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A simple model often used to explain solar system gravitational slingshots is to consider a mass moving to the right with initial velocity v

leading to

In further considering this, I think forcing the signs of the final velocities prejudices the calculation. Obviously, if you set the larger mass' initial and final velocities to be equal in the momentum conservation equation, then the smaller mass' initial and final velocities would also be equal, i.e. no collision.

_{1i}and a much larger mass moving to the left with initial velocity v_{2i}. After the collision, the first mass is moving to the left with velocity v_{1f}and the larger mass continues to move to the left with velocity v_{2f}. As v_{relative,before}= -v_{relative,after}we can write (assuming all v's are positive quantities):v

_{1i}+ v_{2i}= - (-v_{1f}+ v_{2f})leading to

v

It's common to say that since the planet (our large mass here) is so big it's velocity doesn't change much, so it's a good approximation to equate its initial and final velocities, leading to_{1f}= v_{1i}+ v_{2i}+ v_{2f}

v

But the smaller mass gets its velocity boost by "stealing" some from the larger mass. If we impose the condition that the initial and final velocities of the larger mass are exactly equal, why doesn't the equation show that the boost effect is eliminated?_{1f}= v_{1i}+ 2v_{2i}

In further considering this, I think forcing the signs of the final velocities prejudices the calculation. Obviously, if you set the larger mass' initial and final velocities to be equal in the momentum conservation equation, then the smaller mass' initial and final velocities would also be equal, i.e. no collision.

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