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If there is no wind at all, the carrier is still capable of a forward speed to create a headwind for aircraft. The Hornet, and carriers like it were capable of 32 knots.
SR, I believe the P47s that took off from carriers were catapulted off. In these modern times, the Navy had a requirement that the fighter bomber competiton which was eventually won by Super Hornet had to have a zero wind over the deck capability. The Strike Fighter version of the Tomcat submitted by Grumman had that capability while the Super Hornet did not. The Navy waived that requirement. The Tomcat submission also could launch with a full load without using afterburner, the Super Hornet cannot which forces it to refuel after launch.