A spacecraft is in circular Earth orbit at 5400km altitude.
By how much will its altitude decrease if it moves to a new circular orbit where its orbital speed is 20% higher?
To calculate the change in altitude when the spacecraft moves to a new circular orbit with a 20% higher orbital speed, we need to consider the relationship between orbital speed and altitude in a circular orbit.
The orbital speed of a spacecraft in a circular orbit is determined by the gravitational force between the spacecraft and the Earth. In a circular orbit, the gravitational force provides the necessary centripetal force to keep the spacecraft in orbit.
The centripetal force is given by the equation:
F_c = m * (v^2 / r)
where F_c is the centripetal force, m is the mass of the spacecraft, v is the orbital speed, and r is the radius of the orbit (which is equal to the sum of the Earth's radius and the altitude of the spacecraft).
The orbital speed of a circular orbit is given by the equation:
v = sqrt(GM / r)
where G is the gravitational constant and M is the mass of the Earth.
Now, let's assume the initial orbital speed is v1 and the corresponding altitude is h1, and the new orbital speed is v2 and the new altitude is h2.
We are given that the new orbital speed is 20% higher than the initial orbital speed:
v2 = v1 + 0.2 * v1 = 1.2 * v1
We can now equate the expressions for the orbital speed with the corresponding radii:
sqrt(GM / r1) = 1.2 * sqrt(GM / r2)
Squaring both sides of the equation, we get:
GM / r1 = (1.2 * sqrt(GM / r2))^2
GM / r1 = 1.44 * GM / r2
Canceling out GM from both sides of the equation, we get:
1 / r1 = 1.44 / r2
r2 = 1.44 * r1
Since the radius of an orbit is equal to the sum of the Earth's radius and the altitude of the spacecraft, we can write:
r1 = RE + h1
r2 = RE + h2
Substituting these expressions into the equation, we have:
RE + h2 = 1.44 * (RE + h1)
Expanding and rearranging the equation, we get:
h2 = 1.44 * (RE + h1) - RE
Finally, substituting the radius of the Earth (RE) as approximately 6371 km, and the given value of h1, which is 5400 km, we can calculate the change in altitude (Δh):
h2 = 1.44 * (6371 + 5400) - 6371
Simplifying the equation, we have:
h2 ≈ 1.44 * 11771 - 6371
h2 ≈ 16964.24 - 6371
h2 ≈ 10593.24
Therefore, the spacecraft's altitude will decrease by approximately 10593.24 km when it moves to a new circular orbit with a 20% higher orbital speed.