Is Conservation of Mechanical Energy the Key to Solving Projectile Height?

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The discussion revolves around finding the height of a projectile launched against Earth's gravity using conservation of mechanical energy. The initial approach involves deriving a function for motion based on Newton's laws and gravitational equations, specifically considering gravity as a function of distance from the Earth. However, some participants suggest that the complexity of the problem may be unnecessary, as the projectile's height may not require accounting for varying gravity unless specified. The need to consider changes in gravity is emphasized due to the high initial velocities involved. Ultimately, the conversation highlights the balance between simplifying assumptions and the accuracy required for solving the problem.
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Homework Statement



Find the height of a projectile being shot in the oposite direction as the center of the Earth with initial speed v.

Homework Equations



Newtons law of gravity
Kinematic equations?
Possibly conservation of energy equation

The Attempt at a Solution


This problem might be solved by using coservation of mecanical energy. However I was trying to find a function r(t) that described its motion. From Newtons laws we have that g is a function of r, thus g=GM/r^2. This is g as a function of r. I want it as a function of time. So the way to approach this is to solve r``(t)= GM/(r(t))^2. So we have a differential equation that looks like this y``*y=constant. Does this equation have a solution? Am I doing the right approach here?
 
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No. I do not think that you need to go that far. The projectile does not go that far that you have to consider change in gravity. According to me do not complicate until it is mentioned in the question.
 
Actually I have to consider change in gravity. Because they ask me to find velocity very far up and I am working with initial velocities that are very high
 
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