Linear Momentum

The Main Idea

Linear momentum is a vector quantity which is defined by the product of an object's mass, generally denoted as the lowercase "m", and its velocity (a vector), v. Linear momentum is represented by the letter "p" and is generally referred to as momentum for short.

A Mathematical Model

Single Particles

Linear momentum is a vector quantity, like velocity, possessing a direction as well as a magnitude:

[math]\displaystyle{ \mathbf{p} = m \mathbf{v} }[/math]

where p is the vector stating the object's momentum in the three directions of space, and where v is the three-dimensional velocity vector giving the object's movement in each of these directions, and m is the object's mass.

Multiple Particles

The momentum of a system of particles is the sum of both particles' momentum (momenta). If the particles have masses m₁ and m₂, respectively, and velocities v₁ and v₂, the total momentum is:

[math]\displaystyle{ \begin{align} p &= p_1 + p_2 \\ &= m_1 v_1 + m_2 v_2\,. \end{align} }[/math]

This model can be used to measure the momentum of a system of any amount of particles.

A system of particles has a center of mass, which is a point determined by the weighted sum of each of their positions:

[math]\displaystyle{ r_\text{cm} = \frac{m_1 r_1 + m_2 r_2 + \cdots}{m_1 + m_2 + \cdots} = \frac{\sum\limits_{i}m_ir_i}{\sum\limits_{i}m_i}. }[/math]

If all the particles are in motion, the center of mass will most likely be moving as well (unless the particles are in rotation around it). If the center of mass is moving at velocity v_cm, the momentum can be found by using:

[math]\displaystyle{ p= mv_\text{cm}. }[/math]

In Relation to Newton's Second Law

When a force ([math]\displaystyle{ F }[/math]) is applied to a particle for a time interval Δt, the momentum of the particle changes by an amount

[math]\displaystyle{ \Delta p = F \Delta t\,. }[/math]

This quantity is called impulse.

When differentiated, this represents Newton's Second Law. Put simply, rate of change of a particle's momentum is proportional to the force [math]\displaystyle{ F }[/math] acting on it over a certain interval of time. This is symbolized by

[math]\displaystyle{ F = \frac{dp }{d t}. }[/math]

A Computational Model

In this code I have simulated what happens when a cart (visualized by the box) is acted upon by the force of a small gust of wind. Its initial velocity and mass is defined and you can clearly see it being affected by the wind. Due to the wind, its momentum changes because its velocity (both direction and speed) is affected by the wind. https://trinket.io/glowscript/ce43925647

Examples

Scenario

If you are standing at the bottom of a hill and faced with the option of stopping a runaway semi-truck or a runaway bicycle, you would probably choose to stop the bike because the semi truck has more momentum. Momentum simply means mass in motion. The semi truck has larger momentum because it is massive, but also has more speed.

Simple

Find the momentum of a ball that has a mass of 69kg and is moving at <1,2,3> m/s.

Middling

A car has 20,000 N of momentum. How would the momentum of the car change if: a) the car slowed to half of its speed? b) the car completely stopped? c) the car gained its original weight in luggage?

Difficult

You and your friends are watching NBA highlights at home and want to practice your physics. You notice at the beginning of a clip a basketball ball is rolling down the court at 23.5 m/s to the right. At the end, it is rolling at 3.8 m/s in the same direction. The commentator tells you that the change in its momentum is 17.24 kg m/s to the left. Curious at how many basketballs you can carry, you want to find the mass of the ball.

Connectedness

How is momentum used in Electrical Engineering?

Electrokinetic momentum is used in many devices to calculate the voltage necessary to change the current in an inductive circuit. This calculation is utilized within many electrical devices such as resistance grids.

History

Newton's 2nd Law

While Newton's 1st Law was not entirely his own, his 2nd and 3rd are.

Original Latin:

“Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.”

This was translated closely in Motte's 1729:

“Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.”

Essentially:

The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed. In other words, F=ma.