Physics

Problem

A small spacecraft of mass

200\,\text{kg}

is moved radially outward from an orbit of radius

r_{1}=7.0\times10^{6}\,\text{m}

to a higher orbit of radius

r_{2}=1.4\times10^{7}\,\text{m}

around a planet. In the interval considered, the only significant interaction is an attractive radial force of magnitude

F(r)=\dfrac{2.0\times10^{7}}{r^{2}}\,\text{N}

, where

r

is the distance from the planet’s centre. Assume the motion is along the same radial line. a. (i) Starting from the definition of work, derive an expression for the work done by the force when the spacecraft is taken from

r_{1}

r_{2}

. b. (ii) Evaluate the work for the given numerical data, stating its sign clearly. c. (iii) Explain physically why the work has the sign obtained in part (ii).

Intuitive explanation

This problem is about work and energy in a non-uniform force field. The key insight is that you don't need to know the details of how the spacecraft moves—only that it travels from one radius to another against a radial force. Use the work-energy theorem: calculate the work done by the gravitational force (by integrating $F(r)$ over the displacement), then relate it to the change in kinetic energy. The integral $\int F \, dr$ is your main tool, and the $r^{-2}$ form makes it tractable.

Intuitive explanation

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