I think it is important to understand where the equation P = Fv comes from.
P = W / t
and
W = F d
Therefore, P = (F d) / t
and because d / t = v
P = F v
This equation means that as an object experiences a force its velocity will change.
The force is what is kept constant and it is changing the velocity causing it to increase (assuming the force and velocity vectors point in the same direction).
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