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Gravity is a fundamental force that pulls objects toward the center of the Earth. It's what keeps your feet on the ground and what makes objects fall. The acceleration due to gravity on Earth's surface is approximately \(9.8 \, m/s^2\). This value represents the rate at which an object's velocity increases as it falls due to Earth's gravity.
At higher altitudes, the acceleration due to gravity decreases. In the exercise, the main objective is to find the height where this acceleration becomes only 1% of its surface value. To express this change in mathematical terms, we use the formula:\[g' = (0.01)g\] where \(g'\) is the gravitational acceleration at height \(h\) and \(g\) is the standard gravitational acceleration on Earth's surface. Understanding how to manipulate these equations is crucial for solving similar problems in gravitational physics.

The gravitational constant, denoted as \(G\), is a fundamental part of the universal gravitational formula. This constant helps quantify the strength of gravity between two objects. Its approximate value is \( 6.674 \times 10^{-11} \, \text{N}\,\text{m}^2/\text{kg}^2\).
In this exercise, \(G\) allows us to calculate the gravitational force at a height \(h\) above Earth's surface. This is done using the equation:\[ g' = \frac{G \cdot M}{(R+h)^2} \]where \(M\) is Earth's mass, \(R\) is Earth's radius, and \(h\) is the height above Earth's surface.

The radius of the Earth, symbolized by \(R\), is essential for calculations involving gravitational force and acceleration. Generally, Earth's radius is approximately \( 6371 \, \text{km} \).
This value is used whenever you need to assess gravitational forces at different heights. In the context of the problem, the radius of the Earth helps define how the gravitational pull weakens as you move further from the surface.
The gravitational pull decreases with increasing distance from the center of the Earth, emphasizing the significance of understanding the radius in gravitation-related studies.

Mathematical substitution is a technique used to solve equations by replacing one part of the equation with another expression. It simplifies the process of finding unknown variables.
In the step-by-step solution, we used substitution by setting:\[ g' = \frac{G \cdot M}{(R+h)^2} = 0.01 \cdot \frac{G \cdot M}{R^2} \]By simplifying this equation, unnecessary components such as \(G\) and \(M\) cancel out, aiding us in isolating the variable \(h\). This process allows us to find the required height above the Earth's surface where the acceleration due to gravity falls to 1% of its surface value. The simplified form helps in understanding how different variables interact in the context of gravity.