# Introduction to Linear Momentum

"What momentum is. A simple problem involving momentum. Created by Sal Khan."

The linear momentum of an object is defined as

$$\vec{p}=m\vec{v},\tag{1}$$

where $$m$$ is the total mass of the object (given by either $$\sum{m_i}$$ or $$\int{dm}$$), and $$\vec{v}$$ is the object's linear velocity. To understand concepts such as velocity or force, one need only visualize something in their head. But can we vizualize linear momentum? Or, even worse (which is to say, even more abstract), can we vizualize some of the concepts we'll encounter later in our study of mechanics such as the moment of inertia? The concept of linear momentum is like the notion of energy in this respect - both are very abstract and are very hard to "vizualize." A far mor euseful thing to focus on (which will be the focus of most concepts we encounter in physics, especially those which are very abstract) is this: what is the notion of linear momentum good for? Why is it useful in practise? And, what does it tell us about the universe? To answer those questions satisfactorially, we'll need to first derive, using Newton's laws, a principle known as the conservation of linear momentum (which is typically just called the conservation of momentum, for short). Answering those questions will be just a matter of applying the concepts of momentum and the conservation of momentum to many problems. In doing so we'll learn, through examples, the answers to those three questions: linear momentum and the conservation of linear momentum are very useful for analyzing collisions which involve complicated time-varying forces. We'll learn, later on, that this is the same reason why concepts such as impulse and energy are so important. For collisions which involve instantaneous forces acting on objects, none of these concepts are necessary to understand what happens in a collision. One need only apply Newton's laws. But for collisions involving a time-varying force (such as the interaction between an archery bow and an arrow or between a baseball and a bat), it will make life a lot easier to analyze such situations using the notions of momentum and the conservation of momentum.