Compute:
a. The velocity of an object orbiting at height of 2 Re above the surface of Earth.
b. Compute the orbital period of that object.
To compute the velocity of an object orbiting at a certain height above the surface of the Earth, we can use the formula for the velocity of a satellite in circular orbit. This formula is derived from the balance between the gravitational force and the centripetal force.
a. The velocity of an object in orbit can be calculated using the following formula:
v = √(G * M / r)
Where:
- v is the velocity of the object in orbit,
- G is the gravitational constant (approximately 6.674 x 10^-11 N m^2/kg^2),
- M is the mass of the Earth (approximately 5.972 x 10^24 kg),
- and r is the distance between the center of the Earth and the object in orbit (2 Re).
First, note that Re represents the radius of the Earth, which is approximately 6,371 km or 6,371,000 meters.
To calculate the velocity at a height of 2 Re above the surface of the Earth, we need to add the radius of the Earth to the height above the surface. Thus, the distance from the center of the Earth to the object in orbit is:
r = 2 Re + Re = 3 Re
Now we can plug in the values into the formula:
v = √(G * M / r)
v = √(6.674 x 10^-11 N m^2/kg^2 * 5.972 x 10^24 kg / (3 * 6,371,000 m))
By evaluating this equation using a calculator, we find that the velocity of the object in orbit at a height of 2 Re above the surface of the Earth is approximately 9.53 km/s.
b. To compute the orbital period of the object, we can use Kepler's third law of planetary motion. This law relates the orbital period (T) to the semi-major axis (a) of the orbit.
T^2 = (4π^2 / GM) * a^3
Where:
- T is the orbital period of the object,
- π is a constant approximately equal to 3.14159,
- G is the gravitational constant,
- M is the mass of the Earth,
- and a is the semi-major axis of the orbit.
The semi-major axis can be calculated by summing the radius of the Earth (Re) and the height above the surface.
a = Re + height
a = Re + 2 Re
a = 3 Re
Now we can plug in the values into the formula:
T^2 = (4π^2 / GM) * a^3
T^2 = (4π^2 / (6.674 x 10^-11 N m^2/kg^2 * 5.972 x 10^24 kg)) * (3 * 6,371,000 m)^3
By evaluating this equation using a calculator, we find that the orbital period of the object is approximately 2 hours and 7 minutes.