Math Variations

Question

The gravitational force, , between an object and the Earth is inversely proportional to the square of the distance from the object to the center of the Earth. If an astronaut weighs 199 pounds on the surface of the Earth, what will this astronaut weigh 100 miles above the Earth? Assume that the radius of the Earth is 4000 miles. (Round off your answer to the nearest pound.)

Answer 1

In physics it can be helpful to simply write out everything we know (or think we know) from the start. Sometimes the answer or method becomes more apparent once we've done so. (There are more elegant ways to solve this, but this approach will help us learn some things as we do this.)

Consider your starting statement about the relationship of gravitational force and distance.

F_g ¤ 1 / d²

This means "The force of gravity is proportional to the inverse square of the distance." We can add a constant here to maybe be a bit more precise and help us work through the next steps. We'll call it G.

F_g = G / d²

You also have one known data point: The force of gravity on a person on the surface of the earth (4000 miles from the center) is 199 lbs. We can represent this using our formula as follows:

199 lb = G / 4000mi²

With some simple algebra work, we now have a value for G: 199 lb x 4000mi² (or F₁ x earth's radius squared)

Now let's consider our new situation, where the person is 100 miles above the surface of the earth. The formula for this is:

F_g = G / 4100mi²

So, we substitute our value for G from before to solve this:

F_g = 199 lb x 4000mi² / 4100mi² = 199 lb / 1.05 = 189 lb.

If we look closely at this last formula, the more "elegant" solution starts to appear. What we really have there is:

F₂ = F₁ x (d₁² / d₂²)

F₂ / F₁ = (d₁² / d₂²) = (d₁ / d₂)²

So, basically, the ratio of the gravitational forces = square of the ratio of the respective distances. We should be able to plug our numbers into this new formula and arrive at the same answer (and an answer for any distance).

F₂ / F₁ = (d₁ / d₂)²

F₂ / 199lb = (4000 / 4100)²

F₂ = 0.9756² x 199lb = 189 lb

In this way, if you know the distance and force, you can quickly calculate the answer for a related situation at another distance.