Momentum and the Center-of-Mass of a System of Particles

The Inertial System and Conservation of Momentum

A key concept in this study is the inertial system. We can define an inertial system as one where we can "draw a box" around the particles such that no net external force crosses the boundary of that box.

If the net external force $\vec{F}_{\text{ext, net}}$ acting on a system is zero, the total momentum of the system remains constant over time. This is known as the Law of Conservation of Momentum:

$\text{If } \vec{F}_{\text{ext, net}} = 0, \text{ then } \vec{p}_{\text{total}} = \text{constant}$

This means that even if the particles inside the system collide, explode, or push against one another, the sum of their momenta will not change as long as no outside force interferes.

Exam Tip: To determine if momentum is conserved, always identify your system boundaries. If an external force (like friction or gravity acting on a single object) is present, the total momentum of that specific system will change.

Example: The Recoiling Cannon

Consider a 5,000 kg cannon that fires a 50 kg cannonball at 200 m/s. Since the cannon and ball start at rest, the initial momentum is zero. To conserve momentum, the final momentum must also be zero:

$\vec{p}_i = 0$ $\vec{p}_f = M\vec{v}_c + m_b\vec{v}_b = 0$

Solving for the cannon's recoil velocity $\vec{v}_c$: $\vec{v}_c = -\frac{m_b}{M} \vec{v}_b$ $\vec{v}_c = -\frac{50\text{ kg}}{5,000\text{ kg}}(200\text{ m/s}) = -2.0\text{ m/s}$

The negative sign indicates the cannon moves in the opposite direction of the ball.

The Center of Mass (COM)

The center of mass is a unique point that represents the average position of the mass in a system. It allows us to treat a complex collection of particles as a single point mass for the purpose of studying external motion.

The position of the center of mass $\vec{r}_{\text{com}}$ is calculated as:

$\vec{r}_{\text{com}} = \frac{1}{M} \sum_{i=1}^{N} m_i \vec{r}_i = \frac{m_1 \vec{r}_1 + m_2 \vec{r}_2 + \dots}{m_1 + m_2 + \dots}$

Where $M$ is the total mass of the system.

Newton's Law for a System

While individual particles follow $\vec{F}_i = m_i \vec{a}_i$, the entire system follows a simplified version of Newton's Second Law. The net external force acting on a system is equal to the total mass of the system multiplied by the acceleration of its center of mass:

$\vec{F}_{\text{ext}} = M \vec{a}_{\text{com}}$

This explains why an object exploding in mid-air still follows a parabolic trajectory; while the individual pieces fly in many directions, the center of mass of all those pieces continues to move as if it were a single projectile under the influence of gravity.

Collisions: Elastic vs. Inelastic

A collision is an isolated event where bodies exert strong forces on each other for a short duration. In all collisions within an inertial frame, total linear momentum is conserved. However, kinetic energy ($K$) may or may not be conserved.

1. Inelastic Collisions

In an inelastic collision, kinetic energy is not conserved; some energy is transformed into heat, sound, or internal deformation.

• Completely Inelastic Collision: The objects stick together after impact and move with a single common velocity $v'$.

$\vec{p}_i = \vec{p}_f \implies m_1 v_1 + m_2 v_2 = (m_1 + m_2)v'$

2. Elastic Collisions

In an elastic collision, both total momentum and total kinetic energy are conserved.

$\text{Momentum: } m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'$ $\text{Energy: } \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2 = \frac{1}{2}m_1 (v_1')^2 + \frac{1}{2}m_2 (v_2')^2$

For a collision where a moving mass $m_1$ hits a stationary target $m_2$ ($v_2 = 0$), the final velocities are:

$v_1' = \left(\frac{m_1 - m_2}{m_1 + m_2}\right) v_1$ $v_2' = \left(\frac{2m_1}{m_1 + m_2}\right) v_1$

Limiting Cases for Elastic Collisions:

• If $m_1 \gg m_2$: The heavy mass continues at $v_1$, and the light mass shoots forward at $2v_1$.
• If $m_1 \ll m_2$: The heavy mass remains stationary, and the light mass bounces back at $-v_1$.
• If $m_1 = m_2$: The masses exchange velocities; $m_1$ stops and $m_2$ moves forward at $v_1$.

Impulse and Change in Momentum

The concept of impulse relates the force applied to an object to its change in momentum. If a constant average force $F_{\text{avg}}$ is applied over a time interval $\Delta t$, the impulse is:

$\Delta \vec{p} = \vec{F}_{\text{avg}} \Delta t$

This is vital in safety engineering. For example, when a car hits a barrier, the force experienced by the passengers depends on how quickly the momentum changes. A longer impact time $\Delta t$ results in a smaller average force $F_{\text{avg}}$.

Momentum in 2D and 3D

Momentum is a vector, meaning it must be conserved in every dimension independently. In a two-dimensional collision, you must solve for the $x$ and $y$ components separately:

$\sum p_{i,x} = \sum p_{f,x}$ $\sum p_{i,y} = \sum p_{f,y}$

By decomposing the velocity vectors into $\cos \theta$ and $\sin \theta$ components, you can solve for unknown speeds and angles in complex off-center collisions.