Question

Can you tell me an example where you can use both Newton's laws and energy conservation to solve the problem and the priorities of each method?

Answer 1

The solution to the boy sliding down a hemispheric dome is such an example.

Explanation:

We only need two points to describe the motion of an object regardless of its trajectory. We need it's position (r), velocity (v), and the time (t) at these two points. Regardless how complex or simple the motion pathway it is, you solve the problem two points at a time; though you might have to invoke multiple sets of two points.

Conservation of energy nicely ties position and velocity together going from one point to the other. However, it says nothing about the acceleration. You will need Newton's Law to analyze forces that give raise to acceleration. And from the acceleration you can determine how time and velocity changed from one location to the other, i.e., via kinematic equations.