A satellite is in a circular orbit about the earth (ME = 5.98 1024 kg). The period of the satellite is 2.60 104 s. What is the speed at which the satellite travels?
T = orbital period in seconds
µ = Earth's gravitational constant = 3.98x10^14
r = radius of orbit in meters
From T = 2(Pi)sqrt[r^3/µ] r = 18,959.3m.
From V = sqrt[µ/r], V = 4581m/s
To find the speed at which the satellite travels, we can use the formula for the circumference of a circle, which is given by:
C = 2πr
where C is the circumference and r is the radius of the circle.
In this case, since the satellite is in a circular orbit around the Earth, the radius of the circle is equal to the distance between the center of the Earth and the satellite.
The formula for the period of a satellite in a circular orbit is given by:
T = 2π√(r³/GMe)
where T is the period, r is the radius, G is the gravitational constant (6.67430 × 10^-11 m^3 kg^-1 s^-2), and Me is the mass of the Earth.
We can rearrange this formula to solve for the radius:
r = (GT²Me / 4π²)^(1/3)
Once we have the radius, we can calculate the circumference using the first formula, and then divide by the period to find the speed:
v = C / T
Now let's plug in the given values and calculate the speed of the satellite.
1. Calculate the radius:
r = ((6.67430 × 10^-11 m^3 kg^-1 s^-2) × (2.60 × 10^4 s)^2 × (5.98 × 10^24 kg) / (4π²))^(1/3)
2. Calculate the circumference:
C = 2πr
3. Calculate the speed:
v = C / T
By plugging these values into a calculator, we can find the speed at which the satellite travels.