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In the Star Trek: The Next Generation episode "The Most Toys" the crew of Enterprise-D discovers that the android Data may have been stolen while on board another ship, Jovis.
At this point the Jovis, which has a maximum warp factor of 3, has had a 23-hour head start, which the Enterprise-D figures puts her anywhere within a 0.102 light year radius of her last known position.
There is no exact formula for this interval because the quoted velocities are based on a hand-drawn curve; what can be said is that at velocities greater than warp 9, the form of the warp function changes because of an increase in the exponent of the warp factor w.
Due to the resultant increase in the derivative, even minor changes in the warp factor eventually correspond to a greater than exponential change in velocity.
In the episode "Threshold", Tom Paris breaks the warp 10 threshold, but travel beyond the threshold is later discovered to be unacceptably hazardous to biological life.
The "slingshot effect" is first depicted in "Tomorrow Is Yesterday" (1967) as a method of time travel.
A spacecraft equipped with a warp drive may travel at speeds greater than that of light by many orders of magnitude.This successful first trial led directly to first contact with the Vulcans.Later on, a prequel series titled Star Trek: Enterprise describes the warp engine technology as a "Gravimetric Field Displacement Manifold" (Commander Tucker's tour, "Cold Front"), and describes the device as being powered by a matter/anti-matter reaction which powers the two separate nacelles (one on each side of the ship) to create a displacement field.The episode also firmly establishes that many other civilizations had warp drive before humans; First Contact co-writer Ronald D.Moore suggested Cochrane's drive was in some way superior to forms which existed beforehand, and was gradually adopted by the galaxy at large.