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I've been googling this question because I haven't seen the answer anywhere, yet I know it must be calculated with a high degree of precision. Anyone know if it will be faster or slower than its 13.8 km/s approach speed? I would assume faster, but by how much?

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From Wikipedia on New Horizons:

New Horizons is intended to pass within 10,000 km (6,200 mi) of Pluto, with this closest approach date estimated to occur on July 14, 2015 at 11:50 UTC. New Horizons will have a relative velocity of 13.78 km/s (49,600 km/h; 30,800 mph) at its closest approach, and will come as close as 27,000 km (17,000 mi) to Charon, although these parameters may be changed during flight.

For changes to New Horizons flyby trajectory, check New Horizons News Center. That same portal (New Horizons at Johns Hopkins Applied Physics Laboratory) also has a nice overview of New Horizons current position and velocities relative to Pluto and the Sun (Heliocentric velocity), which is currently 14.54 km/s:

Full Trajectory: Side View

 New Horizons Full Trajectory: Side View

This image shows New Horizons' current position along its full planned trajectory. The green segment of the line shows where New Horizons has traveled since launch; the red indicates the spacecraft's future path. Positions of stars with magnitude 12 or brighter are shown from this perspective, which is slightly above the orbital plane of the planets.

Full Trajectory: Overhead View

             New Horizons Full Trajectory: Overhead View

This image shows New Horizons' current position along its full planned trajectory. The green segment of the line shows where New Horizons has traveled since launch; the red indicates the spacecraft's future path. Positions of stars with magnitude 12 or brighter are shown from this perspective, which is above the Sun and "north" of Earth's orbit.

These velocities won't change by much by the time New Horizons comes the closest to Pluto during its flyby, but you can simulate relative velocity change yourself with NASA's Eyes on Pluto. Exact (to within meters per second) relative velocities are yet unknown, since it might be required for New Horizons to perform trajectory corrective maneuvers (last such maneuver, as of the time of writing this answer, was performed on June 13 UTC) and it's still not too late to decide on a different to nominal flyby trajectory (called SHBOT - Safe Haven By Other Trajectory), shall New Horizons detect any debris along its current trajectory that crosses only about 1650 km away from so-called Sailboat Island where it's been calculated that debris could linger in relatively stable S-type orbits between Pluto and Charon. But so far, so good.

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