Typically satellites orbit around 400 above the surface of the earth. If an astronaut weighs 600 on the ground, what will he weigh if he is 400 above the surface?

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 miles.

To calculate the astronaut's weight 400 km above the surface of the Earth, we need to consider the effect of the Earth's gravitational pull weakening with distance.

The weight of an object is determined by the gravitational force acting on it. The formula to calculate weight is:

Weight = Mass × Gravitational Acceleration

The mass of the astronaut remains constant, so we only need to factor in the change in gravitational acceleration.

Close to the surface of the Earth, the standard gravitational acceleration is approximately 9.8 m/s². However, as we move higher above the Earth's surface, the gravitational acceleration decreases. This decrease is due to the inverse square law, which states that the force of gravity decreases with the square of the distance from the center of the Earth.

To find the weight of the astronaut 400 km above the surface, we first need to convert the distance from kilometers to meters:

400 km = 400,000 meters

Next, we need to calculate the new gravitational acceleration at that height. Using the inverse square law, we divide the standard gravitational acceleration by the square of the ratio of the Earth's radius plus the altitude to the Earth's radius:

New Gravitational Acceleration = Standard Gravitational Acceleration / (1 + (altitude / Earth's radius))²

The Earth's mean radius is approximately 6,371 km, which is equivalent to 6,371,000 meters.

Plugging in the values:

New Gravitational Acceleration = 9.8 m/s² / (1 + (400,000 m / 6,371,000 m))²

Calculating this expression gives us:

New Gravitational Acceleration ≈ 7.84 m/s²

Now we can substitute this value into the weight formula along with the astronaut's mass (600 kg):

Weight = Mass × Gravitational Acceleration

Weight = 600 kg × 7.84 m/s²

Calculating this expression gives us:

Weight ≈ 4,704 newtons

Therefore, if the astronaut is 400 km above the surface of the Earth, they would weigh approximately 4704 newtons.