The space shuttle is in a 250 mile high orbit. What are the shuttle's orbital period, in minutes and its speed?
Equate the gravitational force, given by
F=GMm/r²
and the centrifugal force due to the angular velocity ω
F=mrω²
In SI units,
G = 6.674×10−11 N m² kg -2
Mass of the earth = 6*1024 kg (approx.)
r = 250 miles plus the radius of the earth to be converted to metres.
The formula for both gravitation and centrifugal force use r from the centre of the earth.
Solve for ω and calculate the period and speed.
The shuttle's mass, m, is immaterial since it will cancel out on both sides of the equation.
To determine the space shuttle's orbital period and speed, we need to consider several factors.
First, let's start by calculating the orbital period.
The orbital period (T) can be calculated using Kepler's Third Law of Planetary Motion, which states that the square of the orbital period (T^2) is proportional to the cube of the semi-major axis (d^3). In this case, the semi-major axis is the sum of the distance from the center of the Earth to its surface plus the altitude of the orbit.
The Earth's average radius is approximately 3,963 miles. To convert the altitude of the orbit from miles to the Earth's radius, we divide 250 miles by the Earth's radius:
Altitude in Earth's radii = 250 miles / 3,963 miles ≈ 0.063
Now, since we have the altitude in Earth's radii, we can calculate the semi-major axis (d) by adding 1 to the altitude:
Semi-major axis = Altitude in Earth's radii + 1 ≈ 0.063 + 1 = 1.063
Having determined the semi-major axis, we can now calculate the orbital period.
T^2 = d^3
Orbital period (T) = √(d^3)
Orbital period (T) = √(1.063^3)
Orbital period (T) ≈ √1.202 ≈ 1.097
The orbital period of the space shuttle in minutes is approximately 1.097 minutes.
Now let's calculate the shuttle's orbital speed.
The orbital speed (V) can be calculated using the formula:
V = 2πd / T
where π is a constant value of approximately 3.14159.
Orbital speed (V) = 2πd / T = 2π(1.063) / 1.097
Orbital speed (V) ≈ 6.275 miles per minute
Therefore, the space shuttle's orbital period is approximately 1.097 minutes, and its speed is approximately 6.275 miles per minute.