Chapter 9 Linear Momentum and its Conservation

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9.1 Linear Momentum

Let us consider a particle of mass m in motion with velocity as shown in Fig. 9.1 below. The linear momentum of an object is defined as the product of the object's mass and velocity, or:

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Fig. 9.1 A Moving object.

(units: kg.m/s) (9.1)

It is a vector quantity, and we can break both sides into vector components form in x, y, and z as:

(9.2)

Comparing both sides of Eq. (9.2), gives:

(9.3)

From Newton’s second law of motion, we know that time rate of change of linear momentum of particle is equal to the net force acting on the particle, i.e.,

(9.4)

If , implies is constant. Also we know that:

(9.5)

From the KE and momentum formulas i.e., and we have:

(9.6)

which is the relation between KE and linear momentum.

9.2 Conservation of Linear Momentum

Let us consider two particles in motion, each with momentum and , respectively as shown in Fig. 9.2. Applying Newton’s second law to the two moving system of Fig. 9.2, gives:

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Fig. 9.2 Two particles in motion.

and (9.7)

Now using Newton’s third law of motion, which states that for every action, there is an equal and opposite reaction (action-reaction):

(9.8)

Or (9.9)

Substituting Eq. (9.7) into Eq. (9.9), we have

(9.10)

Now define total momentum as:

(9.11)

Þ (momentum conservation) (9.12)

giving rise to: (momentum conservation) (9.13)

(momentum conservation) (9.14)

which states that the total linear moment of a system is conserved. Equation (9.12) can also be written in x, y, and z components as:

and (9.15)

These results are known as the law of conservation of linear motion. A key point to remember is that the total momentum of objects is conserved no matter what the nature of the forces between those objects.

From Eq. (9.13) the change in momentum of the system can also be written as:

(momentum conservation) (9.16)

or (momentum conservation) (9.17)

with vector components given by:

and (9.18)

Equations (9.17) and (9.18) are a consequence of Newton’s third law of motion (action-reaction), that is change in moment of object 1 is equal and opposite to the change in momentum of the second object.