FHSST Physics/Momentum/Properties

What properties does momentum have?

You may at this stage be wondering why there is a need for introducing momentum. Remarkably momentum is a conserved quantity. Within an isolated system the total momentum is constant. No matter what happens to the individual bodies within an isolated system, the total momentum of the system never changes! Since momentum is a vector, its conservation implies that both its magnitude and its direction remains the same.

Momentum is conserved in isolated systems!

This Principle of Conservation of Linear Momentum is one of the most fundamental principles of physics and it alone justifies the definition of momentum. Since momentum is related to the motion of objects, we can use its conservation to make predictions about what happens in collisions and explosions. If we bang two objects together, by conservation of momentum, the total momentum of the objects before the collision is equal to their total momentum after the collision.

Principle of Conservation of Linear Momentum:

The total linear momentum of an isolated system is constant.

or

In an isolated system the total momentum before a collision

(or explosion) is equal to the total momentum after the

collision (or explosion).

Let us consider a simple collision of two pool or billiard balls. Consider the first ball (mass m₁) to have an initial velocity (${\overrightarrow {u_{1}}}$). The second ball (mass m₂) moves towards the first ball with an initial velocity ${\overrightarrow {u_{2}}}$. This situation is shown in Figure 6.1. If we add the momenta of each ball we get a total momentum for the system. This total momentum is then

${\begin{matrix}{\overrightarrow {p}}_{total\ before}=m_{1}{\overrightarrow {u_{1}}}+m_{2}{\overrightarrow {u_{2}}},\end{matrix}}$

File:Fhsst mom2.png

Figure 6.1: Before the collision.

Figure 6.2: After the collision.

After the two balls collide and move away they each have a different momentum. If we call the final velocity of ball 1 ${\overrightarrow {v_{1}}}$ and the final velocity of ball 2 ${\overrightarrow {v_{2}}}$ (see Figure 6.2), then the total momentum of the system after the collision is

${\begin{matrix}{\overrightarrow {p}}_{total\ after}=m_{1}{\overrightarrow {v_{1}}}+m_{2}{\overrightarrow {v_{2}}},\end{matrix}}$

This system of two balls is isolated since there are no external forces acting on the balls. Therefore, by the principle of conservation of linear momentum, the total momentum before the collision is equal to the total momentum after the collision. This gives the equation for the conservation of momentum in a collision of two objects,

$m_{1}{\overrightarrow {u_{1}}}+m_{2}{\overrightarrow {u_{2}}}=m_{1}{\overrightarrow {v_{1}}}+m_{2}{\overrightarrow {v_{2}}}$
m1	: mass of object 1 (kg)
m2	: mass of object 2 (kg)
${\overrightarrow {u_{1}}}$	: initial velocity of object 1 (m.s-1 + direction)
${\overrightarrow {u_{2}}}$	: initial velocity of object 2 (m.s-1 + direction)
${\overrightarrow {v_{1}}}$	: final velocity of object 1 (m.s-1 + direction)
${\overrightarrow {v_{2}}}$	: final velocity of object 2 (m.s-1 + direction)

This equation is always true - momentum is always conserved in collisions.

The chapter `Collisions and Explosions' deals with applications of momentum conservation.