Momentum with respect to external Forces

Editing claimed by estaniforth3 Claimed by vkt3

The Main Idea

Momentum in an open system, is fundamentally different from that within a closed system. No longer do individual elements of a system's momentum equal to each other symettrically to add up to 0, however, they will have to even out to the magnitude of the added Force.

A Mathematical Model

They equation expressing this idea 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.

Examples

Standing on Earth, you throw a small rock with a mass of 0.5 kg into the air. At the instant it leaves your hand, the rock's velocity is v=<0.1,4.0,0.3> m/s Ignore air resistance.

a. Initial Momentum? m=0.5, v=<0.1,4,0.3> p=<0.05,2,0.15> kgm/s

b.Rock's momentum after 0.25 seconds? pf=pi+Fnet(deltat)

pf=<0.05,2,0.15>+<0,(-9.8)(0.5),0>(0.25)=<0.05,2,0.15>+<0,-1.225,0>=pf pf=<0.05,0.775,0.15> kgm/s

c.Calculate the average velocity of the rock from just after it leaves your hand to 0.25 seconds later. p=mv, v=p/m

vf=(pf/m)=(1/0.5)<0.05,0.775,0.15>=vf vf=<0.1,1.55,0.3>m/s vavg=(vi+vf)/2 = (0.5)*[<0.1,4,0.3> + <0.1,1.55,0.3>]= (0.5)<0.2,5.55,0.6>= vavg=<0.1,2.775,0.3>m/s

d. If a rock's initial position just as it leaves your hand is <0,1.2,0>m, find the vector position of the ball after 0.25 seconds.

ri=<0,1.2,0>m rf=ri+vavg(deltat)= <0,1.2,0>+<0.1,2.775,0.3>(0.25)= <0,1.2,0>+<0.025,0.694,0.075>= rf=<0.025,1.894,0.075>

Connectedness

This topic is the basis behind calculating most forms of linear movement with simple forces. While a simple formula, this equation is a powerful tool as that it can include any number of forces acting on a system, and show the change in momentum of an object.

This equation can be used to calculate the simple movements of objects in the vacuum of space with respect to the magnitude of forces acting upon the system.

History

The Momentum Principle was born from Newton's First Law which states that an object at rest will remain at rest, and an object in motion will remain in motion unless acted upon by an external force.