How far from Earth's surface would you have to travel to be 1/4 your weight. Give your answer in Earth radii.
I know you have to travel twice the distance or 1 Earth radii... I just don't know how to mathematically show it. Please help.
since
F = GMm/r^2
when r doubles you have
GMm/(2r)^2 = Gmm/4r^2 = GMm/r^2 * 1/4 = F/4
To mathematically show how far from Earth's surface you would have to travel to be 1/4 your weight, we can utilize the concept of gravitational force and apply the inverse square law.
Let's start by understanding the relationship between weight and distance from the center of the Earth. The force of gravity acting on an object is given by the equation:
F = G * (m1 * m2) / r^2
Where:
F is the gravitational force,
G is the gravitational constant (approximately 6.67430 x 10^-11 Nm^2/kg^2),
m1 and m2 are the masses of the two objects (in this case, the mass of the Earth and the mass of the object),
and r is the distance between the centers of the two objects.
The weight of an object is the force of gravity acting upon it. In this case, we consider the weight of the object on Earth's surface to be its reference weight, denoted as W.
Now, let's consider your weight to be 1/4 of your reference weight (W/4). And let's assume you have traveled a distance of x Earth radii from the Earth's center. This means that the distance between you and the Earth's center would be (1 + x) times the Earth radii (since you are traveling x Earth radii beyond the Earth's surface).
Using the equation above, we can set up the following equation:
(W/4) = G * (me * mo) / (1 + x)^2 * Re^2
Where:
W is your reference weight on the Earth's surface,
G is the gravitational constant,
me is the mass of the Earth,
mo is the mass of the object (which cancels out as we compare relative weights),
x is the distance traveled in Earth radii (beyond the Earth's surface),
and Re is the Earth's radius.
To solve for x, we can rearrange the equation:
(1 + x)^2 = (me / (W/4)) * (G / Re^2)
Taking the square root of both sides:
1 + x = sqrt((me / (W/4)) * (G / Re^2))
Subtracting 1 from both sides:
x = sqrt((me / (W/4)) * (G / Re^2)) - 1
Calculating the value of x using the known values, we have:
x = sqrt((5.972 × 10^24 kg / (W/4)) * (6.67430 x 10^-11 Nm^2/kg^2 / (6,371 km)^2)) - 1
Note: W is the weight in newtons on the Earth's surface, which is equal to the mass of the object times the acceleration due to gravity (9.8 m/s^2).
Once you calculate the value of x, you will have the distance you need to travel beyond the Earth's surface to be 1/4 your weight in terms of Earth radii.