The gravitational slingshot effect. Figure 7-51 shows the planet Saturn moving in the negative x direction at its orbital speed (with respect to the Sun) of 9.6 km/s. The mass of Saturn is 5.69 X 1026 kg. A spacecraft with mass 825 kg approaches Saturn. When far from Saturn, it moves in the +x direction at 10.4 km/s. The gravitational attraction of Saturn (a conservative force) acting on the spacecraft causes it to swing around the planet (orbit shown as dashed line) and head off in the opposite direction. Estimate the final speed of the spacecraft after it is far enough away to be considered free of Saturn's gravitational pull.
Solution 81GP:
We have to determine the final speed with which spacecraft was moving, as its direction of motion was reversed due to gravitational attraction between the spacecraft and Saturn.
Step 1 of 5</p>
Concept:
The interaction between the Saturn and the spacecraft is elastic, because the force of gravity is conservative. Thus kinetic energy is conserved in this interaction.
Relative velocity of body A with respect to body B having velocity respectively, is given as,
Step 2 of 5</p>
We are given that,
Mass of Saturn
Speed of Saturn with respect to the Sun towards left
Mass of the spacecraft
Initial speed of spacecraft when approaching Saturn towards right.
Step 3 of 5</p>
As the mass of the Saturn is so much bigger than the mass of the spacecraft, therefore, the Saturn's speed will not changed during the interaction between the spacecraft and the Saturn. Obviously the spacecraft will have negligible effect on the motion of Saturn.
In the reference frame of Saturn, we can treat this as the equivalent of a small mass "bouncing of "a massive object. Therefore, the relative velocity of the spacecraft in this reference frame will be reversed as the only force acting between them is gravitational force, which is conservative in nature.
We estimate the final speed of the spacecraft from relative speed between the Saturn and the spacecraft.
