Final answer:
To calculate the work needed to adjust a spacecraft's orbit around Mars, one must find the difference in gravitational potential energy between the initial and final orbits, accounting for the planet's mass, the spacecraft's mass, and the respective orbital radii. This involves potential energy calculations with Mars' properties and orbital altitudes converted to SI units, resulting in an energy in joules.
Step-by-step explanation:
Work Required for Orbital Change
To calculate the work to move the spacecraft from a circular orbit at 2500 km above the surface of Mars to a circular orbit that is 3500 km above the surface, we must consider the change in gravitational potential energy (ΔU) of the object. The potential energy of an object in a circular orbit around a planet is given by U = -GMm/r, where G is the universal gravitational constant, M the mass of the planet, m the mass of the object, and r the distance from the center of the planet to the object. Since the spacecraft is already in orbit, we are interested in the difference in potential energy between the two orbits.
For the initial orbit, ri is the sum of Mars' radius and the spacecraft's altitude above its surface, ri = RM + 2500 km. For the final orbit, rf is rf = RM + 3500 km. The work done, W, is the difference in potential energy W = ΔU = Uf - Ui.
Calculating the initial and final potential energies and then finding their difference will provide the amount of work required. Remember to convert the altitude from kilometers to meters to match the mass and radius of Mars, which are given in SI units. The final answer will be in joules (J), the SI unit for work or energy.