8.1 Linear Momentum and Force

Linear Momentum

Definition of linear momentum

Linear momentum ($p$) measures how much "motion" an object carries, combining both how massive it is and how fast it's going. The formula is straightforward:

$p = mv$

where $m$ is mass and $v$ is velocity. The SI unit is kilogram-meter per second (kg·m/s).

Because velocity is a vector, momentum is also a vector quantity. Its direction always matches the direction of the object's velocity. A car driving north has momentum pointing north.

• Greater mass or greater velocity means greater momentum. A slow-moving bowling ball can have the same momentum as a fast-moving bullet because the bowling ball makes up for its lower speed with much more mass.
• For example, a 7 kg bowling ball rolling at 3 m/s has momentum $p = 7 \times 3 = 21$ kg·m/s, while a 0.01 kg bullet traveling at 2100 m/s has the same momentum of 21 kg·m/s.

Momentum in Newton's second law

Newton's second law can be written in terms of momentum rather than acceleration. The net force on an object equals the rate of change of its momentum:

$F_{net} = \frac{dp}{dt}$

When mass is constant, this simplifies to the familiar $F_{net} = ma$, since $\frac{dp}{dt} = m\frac{dv}{dt} = ma$.

This momentum form of Newton's second law is actually the more general version. It still applies even when mass changes (like a rocket burning fuel), while $F = ma$ does not.

Impulse ($J$) connects force and momentum change over a time interval. It's defined as:

$J = F_{net} \Delta t = \Delta p$

The SI unit is the newton-second (N·s), which is equivalent to kg·m/s. Impulse equals the change in momentum.

• When you hit a tennis ball with a racket, the racket exerts a large force over a short time. That impulse changes the ball's momentum from one direction to the other.
• A longer contact time with the same force produces a larger impulse and a bigger momentum change. This is why following through on a swing matters.

Conservation of Momentum

Momentum analysis in collisions

When two objects collide, they exert forces on each other that change each object's individual momentum. However, the total momentum of the system stays constant as long as no external forces interfere. Collisions fall into categories based on what happens to kinetic energy:

• Elastic collisions: Both momentum and kinetic energy are conserved. Billiard ball collisions are close to elastic, and certain atomic-level interactions are truly elastic.
• Inelastic collisions: Momentum is conserved, but some kinetic energy is converted to other forms (heat, sound, deformation). Most real-world collisions are inelastic, like two cars crashing.
• Perfectly inelastic collisions: The objects stick together after impact and move with a common velocity. This type loses the maximum possible kinetic energy while still conserving momentum. Two lumps of clay smashing together is a classic example.

Conservation of momentum in systems

The law of conservation of momentum states that the total momentum of a closed (isolated) system remains constant. A closed system is one where no net external force acts on the objects involved.

For a two-object system, this is written as:

$m_1 v_1 + m_2 v_2 = m_1 v'_1 + m_2 v'_2$

where $v_1, v_2$ are the initial velocities and $v'_1, v'_2$ are the final velocities.

To solve a conservation of momentum problem:

1. Identify the system and confirm that external forces are negligible (or that you're looking at the instant of collision, where external forces have minimal effect).
2. Write out the total momentum before the event.
3. Set it equal to the total momentum after the event.
4. Solve for the unknown quantity (usually a velocity or mass).

This principle applies to collisions, explosions, and any interaction within an isolated system. A rocket launch is a good example: the rocket and exhaust gases form a system, and the forward momentum of the rocket is balanced by the backward momentum of the exhaust.

Force and Momentum Interactions

Newton's third law is the reason momentum conservation works. When two objects interact, object A exerts a force on object B, and object B exerts an equal and opposite force back on A. These forces act for the same duration, so the impulse on A is equal and opposite to the impulse on B.

That means whatever momentum one object gains, the other loses by the same amount. The total momentum of the system doesn't change. This connection between Newton's third law and conservation of momentum is one of the most important ideas in this unit.