# Momentum

## What is Momentum?

The term momentum is associated with moving bodies and is the product of the mass and velocity of a body that is in motion. In general, momentum is represented by p.

In physics, momentum is the mass in motion and represent the overall motion that any moving object exhibit. Thus, is given by the expression,

Momentum (p) = mass (m) * velocity (v)

### Basics of Momentum

The word momentum is very much in our day-to-day life with respect to the daily activities taking place. In general, suppose we talk about a player which is playing a particular sport and is doing very well for the past few years then it is said that he or she is exhibiting or having great momentum.

More specifically, in physics terminology, momentum describes the mass of the moving body meaning amount of mass while in motion. It is represented by the symbol p, which describes the product of mass and velocity of the moving body.

Momentum is a vector quantity. Before elaborating, let us have an idea about

What is a vector quantity?

A vector quantity is one that has both magnitude (i.e., the numerical value) and direction. More simply, in order to completely describe a vector quantity both numerical value and direction are necessary. Unlike, vector quantity another quantity is scalar in nature which is completely represented by a magnitude only.

Examples of vector quantities are displacement, velocity, acceleration, momentum, force, etc.

In order to make a comparison between two vector quantities, both the magnitude and directions of the two quantities must be compared.

### Linear Momentum

The linear momentum of a particle having mass m and velocity v is given as

p = m.v

Basically, the momentum of a particle is associated with the net force acting on that particle.

Taking the derivative of particle’s momentum, we will get,

dp/dt

since, p = mv, so on substituting,

d(mv)/dt

As we know that the mass of the particle is constant, therefore,

m(dv/dt)

since, dv/dt = acceleration (a), hence

ma = Fnet (representing the net force acting on the body)

Therefore, we can write,

Fnet = dp/dt

### Conservation of Linear Momentum

In an isolated system (i.e., the one within which some particles interact with each other only) linear momentum is quite useful. Sometimes we study such systems where the particles within it interact with each other with strong forces.

Consider that within an isolated system there are 2 particles interacting only with each other, then their overall momentum will remain constant.

p1i + p2i = p1f + p2f

: subscript i is used to show initial momenta while f is used to show final momenta.

more simply,

m1i v1i + m2i v2i = m1f v2f + m1f v1f

The above equation is a vector in nature and it is clear that that the momentum of the system is conserved.

It is to be noted here that when the particles within the system move then after a certain point of time, after motion they get close and exert strong forces on each other then get separated and move freely. However, the particles possess similar momenta before and after the collision.

: p1i and p2i are momenta before collision and p1f and p2f are momenta after the collision.

For an isolated system with colliding particles, the total momentum is of conserved nature however, mechanical energy may or may not be conserved. In case if the mechanical energy remains the same before and after collision then such collision is elastic otherwise inelastic.

In case of elastic collision,

½ m1vi12 + ½ m2vi22 = ½ m1vf12 + ½ m1vf12

This is the equation for the conserved momentum for an isolated system where two particles are undergoing elastic collision.