Multi-particle analysis of Momentum

Claimed by mhelmstadter3

The Main Idea

Newton's third law states (on a basic level) that for every action, there is an equal and opposite reaction. The application and importance of this law in terms of momentum is that in a "closed system," momentum is conserved. The meaning of "closed" is that within the system--the actions and reactions on the involved objects--all forces are accounted for. We can model conservation of momentum by looking at the bodies of a system as particles, which is what will be discussed for the remainder of this article. The momentum principle and the multi-particle analysis of it make it easier to analyze situations such as collisions, and can make it much easier to solve problems.

A Mathematical Model

Mathematically, momentum is the product of a body's (in our case, a particle's) mass and velocity. It is also defined as the product of mass, net force, and change in time.

Where p is momentum, F is the net force from the surroundings, and t is time, measured in seconds, The Momentum Principle is formulaically defined as

[math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net} }[/math] (or [math]\displaystyle{ ∆\vec{p} = \vec{F}_{net} * {∆t} }[/math]).

Taking two particles, particle 1 and particle 2, we can define the total momentum as:

[math]\displaystyle{ P = p_1 + p_2 = m_1v_1 + m_2v_2 }[/math]

and deriving from [math]\displaystyle{ F = m * a }[/math], the rate of change is the sum of the vector forces, and therefore:

[math]\displaystyle{ \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 }[/math].

Newton's third law gives that:

[math]\displaystyle{ \frac{d\vec{P}}{dt} = 0 }[/math],

so,

[math]\displaystyle{ m_1\vec{a}\_1 + m_2\vec{a}_2 = 0 }[/math].

[math]\displaystyle{ \vec{F}_{12} = -\vec{F}_{21} }[/math].

[math]\displaystyle{ \vec{F}_{12} = m_2\vec{a}_2 }[/math].

The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.

If i and j are the first two particles out of total N particles,

[math]\displaystyle{ \vec{P} = \sum_{i=1}^N m_i\vec{v}_i }[/math] and [math]\displaystyle{ \frac{d\vec{P}}{dt} = \sum_{j=1}^N m_j\vec{a}_j = \sum_{j=1}^N\sum_{i=1}^N F_{ij} }[/math].

Each term in this sum cancels with another due to the application of "equal and opposite."

If the system is subject to a force that is not included in the system (external),

[math]\displaystyle{ \frac{d\vec{P}}{dt} =\vec{F}_{ext} }[/math].

A Computational Model

Click on the link below to understand the basics of the momentum principle in VPython,

GlowScript

or check out this collision simulation to see a multi-particle analysis of it:

Collision Lab

Examples

Simple

Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.

Answer: 9.4e4 N

Explanation:

With the car as particle 1 and the truck as particle 2, total initial momentum = [math]\displaystyle{ p_1 + p_2 = m_1v_1 + m_2v_2 }[/math]

Initial momentum = Final momentum, therefore total final momentum = [math]\displaystyle{ p_1 + p_2 = m_1v_1 + m_2v_2 }[/math], which is equal to 9.4e4N.

Mid-Difficulty

Answer:

Explanation:

Difficult

In outer space a rock of mass 5kg is acted on by a constant net force <29,-15,40>N during a 4s time interval. At the end of this time interval the rock has a velocity of <114,94,112>m/s. What is the rock's velocity at the beginning of the time interval?

Answer: <90.8,106,80>m/s

Explanation:

[math]\displaystyle{ ∆\vec{p} = \vec{F}_{net} * {∆t} }[/math]

[math]\displaystyle{ \vec{p}_{final} - \vec{p}_{initial} = \vec{F}_{net} * {∆t} }[/math]

[math]\displaystyle{ \vec{p} = m * \vec{v} }[/math]

[math]\displaystyle{ m\vec{v}_{final} - m\vec{v}_{initial} = \vec{F}_{net} * {∆t} }[/math]

(5kg * <114,94,112>m/s) - (5kg * [math]\displaystyle{ \vec{v}_{initial} }[/math]) = <29,-15,40>N * 4s

[math]\displaystyle{ \vec{v}_{initial} }[/math] = <90.8,106,80>m/s

Connectedness

All over the world and at every point in time, interactions are continuously occurring, and I thought it was interesting to see how the Momentum Principle was the most fundamental principle that is used in starting to the analyze the different interactions. Although there is not a direct relationship between the concept of the Momentum Principle and my major in Biochemistry (which would have more connections with the Energy Principle), there are many industrial applications of the Momentum Principle. Again, the Momentum Principle is not directly connected to the applications, but it is used in the process (especially in the beginning) of industrial application. For example, when creating life saving airbags and seat belts for cars, the Momentum Principle is used. The final momentum of a car during an accident would be zero, or would stop, and the initial momentum would be based on the mass and velocity of the car. With the change in momentum fixed, the airbag and seat belt would focus on increasing the time taken for the body's momentum to reach zero (final momentum), which would consequently reduce the force of the collision and protect the body from getting as injured. With the Momentum Principle being applicable in so many areas of my life, I found the concept even more interesting.

History

Although the Momentum Principle is credited as Newton’s second law of motion, it is difficult to just credit Isaac Newton (1643AD – 1727AD) for the development of the principle. As the Momentum Principle is the quantitative and more in-depth representation of Newton’s first law of motion (“An object tends to be at rest or moves in a straight line and a constant speed except to the extent that it interact with other objects”), the development of the first law also serves an important role in the history of the Momentum Principle. Aristotle (384BC – 322BC) initially proposed that objects had the natural tendency to be at rest and that a push (or a force) was absolutely needed to keep the object moving. His proposal was challenged by Galileo (1564AD – 1642AD), who introduced the idea that objects had the natural tendency to travel in a straight line at constant speed unless something (or a force) was interacting with something. Likewise, Descartes (1596AD – 1650AD) also contributed as he proposed three laws of nature in his “Principle of Philosophy,” which actually outlined the later published Newton’s first law of motion. After studying Descartes, Newton adopted Descartes’ principles as his first law of motion, and alongside the famous story of Newton sitting under an apple, Newton was able to create the Momentum Principle, or his second law of motion.

See also

As the Momentum Principle is the first of three fundamental principles of mechanics, the next possible topics to examine would be the other fundamental principles, the Energy Principle and the Angular Momentum Principle. Also, although the Momentum Principle is an extremely important concept that usually signals the start of a momentum related problem, the principle branches out into other momentum topics like Impulse and Iterative Prediction, which are used to solve other types of problems.

External links

[1] https://www.khanacademy.org/science/physics/linear-momentum/momentum-tutorial/v/introduction-to-momentum

[2] https://www.youtube.com/watch?v=ZvPrn3aBQG8

References

[1] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 29 Nov. 2015. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>

[2] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.

[3] Fenton, Flavio. "Momentum and Second Newton's Law." 26 Aug. 2015. Lecture.