 1029 - (960) vs,fx )/ (8.60 x 1024 ) = vp,fx    eq-1

using conservation of kinetic energy

mpv2p,ix + msv2s,ix = mpv2p,fx + msv2s,fx

(8.60 x 1024 ) (- 4.81 x 104)2 + (960) (2.405 x 104)2 = (8.60 x 1024 ) ((- 4.13 x 1029 - (960) vs,fx )/ (8.60 x 1024 ))2+ (960) v2s,fx

vs,fx = - 2.571 x 1014 m/s

using eq-1

(- 4.13 x 1029 - (960) vs,fx )/ (8.60 x 1024 ) = vp,fx

(- 4.13 x 1029 - (960) ( - 2.571 x 1014) )/ (8.60 x 1024 ) = vp,fx

vp,fx = - 4.80 x 104 m/s

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