How far would you have to be above Earth's surface for g to be ½ of its surface value?

• A. 8.55×10⁸ m
• B. 2.64×10⁶ m
• C. 9.01×10⁶ m
• D. 8.13×10⁶ m

Answer: (B)

Verified step by step guidance

1. Start by understanding that the gravitational force decreases with distance from the Earth's center. The formula for gravitational acceleration at a distance r from the center of the Earth is given by: $g=\frac{GM}{r^2}$, where G is the gravitational constant and M is the mass of the Earth.
2. To find the distance where the gravitational acceleration is half of its surface value, set up the equation: $\frac{g}{2}=\frac{GM}{r^2}$. Here, $g$ is the gravitational acceleration at Earth's surface.
3. Rearrange the equation to solve for r: $r^2=\frac{GM}{\frac{g}{2}}$. Then, take the square root of both sides to find r.
4. Calculate the value of r using the known values: $G=6.674\times 10⁻¹¹m³kg⁻¹s⁻²$, $M=5.972\times 10²⁴kg$, and $g=9.81ms⁻²$.
5. Subtract the Earth's radius from the calculated r to find the height above Earth's surface: $h=r-6.371\times 10⁶m$, where $6.371\times 10⁶m$ is the Earth's average radius.