I know this question has actually been answered before in a couple of places such as Here
Unfortunately I am dealing with an irascible and impatient person who simply will not bother to read any more in depth text from any of the innumerable sources out there. He also won't believe me, irrespective of my background in physics. He has the simple formula for the Delta-V in a so-called "slingshot maneuver", but he insists the velocity given for the plant can be any velocity, not matter how it obtained. Obviously, this is complete nonsense. In a simple 2 body problem, where the primary is not in orbit around some other object, the velocity leaving the primary will be quite the same as it was when inbound, He actually agrees, but then he says, "In the frame of reference of the primary. I cannot make him understand a change in velocity is (for any situation) the same for every single frame of reference in the universe.
Can someone here do a better job than me at explaining
Any change in velocity of any object is the same for every reference frame.
For a free large body - like in deep space - a small object passing close by will experience a zero Delta-V, completely irrespective of the constant velocity of the large body with respect to any other object. Clarification Delta-V here is defined as the change in the magnitude of the velocity function of an object after a maneuver as compared to prior to the maneuver. Obviously, the direction of the vehicle changes, but at every given specific altitude on either side of the planet, the speed is exactly the same. The path is perfectly symmetrical. This is NOT true if and only if the large body is significantly influenced by a 3rd body. In the case of a very large 3rd body (like the Sun) in residence, the path of the vehicle is not symmetrical at all, about the large body or otherwise. It will pick up or drop some momentum. So, in point f fact, will the large body. It loses or gains angular momentum. If the initial orbital angular momentum of the large body is zero, the slingshot produces no change in energy for the vehicle on its return path. None.
A gravity assist / braking is a 3 body problem. Expansion In every elastic collision and every gravitational interaction between two objects the paths are perfectly symmetrical. No exceptions. The only way an elastic collision between two objects can result in a net change in energy or momentum is if a 3rd object is involved. In the case of the slingshot maneuver, the additional momentum is "stolen" from the planet's angular momentum. Of course, if the planet were spinning, and the vehicle actually collided with it, the planet could give up some angular momentum that way, but it is not being physically impacted. Barring such a thing, the only angular momentum that can be available is orbital momentum around the third body. Note: the situation can be analyzed by creating two separate reference frames, one involving one pair of objects and the other involving the other pairing, but it still requires three bodies. See: The fundamental concepts of the gravity-assist manoeuvre