# Effect of incoming velocity on change in velocity in a gravity assist

In a gravity assist does the incoming velocity of a space craft effect how much speed it gains. For example say a rocket was approaching Jupiter for a gravity assist with speed v and another rocket was approaching with speed 2v. Would the incoming speeds (v and 2v) effect how much speed they are able to gain from Jupiter (for example, would the rocket at speed v gain more than the one at 2v because it spends more time around the planet and therefore has more time to gain more speed)?

I am looking for a proper quantitive relationship or proof of the effect of the velocity of a spacecraft going into an assist on the ability of the spacecraft to gain or lose speed. I am also curious what the effect on its ability to change direction (deflection angle) would be as well. If anyone can provide the proof or direct me somewhere where it is done it would be greatly appreciated.

• The change in velocity depends on both incoming velocity and "impact parameter" (how close it would pass if it continued along a straight line). You can read about all of this by searching this site for "gravity assist" and "impact parameter" or just clicking on the gravity-assist tag.
– uhoh
Commented Jul 9, 2020 at 2:54
• @uhoh I found a formula for the delta-v and in it is only the gravitational parameter, hyperbolic excess velocity (related to the incoming velocity) and height of periapsis. I was wondering how the impact parameter effects the delta-v - maybe the height of periapsis takes it into account? They are different values however. Commented Jul 9, 2020 at 5:52
• @uhoh I found the answer from answers in these two questions: space.stackexchange.com/questions/6582/… and physics.stackexchange.com/questions/128356/… but I am still trying to understand the derivation for the delta-v formula (if you can help me with that I would greatly appreciate it). Should I answer my question by putting in the links? Commented Jul 10, 2020 at 5:28