Question

An Earth satellite has its apogee at 2,400 km above the surface of Earth and perigee at 900 km above the surface of Earth. At apogee its speed is 5,600 m/s. What is its speed at perigee (in m/s)

Answer 1

The speed of the Earth satellite at perigee is approximately 22,984 m/s.

Step-by-step explanation:

The speed of an Earth satellite at perigee can be determined using the conservation of angular momentum. The angular momentum of a satellite is constant throughout its orbit. At perigee, when the satellite is closest to the Earth, its distance from the center of the Earth is 900 km above the surface. Therefore, using the formula for angular momentum, we can calculate the speed at perigee.

The formula for angular momentum is:

H = mvr

where H is the angular momentum, m is the mass of the satellite, v is the velocity of the satellite, and r is the distance of the satellite from the center of the Earth.

At apogee, the satellite's distance from the center of the Earth is 2400 km + 6370 km (radius of the Earth). Using this information and the given speed at apogee, we can calculate the satellite's mass.

Then, using the calculated mass and the distance at perigee, we can find the speed at perigee.

The speed at perigee is approximately **22,984 m/s**.

Answer 2

The speed of the Earth satellite at perigee is determined as 24,174.66 m/s.

How to calculate the speed at perigee?

The speed at perigee is calculated by applying the principle of conservation of energy as follows.

K.E (apogee) + P.E (apogee) = K.E(perigee) + P.E(perigee)

¹/₂mv₁² - (GMm/r₁) = ¹/₂mv₂² - (GMm/r₂)

¹/₂v₁² - (GM/r₁) = ¹/₂v₂² - (GM/r₂)

v₁² - (2GM/r₁) = v₂² - (2GM/r₂)

v₂² = v₁² - (2GM/r₁) + (2GM/r₂)

v = √ ( v₁² - (2GM/r₁) + (2GM/r₂) )

where;

• v₁ is the speed at apogee
• G is universal gravitation constant
• M is the mass of Earth
• r is the distance

The speed at perigee is calculated as;

v = √ ( v₁² - (2GM/r₁) + (2GM/r₂) )

v = √ ( (5,600)² - (2 x 6.67 x 10⁻¹¹ x 5.97 x 10²⁴/2,400,000) + (2 x 6.67 x 10⁻¹¹ x 5.97 x 10²⁴//900,000) )

v = 24,174.66 m/s