Multi-particle analysis of Momentum

Claimed by mhelmstadter3

The Main Idea

The Momentum Principle is the first fundamental principle of mechanics and describes the relationship between the change in momentum of a system and the total amount of interaction (or total amount of force) within the surroundings. In terms of the system and surroundings, both can be set in any way necessary, where the system may just include a person or the entire Earth. The Momentum Principle can be used in nearly all situations, and it is typically advised to start a problem by first writing out the Momentum Principle and then branching out (by rearranging or substituting values) in order to solve a problem.

A Mathematical Model

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

p is the momentum of the system. In the equation, momentum (measured in kg*m/s) is expressed as the "change in momentum" ([math]\displaystyle{ ∆\vec{p} = \vec{p}_{final} - \vec{p}_{initial} }[/math]), which includes both the magnitude and direction of the momentum.

F is the net force from the surroundings. Force (measured in Newtons, or N) includes the interactions between system and the surroundings, like the gravitational force exerted by the Earth on us or the force that a compressed spring exerts on a mass. In the Momentum Principle, the force includes both the magnitude and direction. Also, it is important to note that the Momentum Principle calls for the net force, which is the sum of all the different forces from the surroundings, like adding both the force of gravity and the force of the spring together to calculate the net force. Because of this, it is even more crucial to pay attention to the direction of the forces as a positive or negative sign error could cause an error in the calculated net force.

t is the time (measured in seconds, or s). Specifically, the Momentum Principle calls for the "change in time" ([math]\displaystyle{ ∆\vec{t} = \vec{t}_{final} - \vec{t}_{initial} }[/math]), or in other words, the duration of the interaction is needed.

The Momentum Principle can be further manipulated to find the change in velocity of the system. This is represented by the formula: Vf = Vi + (Fnet/m)*deltat

Vf is the final velocity of the system, and Vi is the initial velocity of the system. The difference between the two can be shortened to deltaV, and represents the change in velocity of the system.

Fnet is the net force acting on the system.

m is the mass of the system.

deltat is the change in time of the system in which it is acted on by the force.

This manipulation of the momentum principle is very useful when it comes to updating the position of the system because it gives you the change in velocity over the observed time interval.

A Computational Model

Click on the link to see the Momentum Principle through VPython!

Make sure to press "Run" to see the principle in action!

Teach hand-on with GlowScript

Examples

Simple

Two external forces <40,-70,0>N and <20,10,0>N, act on a system. What is the net force acting on the system?

Answer: <60,-60,0>N

Explanation:

[math]\displaystyle{ \vec{F}_{net} = \vec{F}_{1} + \vec{F}_{2} }[/math]

[math]\displaystyle{ \vec{F}_{net} }[/math] = <40,-70,0)N + <20,10,0>N = <60,-60,0>N

Middling

A hockey puck is sliding along the ice with nearly constant momentum <10,0,5>kg*m/s when it is suddenly struck by a hockey stick with a force <0,0,2000>N that lasts for only 3 milliseconds (3e-3s). What is the new (vector) momentum of the puck?

Answer: <10,0,11>kg*m/s

Explanation:

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

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

[math]\displaystyle{ \vec{p}_{final} }[/math] - <10,0,5>kg*m/s = <0,0,2000>N * (3e-3)s

[math]\displaystyle{ \vec{p}_{final} }[/math] = <10,0,11>kg*m/s

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.