The gravitational slingshot effect

[pinkerpikachu]

The gravitational slingshot effect. In the diagram below, the planet Saturn moving in the negative xdirection at its orbital speed (with respect to the Sun) of 9.6 km/s. The mass of Saturn is 5.96 × 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 (aconservative force) acting on the spacecraft causes it to swing around the planet (orbit shown as a 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.m1v1 + m2v2= m1'v1' + m2'v2'

It seems like a simple equation. I know that the speed of Saturn and the mass of Saturn are not going to change. (this is true?) So the focus of this problem should be the spacecraft .

m1v1 = m1'v1' for the spacecraft .

I'm confused about how to factor in the gravitational force of Saturn? and this is obviously (?) important for finding the final speed of the air craft.

i think this is how you would find the gravitation acceleration of saturn:

F = Gm/ r^2; where G is a constant, m= mass of Saturn, and r= the radius of Saturn
= (6.67 X 10^-11 N m^2/kg)(5.96 × 1026 kg)/ (60,268,000^2 m )
= 10.9

is this right? BTW, I'm not so sure about the radius of Saturn...I got different numbers on the web

Okay, now I don't know what to do...

 

[LowlyPion]

I think the key is to recognize what's happening.

Saturn and the object are approaching each other in opposite directions.

But what is the final result?

 

[pinkerpikachu]

I think the key is to recognize what's happening.

Saturn and the object are approaching each other in opposite directions.

But what is the final result?

Okay, as the aircraft approaches Saturn, its going to feel some kind of gravitational pull towards the larger body. Correct? This pull is a force which is stated as conservative. So will it have an impact on the final speed on the air craft? Will this force slow down the aircraft? I know a conservation force is defined something along the lines as: it doesn't matter how many step you take, the work done is the same, but does this apply?

I just realized that only m1v1 = m1'v1' does really make sense. The mass isn't changing so by that equation the velocities would have to be the same.

 

[LowlyPion]

Okay, as the aircraft approaches Saturn, its going to feel some kind of gravitational pull towards the larger body. Correct? This pull is a force which is stated as conservative. So will it have an impact on the final speed on the air craft? Will this force slow down the aircraft? I know a conservation force is defined something along the lines as: it doesn't matter how many step you take, the work done is the same, but does this apply?

I just realized that only m1v1 = m1'v1' does really make sense. The mass isn't changing so by that equation the velocities would have to be the same.

You might also think of a conservative force as what it giveth it taketh away. The speed that it gives inbound it robs out bound. But what about the speed of Saturn itself? Does that affect the final speed?