A satellite in an elliptical orbit has a speed of 9.00km/s when it is at its closes approach to the Earth(perigee). The satellite is 7.00x10^6 m from the center of the Earth at this time. When the satellite is at its greatest distance from the center of the Earth (apogee), its speed is 3.66km/s. Find the distance from the satellite to the center of the Earth at apogee. (assume any energy losses are negligible.)

Question

A satellite in an elliptical orbit has a speed of 9.00km/s when it is at its closes approach to the Earth(perigee). The satellite is 7.00x10^6 m from the center of the Earth at this time. When the satellite is at its greatest distance from the center of the Earth (apogee), its speed is 3.66km/s. Find the distance from the satellite to the center of the Earth at apogee. (assume any energy losses are negligible.)

Answer 1

To solve this problem, we can use the principle of conservation of mechanical energy. The sum of kinetic and potential energies at any point in the satellite's orbit will be constant.

Let's start by determining the kinetic and potential energies when the satellite is at its closest approach (perigee) to the Earth.

Given:
Speed at perigee (v1) = 9.00 km/s
Distance from the center of the Earth at perigee (r1) = 7.00 × 10^6 m

The kinetic energy at perigee (K1) is given by:
K1 = (1/2)mv1^2

Since the mass of the satellite (m) does not change, we can ignore it for now.

Next, we can calculate the potential energy at perigee (U1) using the formula:
U1 = -GMm/r1

where G is the gravitational constant (6.674 × 10^-11 Nm^2/kg^2), M is the mass of the Earth, and r1 is the distance from the center of the Earth.

Now, let's determine the kinetic and potential energies when the satellite is at its greatest distance (apogee) from the Earth.

Speed at apogee (v2) = 3.66 km/s
Distance from the center of the Earth at apogee (r2) = ? (what we need to find)

Since the principle of conservation of mechanical energy states that the sum of kinetic and potential energies is constant, we can equate the two sets of energies.

K1 + U1 = K2 + U2

Plugging in the respective values, we have:
(1/2)mv1^2 - GMm/r1 = (1/2)mv2^2 - GMm/r2

Since the mass of the satellite cancels out, we can simplify the equation to:
(1/2)v1^2 - GM/r1 = (1/2)v2^2 - GM/r2

Now, substituting the given values:
(1/2)(9.00 km/s)^2 - (6.674 × 10^-11 Nm^2/kg^2) / (7.00 × 10^6 m) = (1/2)(3.66 km/s)^2 - (6.674 × 10^-11 Nm^2/kg^2) / r2

Solving for r2 gives us the distance from the satellite to the center of the Earth at apogee.