At what altitude above the Earth's surface would your weight be three-fourths of what it is at the Earth's surface? Assume re = 6.371 10^3 km.
Based on Newton's law of universal gravitation,
F=GMm/r²
which means that the weight would be inversely proportional to the square of the distance from the centre of the Earth.
If H is the altitude when the weight becomes 3/4 of what's on the surface, then
(re/(re+H))^2=3/4
Solve for H.
Solved out for H:
(6371/sqrt(3/4)) - 6371 = H
for other fractions just plug in where 3/4's is.
Hope this helps!
To determine the altitude above the Earth's surface where your weight would be three-fourths of what it is at the Earth's surface, we need to consider the inverse square law of gravitation. According to this law, the force of gravity acting on an object decreases with the square of the distance from the center of the Earth.
Let's denote the weight at the Earth's surface as W0.
According to the inverse square law, the weight at a given altitude h above the Earth's surface can be expressed as:
W = W0 * (re / (re + h))^2
We can rearrange this equation to solve for h:
(re / (re + h))^2 = (3/4)
Taking the square root of both sides, we get:
re / (re + h) = √(3/4)
Now we can solve for h:
re = 6.371 * 10^3 km
W0 = weight at Earth's surface
Plugging in the values, we have:
(6.371 * 10^3 km) / ((6.371 * 10^3 km) + h) = √(3/4)
Squaring both sides, we get:
(6.371 * 10^3 km)^2 / ((6.371 * 10^3 km) + h)^2 = 3/4
Cross-multiplying, we have:
(6.371 * 10^3 km)^2 = (3/4) * ((6.371 * 10^3 km) + h)^2
Now we can solve for h by isolating it:
((6.371 * 10^3 km) + h)^2 = ((6.371 * 10^3 km)^2) * (4/3)
Taking the square root of both sides:
6.371 * 10^3 km + h = sqrt(((6.371 * 10^3 km)^2) * (4/3))
Subtracting 6.371 * 10^3 km from both sides:
h = sqrt(((6.371 * 10^3 km)^2) * (4/3)) - 6.371 * 10^3 km
Evaluating this expression, we get:
h ≈ 3490.25 km
Therefore, your weight would be three-fourths of what it is at the Earth's surface at an altitude of approximately 3490.25 km above the Earth's surface.
To find the altitude above the Earth's surface where your weight would be three-fourths of what it is at the Earth's surface, we can make use of the concept of gravitational force.
First, let's establish the equation for gravitational force:
F = G * (m1 * m2) / r^2
Where:
F is the gravitational force
G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
m1 and m2 are the masses of the two objects
r is the distance between the centers of the two objects
In this case, we can assume m1 as the mass of the Earth, m2 as your mass, and r as the distance from the center of the Earth to your location above its surface. Let's call the altitude h.
Now, we can set up the equation based on the given information:
F_surface = G * (m1 * m2) / r_surface^2
F_altitude = G * (m1 * m2) / r_altitude^2
We are given that F_altitude = 3/4 * F_surface.
Therefore, we can express this relation as:
3/4 * F_surface = G * (m1 * m2) / r_altitude^2
Since the masses m1 and m2 do not change, we can cancel them out:
3/4 * (m1 * m2) / r_surface^2 = G * (m1 * m2) / r_altitude^2
Next, we can cancel the mass factor:
3/4 * 1 / r_surface^2 = G / r_altitude^2
Now, we can rearrange the equation to solve for r_altitude:
r_altitude^2 = r_surface^2 * (4 / 3 * G)
Taking the square root of both sides:
r_altitude = sqrt(r_surface^2 * (4 / 3 * G))
Finally, we can substitute the given values:
r_surface = 6.371 * 10^3 km
G = 6.67430 × 10^-11 m^3 kg^-1 s^-2
Converting r_surface to meters (1 km = 1000 m):
r_surface = 6.371 * 10^6 m
Plug in the values:
r_altitude = sqrt((6.371 * 10^6 m)^2 * (4 / (3 * 6.67430 × 10^-11 m^3 kg^-1 s^-2)))
Calculating this expression gives us the value of r_altitude.