At the nozzle exit, the velocities will be around 1400-1500 ft/sec at the hot nozzle and around 1000 ft/sec at take-off power (just below locally sonic conditions).
For the long cowl mixer nozzle type engines the hot jet rapidly diffuses with the bypass flow.
Further downstream the exhaust will diverge at about 7 degrees.
I don't know if this would hold up for 300m, but if it did the jet would be over 30m in diameter, so about 400 times the area at source and so 1/400th the velocity.
I recall a test some time ago for designing an engine test bed where a pitot rig was used to measure the blast at about 50m behind a jet engine at full power. All of a sudden all the sensors read zero due to the entire rig complete with several blocks of concrete being blown into the next field.
Hope this helps.