Defining Momentum

There are two main types of momentum: linear and angular.

Linear momentum (p), which is primarily used for determining velocities in collisions because the linear momentum is always conserved, is most typically defined to be the product between the mass of an object and its the vector velocity [p = m*v]; therefore making linear momentum a vector as well. By using that same idea we can also find that the units of linear momentum are mass-distance per time (which is equivalent to force per distance), or the SI: kilogram-meters per second (kg*m/s) or Newtons per meter (N/m). Linear momentum can also be found by the force on an object (its mass times its acceleration) per its distance, that is: differentiating force in respect to distance, which shows the more direct reason it can be measured in N/m. This can be taken from the fact that the derivative of velocity in respect to time (dv/dt) is equal to acceleration (a), and that constants-in this case mass (m)-times a derivative of a function are equal to the derivative of the constant times the function. [Force = m*a = m*dv/dt = d(m*v)/dt = dp/dt] In simple terms, the linear momentum of an object is the difficulty for the object to change velocity. That is, the more massive and the faster something is moving the harder it is to change its speed.

Angular momentum (L) only applies to objects that are spinning, but just like linear momentum, it is always conserved, that is, unless some external torque is applied to the object. Also like linear momentum, there are two main ways to define angular momentum: the cross product between the vector distance from some origin (r) and the linear momentum [L = r x p], or the product of the moment of inertia (I) and the angular velocity (w) [L = I*w]. The first definition is a more broad definition because the second only applies to a rotation around a fixed symmetry axis. The second definition uses moment of inertia, which is a complicated number found as the sum of every particle in the object’s mass times the particle’s distance from the rotation axis squared. To simplify this, equations for moment of inertia for typical objects are normally just memorized. Angular velocity (which is written with a lower-case omega and not really a ‘w’) is the change in angle per time. It is most typically found by dividing tangential velocity by the radius. Angular momentum has units of mass-distance-squared per time or force-distance-time which is kilogram-meters-squared per second (kg*m^2/s) or Newton-meter-seconds (N*m*s) in SI. As notable from the cross product involved in finding angular momentum, is an axial vector, or pseudovector, which means it would have direction perpendicular to the plain of rotation, most simply found by the “right hand rule.”