Sensitivity of the jet head position and speed to kinetic power (left) and wind momentum (right) for the Cygnus X-3 runs. Top left: a division of the jet power by 2 decreases all dynamical values, starting from the mean propagation speed. In particular, the acceleration is far less efficient: it shows a gain of ∼40% in speed between the starting point and the peak against a gain of ∼120% in the CygX3 run. Top right: a reduction of Ṁ^⋆ by 25% and v_w by 33% (keeping η constant) lengthens the smooth phase and strengthens the speed fluctuations in the following phases. The jets settle to the same median speed in the turbulent phase. Bottom right: same reduction as in the top right panel, with half the jet power. In the CygX3_mPmW case (right, blue), the initial acceleration phase is followed by a speed plateau.