At what height from the surface of the earth, are the gravitation potential and the value of $g$ are: $-5.4

Question

At what height from the surface of the earth, are the gravitation potential and the value of $g$ are: $-5.4 	imes 10^7 	ext{ J/kg}$ and $6.0 	ext{ ms}^{-2}$ respectively? (Take, the radius of the earth as $6400 	ext{ km}$)

Accepted Answer

1. **Formulas:** 
   - Gravitational Potential at a distance $r$ from the center ($r = R+h$) is $V = -\frac{GM}{r}$.
   - Acceleration due to gravity at distance $r$ is $g_h = \frac{GM}{r^2}$.
2. **Relation:** Dividing the magnitude of potential by the acceleration yields the distance $r$:
   $$ \frac{|V|}{g_h} = \frac{GM/r}{GM/r^2} = r $$
3. **Calculation:**
   $$ r = \frac{5.4 	imes 10^7 	ext{ J/kg}}{6.0 	ext{ ms}^{-2}} = 0.9 	imes 10^7 	ext{ m} = 9 	imes 10^6 	ext{ m} $$
   Converting to kilometers: $r = 9000 	ext{ km}$.
4. **Find Height ($h$):**
   $$ h = r - R = 9000 	ext{ km} - 6400 	ext{ km} = 2600 	ext{ km} $$

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