Linear Momentum and Collisions

Linear Momentum

Our study of kinetic energy showed that a complete understanding of an object’s motion must include its mass and velocity $\mathit{K}=\frac{1}{2}{\mathit{mv}}^2$. However, as powerful as this concept is, it does not include any information about the direction of the moving object’s velocity vector. We’ll now define a physical quantity that includes direction.

Like kinetic energy, this quantity includes both mass and velocity; like kinetic energy, it is a way of characterizing the “quantity of motion” of an object. It is given the name momentum (from the Latin word momentum, meaning “movement”), and it is represented by the symbol $\mathit{\rho }$.

Momentum depends equally on an object’s mass and velocity.

Momentum and Newton’s Second Law

The importance of momentum, unlike the importance of energy, was recognized early in the development of classical physics. Momentum was deemed so important that it was called the “quantity of motion.” Newton actually stated his second law of motion in terms of momentum: The net external force equals the change in momentum of a system divided by the time over which it changes. Using symbols, this law is

${\mathit{F}}_{\mathit{net}}=\frac{\mathit{\Delta p}}{\mathit{\Delta t}}$ or ${\mathit{F}}_{\mathit{net}}=\mathit{ma}$

Conservation of Linear Momentum

If the value of a physical quantity is constant in time, we say that the quantity is conserved. In the image below, Before the collision, the two billiard balls travel with momenta $\overrightarrow{{\mathit{p}}_1}$and  $\overrightarrow{{\mathit{p}}_2}$. The system's total momentum is the sum of these, as shown by the red vector labeled  $\overrightarrow{{\mathit{p}}_{\mathit{total}}}$ total on the left. After the collision, the two billiard balls travel with different momenta  $\overrightarrow{{\mathit{p}}_1^{\text{'}}}$and  $\overrightarrow{{\mathit{p}}_2^{\text{'}}}$. The total momentum, however, has not changed, as shown by the red vector arrow  $\overrightarrow{{\mathit{p}}_{\mathit{total}}^{\text{'}}}$on the right.

Impulse

The connection between momentum and force is described by Newton's second law of motion. This law states that the rate of change of momentum of an object is directly proportional to the net force applied to it and occurs in the direction of the force.

Let  $\mathit{F}\left(\mathit{t}\right)$ be the force applied to an object over some differential time interval $\mathit{dt}$. The resulting impulse on the object is defined as

$d\overrightarrow{\mathit{J}}=\overrightarrow{\mathit{F}\left(\mathit{t}\right)}\mathit{dt}$