Relative motion with local acceleration removed
JPL’s deflection-tool handbook defines incoming v∞ as asteroid velocity relative to Earth with Earth’s encounter acceleration removed. It can be understood as the far-field velocity along the incoming asymptote of the local hyperbola.
Infinity is a mathematical continuation of the two-body encounter model. The real Solar System retains the Sun and other perturbers; an operational parameter does not require physically reaching infinity or eliminating all gravity.
The excess follows an energy relation
In the same unpowered two-body model, local speed v, local escape speed vₑ and excess-speed magnitude satisfy v∞²=v²−vₑ². Since kinetic energy depends on squared speed, v−vₑ is not the excess speed.
For a hypothetical v of 13 kilometers per second and vₑ of 11.2, v∞ is √(169−125.44)=6.6, rather than 1.8. This worked example is a model calculation, not a reported mission measurement.
Incoming, outgoing and impact velocities differ
For an ideal unpowered flyby, incoming and outgoing planet-relative v∞ magnitudes match while directions change. Adding the planet’s motion in a heliocentric frame can nevertheless change solar orbital energy.
The JPL handbook separately defines outgoing spacecraft excess velocity and spacecraft–asteroid encounter velocity. Equal units do not make their reference bodies, epochs or purposes identical; a local surface-impact speed generally includes local gravitational acceleration.