11.1.3 An Analogy with the Energy Principle

You’ve probably noticed that there is a similarity between the momentum principle and the energy principle of Chapters 9 and 10. For a system of one object acted on by a force:

\begin{array}{l} energy principle: & Δ K = W = \int_{x_{i}}^{x_{f}} F_{x} d x \\ momentum principle: & Δ p_{x} = J_{x} = \int_{t_{i}}^{t_{f}} F_{x} d t \end{array}

(11.13)

In both cases, a force acting on an object changes the state of the system. If the force acts over the spatial interval from $x_{i}$ to $x_{f}$ , it does work that changes the object’s kinetic energy. If the force acts over a time interval from $t_{i}$ to $t_{f}$ , it delivers an impulse that changes the object’s momentum. FIGURE 11.5 shows that the geometric interpretation of work as the area under the F-versus-x graph parallels an interpretation of impulse as the area under the F-versus-t graph.

The figure consists of two graphs that illustrate the impact of impulse and work on momentum.

FIGURE 11.5 Full Alternative Text

This does not mean that a force either creates an impulse or does work but does not do both. Quite the contrary. A force acting on an object both creates an impulse and does work, changing both the momentum and the kinetic energy of the object. Whether you use the energy principle or the momentum principle depends on the question you are trying to answer.

In fact, we can express the kinetic energy in terms of momentum as

K = \frac{1}{2} m v^{2} = \frac{{(m v)}^{2}}{2 m} = \frac{p^{2}}{2 m}

(11.14)

You cannot change an object’s kinetic energy without also changing its momentum.