# Conservation of Momentum

## Some Notes on Momentum

Karen Tennennhouse

1. What is momentum? It is not that easy to describe. It is abstract in the sense that it is a calculated quantity. The reason we calculate it (and give it a name) is that it turns out to obey an important conservation law.

2. Definition: An object of mass m, travelling at velocity ${\displaystyle {\vec {v}}}$, is defined to have momentum ${\displaystyle {\vec {p}}=m{\vec {v}}}$.
• Notice, Momentum is a vector
• Units are kg.m/s

3. When and how does the momentum of an object change?
When a force acts on it (for some time).
If the force is constant, the change in momentum ${\displaystyle \Delta p}$ will obey ${\displaystyle \Delta {\vec {p}}={\vec {F}}\Delta t}$.
The quantity is called the Impulse delivered by this force.
NB: Impulse is a vector, and this is a vector equation.

4. For a single object, the ideas of momentum and impulse don’t really give us any more information or convenience than just using Newton’s 2nd law. Momentum becomes more interesting, significant and useful when dealing with a group of objects (system). It is also very useful in handling a system where the details of the internal forces between the objects are complicated or changing.

5. Recall from the energy unit that:
• A system means any group of objects we choose to examine.
• For a given system, we say that a force is internal iff it is exerted BY an object IN the system (on another object in the system.)
• We say a force is external iff it is exerted BY an object which is NOT IN the system (on an object in the system.)

6. Law of Conservation of Momentum:
Given any system, if the net external impulse is zero, then the (vector) total momentum of the system stays constant.

7. In many situations, the net external impulse might not be zero, but could still be small enough to ignore (negligible). This could happen if the external forces themselves are very small OR if the force is small-to-medium but acts for only a very short time interval.
• What is “very small”, “small enough”, etc? It’s small enough if the impulse (hence the change in momentum, ${\displaystyle \Delta {\vec {p}}}$ ) is much smaller than the momentum value ${\displaystyle {\vec {p}}_{total}}$
• This gives us a law like:
If the net external impulse on a system is zero OR negligible then the total momentum is “almost” constant, to as much precision as we need.
i.e. ${\displaystyle {\vec {p}}_{TOT_{i}}={\vec {p}}_{TOT_{f}}}$
• That is the law which applies to all our large, quantitative problems this semester.
• Some examples of situations where this is true (i.e. where net external impulse is negligible) include
• Collisions (where the system includes all the colliding bodies);
• Explosions;
• Events in a system on a level, frictionless surface.

8. As you see, the momentum law is in some ways very similar to the conservation of energy law. But there are important differences.
• The most important difference is that energy and momentum are different physical quantities. (They do not measure the same physical “stuff.”)

In any given situation, the energy might be conserved, but not the momentum. OR momentum might be conserved, but not energy. Or both, or neither.
• For other differences and similarities, see the comparison chart below. Its purpose is to help you understand the various features.

9. Method for solving large 2-Dim Momentum Problems
A. Choose the SYSTEM. You’ll need to be a little more careful here than you did for energy problems.

B. CHECK: Verify that the net external impulse (sum of ) is zero or “small enough”
• Notice, we’re checking the impulse due to external forces only (it does not matter whether they are conservative or not)
• In all our large, quantitative momentum problems this condition will be true. But you should still understand what you are checking for, and why it is true in a certain problem.

C. Write the LAW: If yes to B, then (If no to B, then changes, but we will not do large problems using this.)

D. Large, clear DIAGRAM must include:
• Choose and show AXES
• Before the event, show and label velocity of each object as a vector.
Label the unknowns with suitable symbols, for example, for initial velocity of second object.
• Do the same for the information after the event.
• Label angles and masses (with symbols, if unknown).
• Sketch, and label with symbols, all components.

E. To keep everything organized, it is strongly recommended to use the “table” layout. If you prefer another method, it must be equally organized and explicit. The overall strategy is to apply ${\displaystyle {\vec {p}}_{TOT_{i}}={\vec {p}}_{TOT_{f}}}$ as vector sums, using the component method. So, do all of the following separately for X and Y:
• Calculate the components of velocity and momentum.
• Calculate the total momentum before and the total momentum after.
• Apply the conservation law, i.e. set ${\displaystyle {\vec {p}}_{TOT_{i}}={\vec {p}}_{TOT_{f}}}$
• Solve for desired unknowns.

F. You’re almost finished:
• If the question asks for them, you might need to find some magnitudes or directions, by combining the components in the usual ways. Otherwise, just leave your answers in î, ĵ notation.

10. There are many other situations where the total momentum is not conserved, because the net external impulse is not small enough to ignore. In such a case the change in the total momentum will equal the net external impulse. That is, for any system, ${\displaystyle {\vec {p}}_{TOT_{i}}-{\vec {p}}_{TOT_{f}}=\Delta {\vec {p}}=\sum ({\vec {F}}\Delta t)}$ ${\displaystyle \qquad }$ (vector equation).

11. In general, in a collision, even when the total momentum is conserved, the total kinetic energy may or may not be.
• If the total KE before and after is conserved, then we say that the collision was “elastic”.
If not, the collision was inelastic.
• The confusing term totally inelastic unfortunately does not mean that all KE was lost; it just means the objects stick together after collision

12. Comparing what we know about Energy and Momentum

ENERGY MOMENTUM
is a SCALAR (number).

Has no direction. We can use minus signs to indicate decrease, or less than some zero.

is a VECTOR  !!

In 2 or 3 dims we must remember to use vector operations for it.

Forms Has many different forms

(for example, KE, GPE, heat, Elastic PE, etc), and can transfer from one form to another.

Always one form,
${\displaystyle {\vec {p}}=m{\vec {v}}}$
Measure/cause of changes For constant force,

energy change caused by ${\displaystyle {\vec {F}}}$ will be

Work = ${\displaystyle {\vec {F}}.\Delta {\vec {s}}}$      (dot product)


[ For changing forces, would use an integral, but not used much in our course. ]

For constant force,

momentum change caused by ${\displaystyle {\vec {F}}}$ will be

     Impulse =  ${\displaystyle {\vec {F}}\Delta t}$


[ For changing forces, NOT in our course, we would use an integral ]

The Main Conservation law If net work done by external forces is zero, then total energy will be conserved.

That is the total (scalar) of all forms present in the system.

If net external impulse is zero, then total momentum ( vector total ! ) will be conserved.

That is, vector total of ${\displaystyle m{\vec {v}}}$‘s in the system.

What happens when it is NOT conserved The change in total energy of a system will equal to the net work done by external forces.

That is, change (final minus initial) in the number total of all forms.

The change in total momentum of a system equals the net impulse by external forces.

Notice this is a vector equation: the “change” ${\displaystyle {\vec {p_{f}}}-{\vec {p_{i}}}}$ is a vector subtraction, and the total (net) of the external impulses is a vector sum.

Other important laws for this quantity? Yes. For energy, we are also interested in knowing when certain forms (not just the whole total) will be conserved. We express this information with several other laws.

For example:

• If the work done by non-conservative forces is zero, then the total mechanical energy (TME) is constant. That is, the total of just the kinetic plus all forms of potential energy,( but not other forms such as heat.)
• If work by nonconserv. forces is not zero, then the change in TME will equal the work by nonconservative forces.
• Change in Kinetic energy equals work done by the net force.
Nope.