Questions: A 200 kg satellite is put into a low Earth orbit at an altitude of 300 km . (a) Calculate the orbital speed of the satellite.

Transcript text: A 200 kg satellite is put into a low Earth orbit at an altitude of 300 km . (a) Calculate the orbital speed of the satellite.

Solution

Solution Steps

Step 1: Calculate the orbital radius

The altitude of the satellite is 300 km. We need to add this to the radius of the Earth (6370 km) to find the orbital radius (r).

r = 300 km + 6370 km = 6670 km = 6.67 x 10⁶ m

Step 2: Apply Newton's Law of Universal Gravitation and centripetal force

The gravitational force provides the centripetal force for the satellite's circular motion. Gravitational force = G * (M_E * m)/r² , where G is the gravitational constant (6.67 x 10^-11 Nm²/kg²), M_E is the mass of Earth (5.97 x 10²⁴ kg), m is the mass of the satellite (200 kg), and r is the orbital radius (6.67 x 10⁶ m).

Centripetal force = (m * v²)/r, where v is the orbital speed.

Equating the two gives us: G * (M_E * m)/r² = (m * v²)/r

Step 3: Solve for orbital speed (v)

Simplifying and solving for v: v = sqrt(G * M_E / r) v = sqrt((6.67 x 10^-11 Nm²/kg²) * (5.97 x 10²⁴ kg) / (6.67 x 10⁶ m)) v ≈ 7726 m/s