Conservation Of Momentum

Conservation of Momentum

There is a very useful property of isolated systems, total momentum is conserved.

Lets use a practical example to show why this is the case, let us consider two billiard balls moving towards each other. Here is a sketch (not to scale):

When they come into contact, ball 1 exerts a contact force on the ball 2, $\vec{F}_{\text{B1}}$, and the ball 2 exerts a force on ball 1, $\vec{F}_{\text{B2}}$. We also know that the force will result in a change in momentum:

\[\vec{F}_{net} =\frac{\Delta \vec{p}}{\Delta t}\]

We also know from Newton's third law that:

\begin{align*} \vec{F}_{\text{B1}} & = - \vec{F}_{\text{B2}}\\ \frac{\Delta \vec{p}_{\text{B1}}}{\Delta t} & = - \frac{\Delta \vec{p}_{\text{B2}}}{\Delta t}\\ \Delta \vec{p}_{\text{B1}} & = - \Delta \vec{p}_{\text{B2}}\\ \Delta \vec{p}_{\text{B1}} + \Delta \vec{p}_{\text{B2}} & = 0 \end{align*}

This says that if you add up all the changes in momentum for an isolated system the net result will be zero. If we add up all the momenta in the system the total momentum won't change because the net change is zero. Important: note that this is because the forces are internal forces and Newton's third law applies. An external force would not necessarily allow momentum to be conserved.

In the absence of an external force acting on a system, momentum is conserved.

Optional Activity: Newton's Cradle Demonstration

Momentum conservation

A Newton's cradle demonstrates a series of collisions in which momentum is conserved.

Informal Experiment: Conservation of Momentum

Aim

To investigate the changes in momentum when two bodies are separated by an explosive force.

Apparatus

two spring-loaded trolleys
stopwatch
meter-stick
two barriers

Proper illustration still pending

Method

Clamp the barriers one metre apart onto a flat surface.
Find the mass of each trolley and place a known mass on one of the trolleys.
Place the two trolleys between the barriers end to end so that the spring on the one trolley is in contact with the flat surface of the other trolley
Release the spring by hitting the release knob and observe how the trolleys push each other apart.
Repeat the explosions with the trolleys at a different position until they strike the barriers simultaneously. Each trolley now has the same time of travel t.
Measure the distances ${x}_{1}$ and ${x}_{2}$. These can be taken as measures of the respective velocities.
Repeat the experiment for a different combination of masses.

Results

Record your results in a table, using the following headings for each trolley. Total mass in kg, distance travelled in m, momentum in $\text{kg·m·s$^{-1}$}$. What is the relationship between the total momentum after the explosion and the total momentum before the explosion?

Definition: Conservation of Momentum

The total momentum of an isolated system is constant.

The total momentum of a system is calculated by the vector sum of the momenta of all the objects or particles in the system. For a system with $n$ objects the total momentum is:

\[\vec{p}_T = \vec{p}_1 + \vec{p}_2 + \ldots + \vec{p}_n\]

Example: Calculating the Total Momentum of a System

Question

Two billiard balls roll towards each other. They each have a mass of $\text{0.3}$ $\text{kg}$. Ball 1 is moving at $\vec{v}_{1}=\text{1}\text{ m·s$^{-1}$}$ to the right, while ball 2 is moving at $\vec{v}_{2}=\text{0.8}\text{ m·s$^{-1}$}$ to the left. Calculate the total momentum of the system.

Step 1: Identify what information is given and what is asked for

The questionexplicitly gives

the mass of each ball,
the velocity of ball 1, $\vec{v}_{1}$, and
the velocity of ball 2, $\vec{v}_{2}$,

all in the correct units.

We are asked to calculate the total momentum ofthe system. In this example our system consists of two balls. To findthe total momentum we must determine the momentum of each ball and add them.

\[\vec{p}_{T}=\vec{p}_{1}+\vec{p}_{2}\]

Since ball 1 is moving to the right, its momentum is in this direction,while the second ball's momentum is directed towards the left.

Thus, we are required to find the sum of two vectors acting along thesame straight line.

Step 2: Choose a frame of reference

Let us choose moving to the right as thepositive direction.

Step 3: Calculate the momentum

The total momentum of the system is then the sum of the two momenta taking the directions of the velocities into account. Ball 1 is travelling at $\text{1}$ $\text{m·s$^{-1}$}$ to the right or $\text{+1}$ $\text{m·s$^{-1}$}$. Ball 2 is travelling at $\text{0.8}$ $\text{m·s$^{-1}$ }$ to the left or $-\text{0.8}$ $\text{m·s$^{-1}$}$. Thus,

\begin{align*} \vec{p}_{T}& = {m}_{1}\vec{v}_{1}+{m}_{2}\vec{v}_{2} \\& = \left(\text{0.3}\right)\left(+1\right)+\left(\text{0.3}\right)\left(-\text{0.8}\right) \\& = \left(\text{+0.3}\right)+\left(-\text{0.24}\right) \\& = \text{+0.06}\\& = \text{0.06}\text{ m·s$^{-1}$}~\text{to the right}\end{align*}

In the last step the direction was added in words. Since the result in the second last line is positive, the total momentum of the system is in the positive direction (i.e. to the right).

Conservation Of Momentum