At what altitude above the Earth's surface would your weight be two-thirds of what it is at the Earth's surface? (Assume Re = 6.371 103 km.)
For any one body, the gravitational constant, GM , and your reference mass, m, remain constant. Therefore, you can derive your apparant weight at high altitudes from W = Wo(Ro/R)^2 where Wo = your refernce weight on the body surface, R = the altitude at which the new weight is desired and Ro = the radius of the body surface. Thus, on earth, W = Wo(3963/(3963 + h)^2 where Ro = the earth's radius and h = the height above the earth's surface, in either feet or miles.
I'll let you do the math.
To determine at what altitude above the Earth's surface the weight would be two-thirds of what it is at the Earth's surface, we can use the concept of gravitational force.
The gravitational force acting on an object depends on its mass and the distance from the center of the Earth. At any height above the Earth's surface, the weight of an object can be calculated using the formula:
F = (G * m * M) / r^2
where:
F is the force of gravity,
G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2),
m is the mass of the object,
M is the mass of the Earth,
and r is the distance from the center of the Earth.
At the Earth's surface, your weight is equal to your mass multiplied by the acceleration due to gravity, which is approximately 9.8 m/s^2. Let's assume your weight at the Earth's surface is W.
W = m * g
To find the altitude at which your weight is two-thirds of the Earth's surface weight, we need to solve the equation:
(2/3)W = [(G * m * M) / (r + Re)^2]
Rearranging this equation, we get:
(r + Re)^2 = [(G * m * M) / (2/3)W]
Taking the square root of both sides, we have:
r + Re = sqrt[(G * m * M) / (2/3)W]
Finally, by subtracting Re from both sides of the equation, we can find the value of r:
r = sqrt[(G * m * M) / (2/3)W] - Re
Plug in the given values for Re (6.371 x 10^3 km) and assume W as your weight at the Earth's surface. Calculate m (your mass) based on your weight and the acceleration due to gravity (m = W / g). With these values, you can calculate the altitude above the Earth's surface at which your weight will be two-thirds of what it is at the Earth's surface.