Momentum

Momentum is proportional to mass of object.

Momentum is proportional to velocity of object.

Momentum is EQUAL to the product of an object’s mass and velocity.

p = m v

The units for momentum will be kg m/s (SI). There is no special name for momentum units.

Momentum is a vector quantity. Direction matters.

Momentum (in physics) is not to be confused with a winning football team’s momentum (NOT physics)!

Impulse

Momentum, by itself has only limited usefulness (and excitement!) It gets more interesting when momentum is changing.

To hit a baseball a long ways (by changing its momentum in a big way), you want to increase both the force of the bat on the ball (how hard you swing), and the time of contact between the bat and ball (follow-through.)

Newton defined impulse to be the (average) force on an object multiplied by the time during which that force is applied.

J = F Δt

Note the units for impulse will be N s.

Note relationship between this and Newton’s Second Law:

F = m a = m Δv / Δt multiply both sides by Δt

F Δt = m Δv = Δp

This gives us the:

Impulse-Momentum Theorem

The impulse applied to an object EQUALS the change in momentum of that object.

F Δt = m Δv = Δp

(Newton actually framed his second law in terms of impulse and momentum rather than force and acceleration.)

This concept can make some problems that are difficult to do otherwise easier/possible.

Scene from Apollo 13…

“Cushioning” a catch (of a water balloon, egg, or barehanded softball or baseball).

- To catch a thrown egg, you need to change its

momentum to zero. (A fixed value for a given

initial velocity.)

- “Cushioning” your catch extends the interaction time.

- Increasing the time of impulse decreases the force!

Note units: N s = kg m/s Same!

Momentum (continued)

Momentum is very useful in interactions between objects.

- If object A exerts a force on object B and changes object B’s

momentum, then the equal but opposite force of object B on

object A will change object A’s momentum by exactly the same

amount but in the opposite direction.

- The forces must be the same (Newton’s 3^rd Law), the time of

interaction must be the same (B is touching A as long as A is

touching B), therefore the impulse on B must be exactly the

same as the impulse on A, therefore the change in momentum of

B must equal exactly the change in momentum of A.

- This is quite similar to, but often more useful than,

Newton’s 3^rd Law.

Law of Conservation of Momentum

- If there are no outside forces acting on a system of objects, then

the total momentum of the system stays constant.

- This is a VERY useful and important concept.

- collisions (from car crash analysis to interactions of

colliding galaxies to interactions of atoms and molecules…)

- recoil (guns, cannons…)

- rocket and/or jet propulsion

- ballistics (muzzle velocity of a gun), bomb trajectory

- Problems where forces are hard or impossible to measure

can be solved with this concept.

Collisions

Collision Type	Characteristics	Examples	Kinetic Energy Conserved?	Momentum Conserved?
Perfectly or Totally Elastic	Objects bounce off each other with NO damage or permanent deformation	Atoms and molecules	Yes	Yes
(Approximately) Elastic	Objects bounce off each other with negligible damage or permanent deformation	Pool balls, steel ball off marble, rubber bouncy balls, golf ball off club	Very Nearly (Yes, for our problems)	Yes
Inelastic	Objects bounce off each other with permanent damage or deformation	Most car crashes, people bumping, any collision that isn’t perfectly inelastic but KE losses are large^*	No	Yes
Perfectly or Totally Inelastic	Any collision where objects stick together	Bullet imbedded in an object, a football tackle, trains coupling, jumping into a boat, catching a ball…	No	Yes

* There is a continuum between collisions that are approximately elastic and inelastic, depending on how much KE is conserved. A golf club hitting a golf ball can have more than 80% of KE conserved (transferred) whereas a “dead” tennis ball off a racket may have less than 40% KE conserved.

Note that in all these cases, momentum is conserved! The conservation of momentum makes it possible to solve a large number of different types of problems.

Conservation of Momentum (continued)

Possibly the easiest way to set up these problems is to

1) Identify clearly the interaction in question.

2) Set up the (vector) sum of all momenta before the interaction.

3) Set up the (vector) sum of all momenta after the interaction.

4) Equate the two sums!

Σ p_before = Σ p_after

All conservation of momentum problems can be set up this way.

Elastic collisions may require simultaneously solving conservation of KE equation with conservation of momentum equation.

2-D problems require that you isolate x and y components of the momentums, but the concept is the same!

Σ p_x-before = Σ p_x-after

Σ p_y-before = Σ p_y-after