A spy satellite is in circular orbit around Earth. It makes one revolution in 5.99 hours.

(a) How high above Earth's surface is the satellite?

(b) What is the satellite's acceleration?

To find the answers, we can use the formulas for the height above the Earth's surface and the acceleration of an object in circular motion.

(a) To find the height above the Earth's surface, we need to know the radius of the orbit. The radius of the orbit is equal to the distance from the center of the Earth to the satellite's location.

To calculate the radius, we'll use the formula for the period of a circular orbit:

T = 2π√(r^3/GM)

Where:
T is the period of the orbit (5.99 hours)
π is a mathematical constant (approximately 3.14159)
r is the radius of the orbit (unknown)
G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
M is the mass of the Earth (approximately 5.972 × 10^24 kg)

Rearranging the formula, we have:

r = (T^2 * GM) / (4π^2)

Plugging in the given values:

r = (5.99 hours)^2 * (6.67430 × 10^-11 m^3 kg^-1 s^-2) * (5.972 × 10^24 kg) / (4π^2)

Calculating this expression, we find that the radius (and therefore the satellite's height above Earth's surface) is approximately 3,592 kilometers.

(b) The satellite's acceleration is given by the formula:

a = (GM) / r^2

Where:
a is the acceleration
G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
M is the mass of the Earth (approximately 5.972 × 10^24 kg)
r is the radius of the orbit (3,592 kilometers = 3,592,000 meters)

Plugging in the values:

a = (6.67430 × 10^-11 m^3 kg^-1 s^-2) * (5.972 × 10^24 kg) / (3,592,000 meters)^2

Evaluating this expression, we find that the satellite's acceleration is approximately 0.216 m/s^2.

To calculate the height above Earth's surface and the acceleration of the spy satellite, we will use the following formulas:

(a) Height above Earth's surface is given by the equation:
h = R + r
where R is the radius of the Earth and r is the height of the satellite above the Earth's surface.

(b) Acceleration is given by the equation:
a = v² / r
where v is the orbital velocity and r is the distance from the center of the Earth to the satellite.

Let's calculate each step:

Step 1: Calculate the radius of the Earth (R):
The radius of the Earth, R, is approximately 6,371 km.

Step 2: Calculate the height above Earth's surface (h):
Using the equation h = R + r, we need to determine the value of r.
Since the spy satellite is in a circular orbit, the height of the satellite above the Earth's surface is the same as the radius of its orbit.
The satellite completes one revolution in 5.99 hours, which is equivalent to 5.99 * 3600 seconds.
The distance traveled in one full revolution is the orbit's circumference:
C = 2πr
The time taken for one revolution is equal to the distance divided by the speed:
Speed = Distance / Time
v = C / Time
Solving for C:
C = v * Time
C = v * 5.99 * 3600 s
Since v = C / (2πr):
2πr = v * 5.99 * 3600 s
r = (v * 5.99 * 3600 s) / (2π)
Now, we can substitute this value of r into the formula for h:
h = R + r = R + (v * 5.99 * 3600 s) / (2π)

Step 3: Calculate the acceleration (a):
Using the equation a = v² / r, we can substitute the values of v and r:
a = (v²) / r = (v²) / ((v * 5.99 * 3600 s) / (2π))

Let's calculate both the height above Earth's surface and the acceleration using the given information.