Linear Momentum

Linear momentum is the product of the mass and velocity of an object, it is conserved in elastic and inelastic collisions.

Learning Objective

Calculate the momentum of two colliding objects

Key Points

Like velocity, linear momentum is a vector quantity, possessing a direction as well as a magnitude.
Momentum, like energy, is important because it is a conserved quantity.
The momentum of a system of particles is the sum of their momenta. If two particles have masses m₁ and m₂, and velocities v₁ and v₂, the total momentum is $\begin{aligned} p &= p_1 + p_2 \\&= m_1 v_1 + m_2 v_2\, \end{aligned}$ .

Terms

elastic collision
An encounter between two bodies in which the total kinetic energy of the two bodies after the encounter is equal to their total kinetic energy before the encounter. Elastic collisions occur only if there is no net conversion of kinetic energy into other forms.
inelastic
(As referring to an inelastic collision, in contrast to an elastic collision. ) A collision in which kinetic energy is not conserved.
conservation
A particular measurable property of an isolated physical system does not change as the system evolves.

Full Text

In classical mechanics, linear momentum, or simply momentum (SI unit kg m/s, or equivalently N s), is the product of the mass and velocity of an object. Mathematically it is stated as:

$\mathbf{p} = m \mathbf{v}$ .

(Note here that p and v are vectors. ) Like velocity, linear momentum is a vector quantity, possessing a direction as well as a magnitude. Linear momentum is particularly important because it is a conserved quantity, meaning that in a closed system (without any external forces) its total linear momentum cannot change.

Because momentum has a direction, it can be used to predict the resulting direction of objects after they collide, as well as their speeds. Momentum is conserved in both inelastic and elastic collisions. (Kinetic energy is not conserved in inelastic collisions but is conserved in elastic collisions. ) It important to note that if the collision takes place on a surface with friction, or if there is air resistance, we would need to account for the momentum of the bodies that would be transferred to the surface and/or air.

Let's take a look at a simple, one-dimensional example: The momentum of a system of two particles is the sum of their momenta. If two particles have masses m₁ and m₂, and velocities v₁ and v₂, the total momentum is:

$p = p_1 + p_2 = m_1 v_1 + m_2 v_2\,$ .

Keep in mind that momentum and velocity are vectors. Therefore, in 1D, if two particles are moving in the same direction, v1 and v2 have the same sign. If the particles are moving in opposite directions they will have opposite signs.

If two particles were moving on a plane we would choose our xy-plane to be on the plane of motion. We can then write the x and y component of the total momentum as:

$p_x = p_{1x} + p_{2x} = m_1 v_{1x} + m_2 v_{2x}\, \\ p_y = p_{1y} + p_{2y} = m_1 v_{1y} + m_2 v_{2y}.$

If the 2D momentum vector is decomposed into two components, the equations for each component are reduced to its 1D equivalents.

Momentum, like energy, is important because it is conserved. "Newton's cradle" shown in is an example of conservation of momentum. As we will discuss in the next concept (on Momentum, Force, and Newton's Second Law), in classical mechanics, conservation of linear momentum is implied by Newton's laws. Only a few physical quantities are conserved in nature. Studying these quantities yields fundamental insight into how nature works.

Newton's Cradle

Total momentum of the system (or Cradle) is conserved. (neglecting frictional loss in the system. )

[ edit ]

Prev Concept

Potential Energy Curves and Equipotentials

Momentum, Force, and Newton's Second Law

Next Concept