Also see our lesson on elastic and inelastic collisions.
Anytime objects collide, or break apart,
a special quantity is conserved: momentum.
Here’s an inelastic collision:
Each object has it’s own m and v.
If we multiply them, we get a number.
Doesn’t seem special.
But now add the m·v of all objects before the collision
Compare to the m·v of all objects after the collision:
this sum stays the same … for every collision.
In fact, this sum stays the same for every collision in the universe, under every possible circumstance!
So this conserved quantity is a powerful tool for analyzing any kind of physical phenomenon.
Take anything – and let it interact/smack into/bounce off – anything else.
Look at all the moving parts before they meet:
add up the m·v for all these parts
Then look at all parts afterwards:
and add up the m·v for all these parts
Sum total of m·v (before) = Sum total of m·v (after)
Σ m·v (before) = Σ m·v (after)
Sometimes results are not intuitive 🙂 Colin Sullender writes:
When dropped individually, these balls fail to reach their initial heights,
because energy is lost as they impact on the ground.
Because of Conservation of Energy, the upwards bouncing ball
no longer has the kinetic energy necessary to reach the same height.
However, when stacked on top of each other,
the golf ball is sent flying because of conservation of momentum
Upon impact on the ground, the basketball transferred all its
momentum into the blue ball,
which in turn transferred all its momentum into the golf ball.
Because momentum is related to mass, these larger balls
have much more momentum than the golf ball.
This means that in order for momentum to be conserved,
the upwards velocity of the golf ball is massively increased,
sending the golf ball flying high.
Momentum formula p = m•v
Why abbreviate as p?
We can’t use m (used for ‘mass’ and ‘meters’)
P is from Latin petera “to seek”, related to the word “impulse”
Impulses change momentum
If you put a force on an object, for an amount of time,
then you change it’s momentum:
This is an impulse.
impulse = F•Δt (force x time)
impulse = m·Δv (change of momentum)
Relate the two:
We might imagine that collisions are “instant”,
but all interactions do take some Δt (amount of time.)
Bullet impacting a steel wall
Ball bouncing on a billiards table
If Δp occurs over a long period time
then the force of impact is small.
if Δp occurs over a short period time
then the force of impact is large
very large Δp in a small Δt
If the Δt is large,then the force is dissipated over a longer period of time.
If the Δt is small then the same force is dissipated
over a smaller amount of time.
Yet it is the same impulse!
The impulse required to bring an object to a stop,
and then throw it back again,
is greater than the impulse needed to bring it to a stop:
Shoshana imparts a large impulse to the bricks in a short time
– this produces considerable force.
Her hand bounces back, yielding as much as twice the impulse to the bricks.
Let’s look at this in slow motion:
Conservation of momentum
“p before firing is zero. After firing, net p is still zero,
because the p of the cannon is equal and opposite to the p of the cannonball.”
The force on the cannonball inside the cannon barrel
is equal and opposite to the force causing the cannon to recoil.
Before firing, p = 0
After firing, net p = 0
p is neither gained nor lost.
P has both direction and magnitude : a vector
The cannonball gains p,and the recoiling cannon gains p in opposite direction.
The system, as a whole, gains no p.
The p of the cannonball and the cannon are equal in magnitude and opposite in direction.
No net force acts on the system, so there is no net impulse on the system, and there is no net change in p.
In every case, p of a system cannot change unless it is acted on by external forces.
The net p of a system before and after the event is the same. Examples are:
atomic nuclei undergoing radioactive decay,
pool balls colliding on billiards table
Does p disappear?
After a car crash, eventually all the pieces come to a stop. The motion of every piece, eventually becomes zero.
The p=m•v of every piece becomes 0 … so where did the momentum go?
Is it “lost”? Does it disappear? Nope!
p is transferred to vibrations of individual molecules in the car pieces,
and vibrations of pavement molecules.
Pieces of debris slow down – but p is transferred to molecules in the road, air, etc.
When molecules are warmer, rotating and vibrating,
it may increase their p.
This molecule became heated during the car collision –
notice how it’s moving? It has rotational velocity
As the large pieces come to a stop (seemingly “losing momentum”)
p is really transferred on a microscopic level to molecules.
Momentum paradox? question
“At first glance this looks impossible. In physics terms, a system (in this case a canoe + two occupants) cannot change its own overall momentum because all internal forces ultimately cancel out. In other words, where is the newfound linear momentum coming from that gets them out of this predicament?”
objects collide without being (permanently) deformed,
and without generating (much) heat
objects collide and are (a) deformed,
or (b) seem to lose p, by generating heat.
Below we see 3 bullets, with equal mass,
running into 3 blocks of wood with equal mass.
The first bullet passes through the block
and maintains much of its original p.
As a result, little p gets transferred to the block
The second bullet, expands as it enters the block of wood,
which prevents it from passing all the way through it.
So most of its p transfers to the block (totally inelastic collision.)
The third bullet (rubber) bounces off the block,
transferring all of it’s p
– and then borrowing some more from the block.
This has the most p transferred to the block. (elastic collision.)
image from Science Joy Wagon
Let’s look at train cars colliding, examined more in our PowerPoint:
Read examples in your textbook, and look at the examples below from PhysicsClassroom.com
Momentum and Collisions: PhysicsClassroom.com
The Cart and The Brick
The Fish Catch
1-Dimensional conservation of p
Something else besides momentum is conserved
Consider Newton’s cradle: Lift one steel ball, drop it, see what happens. Lift two steel balls, see what happens, and so on.
Many possible outcomes are possible, in which momentum is conserved, yet which never actually happen.
Some principle must be restricting the real-world outcomes – we infer that some other conservation principle exists.
The following image is from HyperPhysics, by Carl R. Nave
Momentum paradox? solution!
Solution to the above conundrum!
By bouncing up and down they create spreading waves.
By falling onto the leading edge of a wave,
the canoe pushes the wave backward, which in turn pushes the canoe forward.
In other words, the canoe is getting its momentum by propelling water in the opposite direction,
thereby conserving momentum overall.
“A Tale of Momentum & Inertia” (House Special CGI animators, Director Kameron Gates)
Cartoon laws of physics. Intuitive ideas about motion – that are totally wrong. Inspired by classic Warner Brothers cartoons.
Goal: Study conservation of momentum, in 1-dimension, in collisions of two bodies.
Use air-carts floating on an aluminum track, suspended by a cushion of air so as to be nearly frictionless.
How was conservation of momentum discovered?
2016 Massachusetts Science and Technology/Engineering Curriculum Framework
HS-PS2-2. Use mathematical representations to show that the total momentum of a system of interacting objects is conserved when there is no net force on the system. Emphasis is on the qualitative meaning of the conservation of momentum and the quantitative understanding of the conservation of linear momentum in
interactions involving elastic and inelastic collisions between two objects in one
HS-PS2-3. Apply scientific principles of motion and momentum to design, evaluate, and refine a device that minimizes the force on a macroscopic object during a collision. Clarification Statement: Both qualitative evaluations and algebraic manipulations may be used.
Common Core Math
- CCSS.MATH.CONTENT.7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
- CCSS.MATH.CONTENT.8.EE.C.7 Solve linear equations in one variable
- CCSS.MATH.CONTENT.HSA.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (including isolating a variable)
- CCSS.MATH.CONTENT.HSA.CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.