How to integrate acceleration due to gravity with height? If I project a body at a very large velocity such that it reaches a height h which is comparable to Earth's radius, then how to derive an equation for 'h' in terms of initial velocity 'u' and other constants.This is where I ended up with this problemS = u²/2gBut g is variable with height as h approaches RSo dg = GM/(R+dr)² --- this is where I think I might be wrongHence dS = u²(R+dr)²/GMSo how do I go ahead?

Question

How to integrate acceleration due to gravity with height? If I project a body at a very large velocity such that it reaches a height h which is comparable to Earth&#039;s radius, then how to derive an equation for &#039;h&#039; in terms of initial velocity &#039;u&#039; and other constants.This is where I ended up with this problemS = u²/2gBut g is variable with height as h approaches RSo dg = GM/(R+dr)² --- this is where I think I might be wrongHence dS = u²(R+dr)²/GMSo how do I go ahead?

Madelyn Lynch · Accepted Answer

As I cannot fully understand your question, I will somewhat provide you with the tools so that you can answer your own question.  g  =            G      M              R      2      The above mentioned equation gives you the acceleration due to the gravitational at some point point that is R units of distance away from the center of any body of mass M. This can also be called as the gravitational field.For earth,   M  =  6  ∗      10          24      (approximately), and for any point on the surface of the earth   R will be approximately   6400  k  mNow, only using the variables..Let   g be the acceleration due to gravity on the earth&#039;s surface. Therefore,  g  =            G      M              R      2      , where R = radius of earth.Let   g  ∗ be the acceleration due to gravity at a height h from the earth&#039;s surface. Now,  g  ∗  =            G      M              (      R      +      h              )        2            With the ratio             g      ∗        g  , you can compare the two fields. For h &amp;lt;&amp;lt; R, use binomial expansion for negative index and approximate. (Higher powers of   h      /    R are negligible).If you want to find the potential energy, integrate with limits from infinity to the desired distance (  R  +  h) the following expression.  W  =  ∫  F  d  x  =      ∫    ∞          (      R      +      h      )                  G              m        1                    m        2                    x      2        d  xNote: Gravitational potential energy at a point is defined as the work done to bring a mass from infinity to that point.Hope I am right and hope this helps you.

How to integrate acceleration due to gravity with height? If I project a body at a very large velo

Answered question

Answer & Explanation

New Questions in Force, Motion and Energy