the space shuttle is in a 250 mile high orbit. What are the shuttle's orbital period, in minutes, and it speed?
the space shuttle is in a 250 mile high orbit. What are the shuttle's orbital period, in minutes, and it speed?
The earth's radius r = ~3963 miles.
The orbital speed is given by V = sqrt(µ/r) ft./sec. where µ = the gravitational constant of the earth = 1.407974x10^16 ft.^3/sec.^3.
The orbital period derives from
T = [2(3.14)sqrt(r^3/µ)]/60 minutes.
To calculate the space shuttle's orbital period and speed, we need to use the following formulas:
1. Orbital Period Formula:
Orbital Period = 2 * π * √(r^3 / GM)
Where:
- r is the radius of the orbit (distance from the center of the Earth)
- G is the gravitational constant (approximately 6.67430 × 10^-11 m^3/kg/s^2)
- M is the mass of the Earth (approximately 5.972 × 10^24 kg)
2. Orbital Speed Formula:
Orbital Speed = √(GM / r)
Where:
- G is the gravitational constant (approximately 6.67430 × 10^-11 m^3/kg/s^2)
- M is the mass of the Earth (approximately 5.972 × 10^24 kg)
- r is the radius of the orbit (distance from the center of the Earth)
Now, let's calculate the orbital period and speed of the space shuttle:
1. Determine the radius of the orbit:
Given that the space shuttle is in a 250-mile high orbit, we need to add the Earth's radius to this value to find the total distance from the center of the Earth.
Earth's radius = 3959 miles (approximately)
Orbit radius = 250 miles + 3959 miles
2. Calculate the orbital period:
Using the formula: Orbital Period = 2 * π * √(r^3 / GM)
Convert the radius from miles to meters: 4209 meters (1 mile = 1609.34 meters)
Plug in the values to calculate the orbital period in seconds.
3. Convert the orbital period to minutes:
Divide the orbital period in seconds by 60 to get the value in minutes.
4. Calculate the orbital speed:
Using the formula: Orbital Speed = √(GM / r)
Convert the radius from miles to meters: 4209 meters (1 mile = 1609.34 meters)
Plug in the values to calculate the orbital speed in meters per second.
Now, using these steps, let's calculate the space shuttle's orbital period and speed:
Step 1: Determine the radius of the orbit:
Orbit radius = 250 miles + 3959 miles
Step 2: Calculate the orbital period:
Orbital Period = 2 * π * √(r^3 / GM)
Step 3: Convert the orbital period to minutes:
Orbital Period (minutes) = Orbital Period (seconds) / 60
Step 4: Calculate the orbital speed:
Orbital Speed = √(GM / r)
Following these steps, we can find the space shuttle's orbital period and speed.
To determine the space shuttle's orbital period and speed, we can make use of some key formulas related to orbital mechanics.
1. Orbital Period (T) Formula:
The formula to calculate the orbital period of an object in orbit is given by:
T = 2π * √(r^3 / GM)
Where:
T = Orbital Period (in seconds)
π = Pi (approximately 3.14159)
r = Radius of the orbit from the center of the Earth
G = Gravitational constant (approximately 6.67430 x 10^-11 m^3 kg^-1 s^-2)
M = Mass of the Earth (approximately 5.972 x 10^24 kg)
2. Velocity (v) Formula:
The formula to calculate the velocity of an object in orbit is given by:
v = √(GM / r)
Where:
v = Velocity (in meters per second)
G = Gravitational constant (approximately 6.67430 x 10^-11 m^3 kg^-1 s^-2)
M = Mass of the Earth (approximately 5.972 x 10^24 kg)
r = Radius of the orbit from the center of the Earth
Now, let's calculate the orbital period and speed of the space shuttle.
Given:
Radius of the orbit (r) = 250 miles = 402336 meters
1. Orbital Period (T) Calculation:
Using the formula: T = 2π * √(r^3 / GM)
Substituting the values:
T = 2π * √((402336^3) / (6.67430 x 10^-11 * 5.972 x 10^24))
Calculating this expression yields the orbital period in seconds.
2. Velocity (v) Calculation:
Using the formula: v = √(GM / r)
Substituting the values:
v = √((6.67430 x 10^-11 * 5.972 x 10^24) / 402336)
Calculating this expression yields the velocity in meters per second.
Finally, convert the orbital period from seconds to minutes by dividing the calculated value by 60.
By following these steps, you can calculate the space shuttle's orbital period in minutes and its speed in meters per second.