Final answer:
The total work done by gravity on the satellite fragment is the change in gravitational potential energy from its initial orbit to the Earth's surface. We use the formula U = -G M m / r for potential energy, where G is the gravitational constant, M is the Earth's mass, m is the fragment's mass, and r is the radial distance. The work done by gravity is this change in potential energy, and if not accounted for in the final kinetic energy, it is converted into heat and other forms.
Step-by-step explanation:
To find the total work done by gravity on the satellite fragment, we first need to determine the initial and final gravitational potential energy since the fragment was initially in orbit and falls directly to Earth. The work done by gravity will be the difference in potential energy at the two positions. The formula for gravitational potential energy at a distance r from the center of the Earth is U = -G M m / r, where G is the gravitational constant, M is the mass of the Earth, m is the mass of the fragment, and r is the radial distance from the center of the Earth to the object.
We calculate the initial potential energy when the object was at the orbit radius (orbital distance from the Earth's center = 1.015 × Earth's radius) and final potential energy when it hits the ground (distance from the Earth's center = Earth's radius). The total work done by gravity will then be the change in potential energy, which is the final energy minus the initial energy. Any difference not accounted for by the final kinetic energy on impact (provided as 357.0 m/s for 73.0 kg mass) would be converted into other forms, like heat and sound, during the descent through the Earth's atmosphere.