14.5 Conservation of linear momentum

We have briefly defined linear momentum, while describing Newton's second law of motion. The law defines force as the time rate of linear momentum of a particle. It directly provides a measurable basis for the measurement of force in terms of mass and acceleration of a single particle. As such, the concept of linear momentum is not elaborated or emphasized for a single particle. However, we shall see in this module that linear momentum becomes a convenient tool to analyze motion of a system of particles - particularly with reference to internal forces acting inside the system.

It will soon emerge that Newton's second law of motion is more suited for the analysis of the motion of a particle like objects, whereas concept of linear momentum is more suited when we deal with the dynamics of a system of particles. Nevertheless, we must understand that these two approaches are interlinked and equivalent. Preference to a particular approach is basically a question of suitability to analysis situation.

Let us now recapitulate main points about linear momentum as described earlier :

(i) It is defined for a particle as a vector in terms of the product of mass and velocity.

$$\begin{array}{l}\mathbf{p}=m\mathbf{v}\end{array}$$

The small " p " is used to denote linear momentum of a particle and capital " P " is used for linear momentum of the system of particles. Further, these symbols distinguish linear momentum from angular momentum ( L ) as applicable in the case of rotational motion. By convention, a simple reference to "momentum" means "linear momentum".

(ii) Since mass is a positive scalar quantity, the directions of linear momentum and velocity are same.

(iii) In physical sense, linear momentum is said to signify the "quantum or quantity of motion". It is so because a particle with higher momentum generates greater impact, when stopped.

(iv) The first differentiation of linear momentum with respect to time is equal to external force on the single particle.

$$\begin{array}{l}{\mathbf{F}}_{\mathrm{Ext.}}=\frac{d\mathbf{p}}{dt}=m\mathbf{a}\end{array}$$

Momentum of a system of particles

The concept of linear momentum for a particle is extended to a system of particles by summing the momentum of individual particles. However, this sum is a vector sum of momentums. We need to either employ vector addition or equivalent component summation with appropriate sign convention as discussed earlier. Linear momentum of a system of particles is, thus, defined as :

Momentum of a system of particles

The linear momentum of a system of particles is the vector sum of linear momentums of individual particles.

Momentum of a system of particles

$$\begin{array}{l}\mathbf{p}=m_1{\mathbf{v}}_1+m_2{\mathbf{v}}_2+................+m_{\mathrm{n}}{\mathbf{v}}_{\mathrm{n}}\end{array}$$

$$\begin{array}{l}\Rightarrow \mathbf{p}=\sum m_i{\mathbf{v}}_i\end{array}$$

From the concept of "center of mass", we know that :

$$\begin{array}{l}M{\mathbf{v}}_{\mathrm{COM}}=m_1{\mathbf{v}}_1+m_2{\mathbf{v}}_2+................+m_{\mathrm{n}}{\mathbf{v}}_{\mathrm{n}}\end{array}$$

Comparing two equations,

$$\begin{array}{l}\mathbf{P}=M{\mathbf{v}}_{\mathrm{COM}}\end{array}$$

The linear momentum of a system of momentum is, therefore, equal to the product of total mass and the velocity of the COM of the system of particles.

External force in terms of momentum of the system

Just like the case for a single particle, the first differentiation of the total linear momentum gives the external force on the system of particles :