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The answer to question below is B please tell me why? I thought the answer would be C because by Keplers law and conservation of energy. The total mechanical energy = gravitational potential energy + the kinetic energy, at every point in time energy would be the same because there's not external force. I understand that since there's no net external torque the angular momentum is conserved throughout the whole orbit. Angular momentum is also defined to be the linear momentum times the radius. By Kepler's Law the further away the planet is the slower it will move. Since they sweep out equal area from the planet to the moon in equal time periods. Therefore for angular momentum (Mass x Velocity x Radius) to be conserved if R increase/decrease V does the opposite by the same factor. Please correct me if I'm wrong and explain to me why the answer is B. enter image description here

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The force of gravitational attraction is an example of a central force because it always points towards a "fixed" point which in this case is the centre of mass of the planet.
This means the the force on the moon due to the planet exerts no torque about the centre of the planet.
If there is no torque exerted on the moon then its angular momentum would be conserved.

Any response in terms of energy, which is also conserved, would require both the kinetic energy and the gravitational potential energy to be known at positions $A$ and $B$.
You would not be able to find a solution "in terms of the given quantities", $(u_{\rm A}, u_{\rm B}$ and $r_{\rm A})$, as demanded by the question.

Option (C) is incorrect because energy is always conserved irrespective of the direction of any forces.

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In general, you can calculate the properties either by conservation of energy or by conservation of angular momentum. The question here is, do you have enough information to do it by energy. You only know the speeds at A and B and the Radius at A. Try to calculate it both ways and have a look if you have enough data to do it via energy.

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