Why is a jet engine most efficient when its exhaust speed is close to the aircraft’s flight speed?

A jet engine achieves maximum propulsive efficiency when its exhaust velocity ($V_j$) is as close as possible to the aircraft’s flight speed ($V_0$).1

While a jet engine must produce an exhaust speed greater than the flight speed to generate the necessary thrust, a large discrepancy between these two velocities is detrimental to efficiency. This is because propulsive efficiency is defined as the portion of the jet stream's total energy that is usefully applied to propel the aircraft. When the exhaust velocity significantly exceeds the flight speed, a large amount of energy remains in the high-velocity exhaust stream that is not effectively utilized for propulsion.12

To optimize performance, engineers design different types of engines to match specific flight speed ranges. For example, slow-moving aircraft use engines with lower jet velocities, while high-speed aircraft require engines capable of higher exhaust velocities. By minimizing the velocity excess while still maintaining sufficient thrust, the engine converts fuel energy into propulsive force more effectively.21