A satellite follows the elliptical orbit shown in FIGURE P12.77. The only force on the satellite is the gravitational attraction of the planet. The satellite's speed at point 1 is 8000 m/s. b. What is the satellite's speed at point 2?

Verified step by step guidance

1. Identify the conservation of mechanical energy in the system, which states that the total mechanical energy (kinetic energy plus potential energy) of the satellite remains constant if the only force acting is conservative, such as gravity.
2. Write the expression for the total mechanical energy at point 1 and point 2. Since the mechanical energy is conserved, set these two expressions equal to each other: \(E_1 = E_2\).
3. Express the kinetic energy at each point as \(K = \frac{1}{2}mv^2\) and the gravitational potential energy as \(U = -\frac{GMm}{r}\), where \(m\) is the mass of the satellite, \(v\) is its speed, \(r\) is the distance from the center of the planet to the satellite, \(G\) is the gravitational constant, and \(M\) is the mass of the planet.
4. Substitute the known values and expressions into the conservation of energy equation. Solve this equation for the unknown speed \(v_2\) at point 2.
5. Remember to consider the distances from the planet to the satellite at points 1 and 2, as these will affect the potential energy values at each point.