I don't think the mathematics could work. Consider accelerating a ship in which you were riding, so that instead of the sphere zipping by you at "thousands of miles per second", the ship, you and the sphere were all moving in the same direction and the same speed, so that the sphere appeared stationary from your prospective. The only way to accelerate you and the ship in this way would be to throw out mass in the opposite direction. Gravity sling-shotting works in the same way, you move faster along a chosen vector, and as a result, some large body is slowed down (or accelerated in the opposite direction).

If you are looking for an example of momentum disappearing, consider two sticky balls of goo, moving toward each other and colliding. The balls hit and if they have the same mass, the resulting two-ball stuck-together system will have zero momentum.

I don't think the mathematics could work. Consider accelerating a ship in which you were riding, so that instead of the sphere zipping by you at "thousands of miles per second", the ship, you and the sphere were all moving in the same direction and the same speed, so that the sphere appeared stationary from your prospective. The only way to accelerate you and the ship in this way would be to throw out mass in the opposite direction. Gravity sling-shotting works in the same way, you move faster along a chosen vector, and as a result, some large body is slowed down (or accelerated in the opposite direction).

If you are looking for an example of momentum disappearing, consider two sticky balls of goo, moving toward each other and colliding. The balls hit and if they have the same mass, the resulting two-ball stuck-together system will have zero momentum.