A spy satellite is in circular orbit around Earth. It makes one revolution in 5.95 hours. (Radius of the Earth=6.371 106 m)
(a) How high above Earth's surface is the satellite?
(b) What is the satellite's acceleration?
velocity=2PI*(re+h)/period
but velocity can be found..
9.8(re/(re+h))^2=v^2/(re+h)
or v=sqrt (9.8 (re^2/(re+h))
then put that v into the first equation, and solve for h.
This is a form of Keplers' law.
To answer these questions, we need to use some key equations related to circular motion. The first equation we will use is the formula for the period of a satellite in circular orbit:
T = 2π √(r³/GM)
Where:
T = period of the satellite (given as 5.95 hours)
r = radius of the satellite's orbit (unknown)
G = gravitational constant (6.674 × 10⁻¹¹ N m²/kg²)
M = mass of the Earth (5.972 × 10²⁴ kg)
By rearranging this equation, we can solve for the radius (r) of the satellite's orbit:
r = [(T²GM)/(4π²)]^(1/3)
(a) To find the height above Earth's surface, we need to subtract the radius of the Earth from the radius of the satellite's orbit:
Height above surface = r - radius of the Earth
(b) To find the satellite's acceleration, we will need to use the centripetal acceleration equation:
a = v²/r
Where:
a = acceleration of the satellite
v = velocity of the satellite (unknown)
To find the velocity (v), we can use the formula for linear speed in circular motion:
v = 2πr/T
Now we have all the necessary equations to find the answers. Let's calculate them step by step.
(a) Calculate the height above the Earth's surface:
First, calculate the radius of the satellite's orbit using the given period (T) and the known values of G and M.
Then, subtract the radius of the Earth from the calculated radius to get the height above the Earth's surface.
(b) Calculate the satellite's acceleration:
First, calculate the velocity of the satellite using the formula for linear speed in circular motion using the calculated radius and the given period.
Then, use the centripetal acceleration formula to find the satellite's acceleration using the calculated velocity and the radius of the satellite's orbit.