A ship goes to warp with us only when all of its hull is inside the field. One cut by the field's edge blocks the jump; ships outside are left behind.
Every drive: power = max power × warp ÷ max warp × (1 + (max warp − 1) × stretch²), where stretch is the extension over the most the drive can add (3.57 m per PU of max power). At full stretch even warp 1 takes all of the drive's power.
The normal field is half the hull plus 20 m. Supply is what the bus gives the drive; less than its max power lowers the warp it can reach.
Stand-ins for the Helm, which sets the warp factor and goes to warp. This console only shows the limit its field puts on the drive, and which ships the field will carry.
The same for every warp drive: how many metres of extension each PU of max power buys. 3.57 m gives the 140 PU Mark I its 500 m.