8.2. Key Equations and Definitions http://www.ck12.org
8.2 Key Equations and Definitions
We start with a definition of momentum. Since mass is a scalar and velocity is a vector, momentum — their product
— is a vector. The momentum of an object with massmtraveling at a velocityvis:
~p=m~v[1]It points in the direction of an object’s velocity, and has a magnitude equal to the object’s mass times its speed. For
a system of many objects, the momentum of the system is equal to the sum of the individual momentum vectors:
p~sys=∑~pi[2]
Newton referred to momentum in his Second Law; in his terms, if an object ofmexperiences a net forceFnetin a
period∆t, the following relationship holds:
F~net=∆~p
∆t=m
∆~v
∆t=m~a[3]In other words,an unbalanced force changes an object’s momentum, with a change equal to
∆~p=F~net∆t[4]Again, momentum is such an important quantity that Newton defined his Second Law in terms of it.
Finally, according to the law of conservation of momentum, the final momentum of a closed system — like its energy
— is equal to its initial value. Using [2], we can write:
∑pinitial~ =∑~p
final [5]
An important point is that since momentum is a vector, both its magnitudeanddirection are conserved. The (vector)
sum of the initial momentum vectors will be equal to the sum of the final vectors.
Much like forces cannot affect motion in direction perpendicular to them—think horizontal velocity of projectiles
in 2-D, perpendicular components of the momentum vector are independent. This means that [3] implies that any
mutually perpendicular components of momentum would have to be conserved as well, in particular:
∑p~yi=∑p~yf[6]
∑p~xi=∑p~xf[7]This fact is useful in two dimensional problems, where you can set up equations for each component.