Animated Solution for Physics - Gravitation: The value of acceleration due to gravity at earth's surface is $9.8{\text{ ms}}^{-2}$. The altitude above its surface at which the acceleration due to gravity decreases to $4.9{\text{ ms}}^{-2}$, is close to (Take, radius of earth = $6.4\times 10^6\text{ m}$ )

The universe operates on beautifully simple rules, and one of the most profound is Newton's Inverse Square Law of Gravitation. In this problem, we are going to explore exactly how gravity fades away as we travel further from the surface of the Earth.

Imagine you are standing on the surface of the Earth. You feel your normal weight, and the acceleration due to gravity is a familiar $9.8{\text{ m/s}}^2$. Now, imagine boarding a spacecraft and flying straight up. As you get further from the center of the Earth, the gravitational pull weakens. We want to find the exact altitude where this acceleration is cut exactly in half, down to $4.9{\text{ m/s}}^2$.

A common trap many students fall into is using the approximation formula $g_h=g(1-\frac{2h}{R_e})$.

This is a massive red flag! That approximation is only valid when the height $h$ is very small compared to the radius of the Earth (typically less than $300\text{ km}$). Here, the gravity has halved, which means we are dealing with a very large distance. We must use the exact, fundamental formula for the variation of gravity with altitude.

The true relationship for acceleration due to gravity at an altitude $h$ is given by:

This equation tells us that gravity decreases with the square of the distance from the center of the Earth. Let's plug in the values we know. We are given $g=9.8{\text{ m/s}}^2$ and we want to find the height where $g_h=4.9{\text{ m/s}}^2$.

Now, it's just a matter of careful algebra. Let's rearrange the equation to isolate the term with $h$:

Notice how perfectly the numbers are chosen! $9.8$ is exactly double $4.9$. This simplifies our equation beautifully:

To get rid of the square, we take the square root of both sides. Remember, since distance must be positive, we only consider the positive root:

We know that $\sqrt{2}\approx 1.414$. Subtracting $1$ from both sides gives us the ratio of our altitude to the Earth's radius:

Finally, to find the actual altitude $h$, we multiply this ratio by the radius of the Earth, $R_e=6.4\times 10^6\text{ m}$:

Looking at our options, the closest value is $2.6\times 10^6\text{ m}$.

This means you have to travel over $2,600\text{ kilometers}$ into space just to reduce your weight by half! It's a fantastic reminder of just how far the Earth's gravitational influence extends.