Summary

General Principles

Law of Conservation of Momentum

The total momentum $\vec{P} = \vec{p}_{1} + \vec{p}_{2} + \dots$ of an isolated system is a constant. Thus

\vec{P}_{f} = \vec{P}_{i}

Newton’s Second Law

In terms of momentum, Newton’s second law is

\vec{F} = \frac{d \vec{p}}{d t}

Solving Momentum Conservation Problems

Model Choose an isolated system or a system that is isolated during at least part of the problem.

Visualize Draw a pictorial representation of the system before and after the interaction.

Solve Write the law of conservation of momentum in terms of vector components:

\begin{array}{l} {(p_{f x})}_{1} + {(p_{f x})}_{2} + . . . = {(p_{i x})}_{1} + {(p_{i x})}_{2} + . . . \\ {(p_{f y})}_{1} + {(p_{f y})}_{2} + . . . = {(p_{i y})}_{1} + {(p_{i y})}_{2} + . . . \end{array}

Review Is the result reasonable?

Important Concepts

Momentum $\vec{p} = m \vec{v}$

The figure shows the velocity and momentum vectors associated with an object.

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Impulse $J_{x} = \int_{t_{i}}^{t_{f}} F_{x} (t) d t = area under$ force curve

Impulse and momentum are related by the momentum principle

Δ p_{x} = J_{x}

The figure illustrates an impulse graph drawn on a coordinate plane .

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The impulse delivered to an object causes the object’s momentum to change. This is an alternative statement of Newton’s second law.

Momentum bar charts display the momentum principle $p_{f x} = p_{i x} + J_{x}$ in graphical form.

The figure illustrates a momentum bar chart to show the impact of impulse on the momentum of a system.

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System A group of interacting particles.

Isolated system A system on which there are no external forces or the net external force is zero.

The figure illustrates a system of three particles that interact with each other in an isolated environment.

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Applications

In a perfectly inelastic collision, two objects stick together and move with a common final velocity. In a perfectly elastic collision, they bounce apart and conserve mechanical energy as well as momentum.

The figure illustrates two spherical objects of different sizes interacting with each other to show the effect of collisions on momentum conservation.

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In an explosion, two or more objects fly apart from each other. Their total momentum is conserved.

The figure illustrates three spherical objects labeled 1, 2, and 3, respectively, flying apart from each other during an explosion.

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The same ideas apply in two dimensions. Both the x- and y-components of $\vec{P}$ must be conserved. This gives two simultaneous equations to solve.

The figure shows the path of two spherical objects labeled 1 and 2 colliding and moving in a two-dimensional plane.

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Rockets

The momentum of the exhaust-gas $+$ rocket system is conserved. Thrust is the product of the exhaust speed and the rate at which fuel is burned.