In classical mechanics, the power (P) exerted by a force (F) on an object is indeed calculated using the formula P = F * v, where v represents the velocity or speed of the object. However, it is important to note that this formula applies in the reference frame where the force is being applied. If you switch to another inertial reference frame moving at a different velocity, the power calculation will change as well.
Let's consider the scenario you mentioned with a car. In the car's reference frame, the power generated by the engine is calculated using the formula P = F * v, where F represents the force generated by the engine and v is the velocity of the car. If the car's speed doubles while the force remains the same, the power output of the engine will also double. This is because the force and speed are directly proportional in this reference frame.
However, if you shift to another inertial reference frame moving at a different velocity relative to the car, the power calculation will change in that frame. From the perspective of this new reference frame, the car's velocity will appear different, and therefore the power calculation will be different as well.
In summary, the power calculation depends on the reference frame from which it is observed. Different reference frames may yield different values for power, even if the force applied and the power output of the engine remain constant.