Multi-particle Analysis of Momentum

claimed by nacharya7

The Main Idea

The Momentum Principle is a foundation of classical physics that is applicable to almost any system of objects at any scale, micro- or macroscopic. As such it can be used to analyse systems of point particles as well as multi-particle systems; however, there are a few subtleties one needs to take note of when choosing to analyze a multi-particle system that this page will detail.

A Mathematical Model

The Momentum Principle is [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net} }[/math] where p is the momentum of the system and F is the net force from the surroundings.

The Energy Principle is [math]\displaystyle{ {Δ{E}} = {W} }[/math] where E is the total change of a system's energy ans W is the work done on the system by the surroundings.

You will also need to know how to find the center of mass: [math]\displaystyle{ {cm} = {\frac{\sum{mr}}{{M}_{tot}}} }[/math], where cm is the location of the center of mass relative to an origin, mr is a fractional mass and length product summed up to infinity, and M is the total mass of the system.

Procedure

In order to analyze the motion of a multi-particle system, we need to apply both the energy principle and momentum principle.

Step 0: Identify the system and the surroundings of the system.

Step 1: Compress the system into a point-particle located at the system's center of mass.

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Step 2: Apply all forces acting on the system with their tails connected to the center of mass, retaining direction and magnitude.

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Step 3: Use the energy principle on the point particle.

[math]\displaystyle{ Δ{K} = Δ{W}_{trans} = \vec{F} * Δ{r}_{cm} }[/math]

Step 4: Return to the real-system and visualize the initial and final states of the systems.

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Step 5: Calculate the work that each force does

Step 6: Set up the Energy Principle for the problem.

[math]\displaystyle{ Δ{K}_{trans} + Δ{K}_{rot} + Δ{U} = {W}_{surr} + {Q} }[/math]

Problem Description

Two wooden blocks of mass [math]\displaystyle{ {\vec{m}} }[/math] are shot by a bullet on a vertical apparatus such that the first wooden block along the axis of its center of mass while the other is hit slightly to the left. The bullet travels at speed [math]\displaystyle{ {\vec{v}} }[/math]. Which block travels further?

Solution

Step 0: System: Block, Bullet Surroundings: Earth

Step 1:

Compress block to point-particles centered at center of mass.

Step 2: Apply all forces to point particle (gravity as well)

[math]\displaystyle{ {F}_{net,1} = {{d}{\vec{p}} / {dt}} + {mg} = {F}_{net,2} }[/math]

Step 3: Apply Energy Principle to this case.

[math]\displaystyle{ {K}_{trans} = {W}_{surr} = {F}_{net} * {r}_{cm} }[/math]

From here we see that the translational kinetic energy for both blocks is the same, meaning they travel the same vertical distance, however by following through in our analysis we can see why this counter-intuitive answer is such.

Step 4: Find the initial and final states of the real system.

Initial:

Final:

Step 5: Each block goes up height [math]\displaystyle{ {h} }[/math], therefore
[math]\displaystyle{ {W}_{surr} = ({F}_{net,1} + {F}_{net,2}) * {r}_{cm} }[/math]

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