Physics
Physics
Problem
A research satellite of negligible mass follows a closed elliptical orbit around Earth. A verbal description of a potential-energy diagram for this system is given below. • The horizontal axis represents the satellite’s distance r from Earth’s center, marked at regular increments from 6.5×10^6 m to 4.2×10^7 m. • The vertical axis represents gravitational potential energy U relative to zero at infinite separation. • The curve is a smooth, monotonically increasing function of r; its magnitude is largest (most negative) at the left edge of the axis and approaches 0 J asymptotically as r increases. • Two vertical dashed lines identify the satellite’s perigee distance r_p = 7.0×10^6 m and apogee distance r_a = 4.0×10^7 m. • At each dashed line a dot on the potential-energy curve is connected to a point higher on the same vertical line; the vertical segment between these two points represents the satellite’s kinetic energy at that location. Answer the following questions about the motion and energies of the satellite. (Neglect any resistive forces.) (a) Describe how the satellite’s kinetic and potential energies vary as it travels from apogee to perigee. (b) Explain why the total mechanical energy of the Earth–satellite system is represented by a horizontal line on this diagram, and state whether that line is above, below, or coincident with the potential-energy curve. (c) The satellite’s speed at apogee is observed to be 1.0 km/s. Without doing numerical calculations, explain qualitatively how its speed at perigee compares to this value and why. (d) Suppose mission control gradually lowers the apogee to 3.0×10^7 m while keeping the perigee distance unchanged. Describe qualitatively how each of the following would change, if at all: the satellite’s total mechanical energy; its speed at apogee; its speed at perigee.Intuitive explanation
Gravitational potential energy is always negative and grows less negative (increases) as you move farther from Earth—this is the shape of the curve. The key insight is that total mechanical energy is conserved in orbit, so at any distance , the total energy equals potential energy plus kinetic energy. The vertical segments shown tell you directly how much kinetic energy remains at perigee and apogee: wherever the potential energy is more negative, less energy is "left over" for motion, so the satellite moves slower there. Use this energy-conservation principle to connect orbital speeds, distances, and the total energy budget.
Intuitive explanation
Gravitational potential energy is always negative and grows less negative (increases) as you move farther from Earth—this is the shape of the curve. The key insight is that total mechanical energy is conserved in orbit, so at any distance , the total energy equals potential energy plus kinetic energy. The vertical segments shown tell you directly how much kinetic energy remains at perigee and apogee: wherever the potential energy is more negative, less energy is "left over" for motion, so the satellite moves slower there. Use this energy-conservation principle to connect orbital speeds, distances, and the total energy budget.