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There is the this question:

Imagine you are projecting a ball with $3/4$th the escape velocity from the surface of the earth. What is the farthest distance will it reach it from the centre of earth? ($R=$ Radius of the earth)

The answer given in my workbook is $16R/7$

I could solve this question using the conservation of energy theorem. Why can I not use the third equation of motion

$$v^2 - u^2 = 2as~?$$

$s=$ distance from the surface of the earth

I tried solving through the equation of motion method but my answer is widely different.

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    $\begingroup$ Escape velocity is the speed at which an object can coast to an arbitrary ("infinite") distance from a massive body (i.e. it leaves and never comes back). If you take the limit of that equation as s goes to infinity, you get v going to infinity as well - but that contradicts the very idea of an escape velocity, that there is a finite speed which is sufficient to escape a massive body. $\endgroup$
    – Nuclear Hoagie
    Commented May 21, 2021 at 17:05
  • $\begingroup$ What value did you use for $a$? $\endgroup$
    – jacob1729
    Commented May 21, 2021 at 18:02
  • $\begingroup$ I used -9.8 for a due to the fact that g is a downward force and I utilised the convention using a negative notation. $\endgroup$
    – Snehal Saurabh
    Commented May 21, 2021 at 19:38

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That equation is for constant acceleration motions. The value of $g$ decreases with altitude according to $g=\frac{GM}{r^2}$. In this problem, the decrease in $g$ is significant, as you could expect the projectile to reach a great height.

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  • $\begingroup$ So, am I not supposed to use the equations of motion in these questions? How do I determine if I can use these equations as I had been previously using these equations in other questions and they worked but in this one they didn't. $\endgroup$
    – Snehal Saurabh
    Commented May 21, 2021 at 16:59
  • $\begingroup$ You can use equations of motion, but they have to be the correct ones. To see which ones are correct you need to understand their derivations. $\endgroup$
    – mike stone
    Commented May 21, 2021 at 17:06
  • $\begingroup$ @SnehalSaurabh $v^2-u^2=2as$ Doesn't describe all types of motion. It only describes one specific type (constant acceleration). Most motion is not constant acceleration motion, and so this equation doesn't apply most of the time. $\endgroup$
    – BioPhysicist
    Commented May 21, 2021 at 18:01
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    $\begingroup$ In my opinion it is regrettable that some teachers and textbooks call this set of equations equations of motion. The name makes it sound as if the equations were generally applicable. I much prefer suvat equations or constant acceleration equations. $\endgroup$
    – Philip Wood
    Commented May 21, 2021 at 18:09
  • $\begingroup$ Thank you. Helps a lot. $\endgroup$
    – Snehal Saurabh
    Commented May 21, 2021 at 19:26

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