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Air friction is roughly proportional to the (magnitude of the relative) velocity squared. Power is equal to force times velocity (the time derivative of work, which is force times distance). Since the force is proportional to the velocity squared, then the power will be proportional to the velocity cubed.


It turns out you cannot use an accelerometer to determine the attitude of a rocket at any time other than when it is experiencing the normal force from the earth. This is explained in some detail in this article: Thinking About Accelerometers and Gravity. The key point from the article is this line: "An accelerometer never senses gravitational acceleration ...


As @EntropicallyDriven mentions, matter-wave interferometry can be used for inertial navigation. Better clocks (in terms of both performance and size / weight / power) would help with pulsar navigation.


One possible application of AMO physics would be inertial guidance systems based on atomic interferometers--similar systems are currently being investigated for missile guidance and have shown much higher accuracy than other methods. Inertial navigation systems don't rely on a network of GPS satellites, which obviously wouldn't be present on Mars (yet!)


Many factors determine our ability to maneuver missiles. Assuming your scenario involves a war-head missile, is it incoming or outgoing? If it is outgoing then yes, we have control of the missile until it ends its course. If it is an incoming enemy missile then - who knows? If our intelligence is good enough perhaps we have the necessary codes to affect ...


Probably the best way to think about this is to say that $$p = mv\\ F=\frac{dp}{dt}=m\frac{dv}{dt}+v\frac{dm}{dt}$$ (Using the usual product rule for differentiation - thanks @ja72 for the suggestion). If velocity is constant the first term vanishes and your result follows.

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