Momentum vs Inertia
Inertia and momentum are two concept involved in the study of motion of solid bodies. Momentum and inertia are used to describe the current state of the object. Both inertia and momentum are concepts related to the mass of the object. Inertia and momentum are relativistic variants, which mean the equations for calculating these properties vary when the velocity of the object approaches the speed of light. However, Inertia and momentum plays a very important role in both Newtonian mechanics (classical mechanics) and relativistic mechanics.
Momentum
Momentum is a vector. It is defined as the product of the velocity and the inertial mass of the object. Newton’s second law is mainly focused on momentum. The original form of the second law, force = mass x acceleration, can be written in terms of velocity change as: force = (mass x final velocity – mass x initial velocity)/time. In a more mathematical form, this can be written as change of momentum/time. The acceleration described in Newton’s formula is actually an aspect of momentum. The momentum is said to be conserved if no external forces act on a closed system. This can be seen in the simple instrument “balance balls”, or scientifically known as Newton’s cradle. Momentum takes the forms of linear momentum and angular momentum. The total momentum of a system is equal to the combination of linear momentum and angular momentum
Inertia
Inertia is derived from the Latin word “iners”, which means idle or lazy. Inertia is a measurement of how lazy the system is. Inertia of a system gives us the meaning how hard it is to change the current state of the system. The higher the inertia of a system, harder it is to change the velocity, acceleration, direction of the system. Objects having higher masses have higher inertia. That’s why they are hard to move. Given that it is on a frictionless surface, a moving higher mass object would be hard to stop too. Newton’s first law gives a very good idea about inertia of a system. It states “an object not subject to any net external force, moves at a constant velocity”. This tells us that an object has a property that is not changed, unless, there is an external force acting upon it. An object at rest can also be considered as an object having null velocity. In relativity, the inertia of an object tends to be infinity when the speed of the object reaches the speed of light. Hence an infinite force is required to increase the current velocity. It can be proved that no mass can reach the speed of light.
What is the difference between momentum and inertia?  Momentum is a physically calculable property, while inertia cannot be calculated using a formulae.  Inertia is just a concept to help us understand and define mechanics better.  While, momentum comes in the forms of linear momentum and angular momentum, the inertia comes only in one form.  Momentum is conserved in some cases. Momentum conservation can be used to solve problems. Inertia doesn’t have to be conserved in any case.
