91Ó°ÊÓ

Understanding Earth's radius is essential for many calculations related to gravity and space. Earth's radius is the distance from the center of the Earth to its surface which in this exercise is given as 6400 kilometers. This measurement is crucial in determining how gravitational forces change as you move away from the surface.

The radius is a baseline for calculations, as it allows us to estimate distances and forces acting upon objects when they are moved vertically from the Earth. Whether you're working with satellites or simply calculating the impact of gravity at different heights, knowing Earth's radius gives a starting point to determine how height alters gravitational pull.

The gravitational formula is central to understanding how planets and objects interact through the force of gravity. This formula is expressed as: \[ F = \frac{G M m}{r^2} \] where:

• \( F \) is the gravitational force
• \( G \) is the gravitational constant
• \( M \) is the mass of the larger object (like Earth)
• \( m \) is the mass of an object experiencing the force
• \( r \) is the distance from the center of the mass

This formula shows that gravitational force is inversely proportional to the square of the distance between two objects. This means that as you move farther from the Earth, the gravitational force decreases quickly. Using this formula, one can calculate the forces exerted on objects at varying distances from Earth.

When you need to determine how much gravity decreases at a height above Earth's surface, using a forces ratio helps simplify calculations. Suppose the gravitational force on Earth's surface is \( F_0 \). At a certain height, it becomes less than \( F_0 \)—this is where the forces ratio is applied. Such tasks involve expressions like:
\[ F_h = 0.9 F_0 \] where \( F_h \) is the force at height \( h \) above Earth.

To use the forces ratio effectively, we adapt the formula given for gravity at Earth's surface and at height \( h \): \[ \frac{G M m}{(6400 + h)^2} = 0.9 \frac{G M m}{6400^2} \]. By simplifying these equations, you eliminate common variables to focus on the distance and proportional change of forces.

Calculating the height at which the force of gravity decreases by a certain percentage can be intriguing. Once we have arranged the forces ratio, the next step is to solve for \( h \). You find \( h \) by rearranging the equation as:\[ (6400 + h)^2 = \frac{6400^2}{0.9} \].

After setting the equation, taking the square root helps isolate \( h \): \[ 6400 + h = 6400 / \sqrt{0.9} \]. Solving for this value, you find that \( \sqrt{0.9} \approx 0.9487 \). Simplifying gives: \[ 6400 + h \approx 6745.693 \]. Therefore, \( h \approx 345.693 \text{ km} \), which when analyzed against given options, most closely matches with 345.60 km. Understanding each step in the calculation helps students grasp not only the final result but also the significance behind the math.