Relationship Between Aircraft Thrust and Jet Exhaust Velocity?
The thrust produced is propotional to the exhaust velocity and mass flow, in simple terms. The thrust equation can be written as,

$T$ = $\dot{m}_{e} v_{e} - \dot{m}_{o} v_{o} - (P_{e} - P_{o}) A_{e}$

where, $\dot{m}$ is the mass flow rate $v$ is the velocity, $P$ is the pressure and $A$ is the (exhaust) area. The subscripts 'o' and 'e' denote the freestream and exhaust values respectively. 

There are multiple ways to go about noise reduction in aicraft engines, all with some kind of penalty- the Boeing method is to use chevron shaped exhausts (Interestingly, the type chosen is the one with lesser thrust penalty, not the most effective in noise reduction). However, Airbus has cited higher fuel consumption as the main reason for them not using it. Airbus uses acoustic liners in the engine to reduce noise:


  The thrust reverser structure was acoustically treated along approximately
  half of its internal surface. It was clear that increasing this acoustic area would increase the acoustic sound absorption and hence decrease the overall measurable exhaust noise.