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I've worked through many example problems in my college physics text in the section on kinetic energy and work. I noticed that if the desired quantity is velocity or force, they can be solved entirely with kinematics formulas and the second law. For non-constant acceleration, the acceleration function can be appropriately integrated to solve for velocity and distance with appropriate initial conditions.

So is the work and kinetic energy concepts needed, if one is only after distance, velocity and force? If yes, please provide an example where energy and the work-kinetic energy theorem must be used to solve it.

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What you are describing are, to me, different way to tackle a problem. The way you should choose is the shortest one, although it might be hard to have an intuition of which is actually shorter, and I guess this come with experience.

As an example, consider this problem I came across many years ago

Suppose the Moon stops abruptly in its orbit around the Earth. Let $T_f$ be the time it would take to the Moon to fall on Earth, and let $T_o$ be the original orbital period of the Moon around the Earth. Determine the ratio $T_f/T_o$.

You could use kinematics here, by determining the force (hence the acceleration) acting on the Moon and integrate that, but you need to write just one equation down if you use Kepler's third law.

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