Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 125962
Conservation of Linear Momentum
This is an application based activity that allows students to question and explore the Conservation of Momentum and how it governs the natural
world. It is designed for students who have a firm grasp on physical concepts of nature and mathematical derivations and manipulations. In this
activity the teacher will use an Online Simulation titled "2D Elastic Collisions of Two Hard Spheres" to model idealistic elastic collisions and describe how
mass and initial velocities can affect the post-collision momentum for each mass. The students will also be introduced to inelastic collisions and will
compare these to elastic collisions. Students will fill out the attached lab worksheet and perform calculations based on manipulating the mathematical
equation for Momentum Conservation.
Subject(s): Science
Grade Level(s): 11, 12
Intended Audience: Educators
Suggested Technology: Computer for Presenter,
Computers for Students, Internet Connection,
Interactive Whiteboard, Basic Calculators, Adobe Flash
Player
Instructional Time: 2 Hour(s)
Keywords: Conservation of linear momentum
Resource Collection: FCR-STEMLearn Physical Sciences
ATTACHMENTS
Student Handout.docx
Student Handout Key.docx
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What will students know and be able to do as a result of this lesson?
1) Students will learn that momentum is always conserved in elastic collisions.
2) Students will learn that in inelastic collisions, momentum as well as energy is still conserved in a closed system. However, some of the system's kinetic energy will
be converted to heat energy and lost to the surrounding environment.
3) Students will be instructed that these collisions will be taking place under extremely idealized conditions. The calculations involved will be related to the to the
instants just before and just after collisions, neglecting friction's effect on the entire system. In addition, complications to calculations arise when collisions occur within
non-inertial reference frames and/or at relativistic speeds.
Prior Knowledge: What prior knowledge should students have for this lesson?
1) An understanding of how to calculate momentum vectors ( p = mass *velocity vector)
2) An understanding of adding vectors
3) The ability to algebraically manipulate equations
page 1 of 5 4) Internet access to Geogebra.org (not necessary if the instructor has Internet access as well as a smart board or projector to do a whole-class demonstration).
Online Simulation : 2D Elastic Collision Simulation
http://tube.geogebra.org/material/show/id/50589
Guiding Questions: What are the guiding questions for this lesson?
1. What conditions are required for momentum to be conserved?
A well defined reference frame within an inertial system.
Does the environment surrounding the collision affect the collision?
Yes, different mediums can provide friction, slowing objects after the collisions. Also, collisions in different mediums can lose energy due to heat. This is why we
use the instant just before and just after the collision. Also, non-inertial reference frames (ie. a collision close to a strong gravitational force or in an accelerating
reference frame) further complicate calculations.
Are momentum vector sums constant only in head-on collisions? Opposite Directions? Glancing collisions?
No, they are always constant, no matter the type of collision
How can the conservation govern the particles in an explosion?
Since the initial velocity of the object to be exploded is 0, its momentum is therefore 0. Then for every particle that gets ejected in one direction, another particle
must be ejected in the exact opposite direction with the same speed. This is what gives explosions their spherical shape as they propagate. Military "shape charges"
use angular chambers within the explosives body to manipulate the direction of some of the ejected energy, and thus creating a one-dimensional (or very focused
two-dimensional) explosion. Although the sum total of the momentum of individual particles will still equal zero, the chambers will "turn" some particles in other
directions. Since this is done in separate stages after the collision, so called "shape charges" still follow the conservation of momentum law.
2. If two equal masses collide head-on, should their final velocities be equal and opposite?
Not necessarily. The final velocities depend on the initial velocities.
How can a mass' momentum vector be converted into two, distinct momenta in the x and y directions?
For an object of velocity v and angle theta with respect to the coordinate system in place, Momentum (P) in the x direction can be found using Px = m* v
cos(theta) and momentum in the y direction can be found using Py = m *v sin(theta).
How can you add the two initial momentum vectors to find the resultant constant that will govern the final momentum vectors?
Px1 + Px2 = Px total and Py1 + Py2 = Py total
3. How can a final momentum be calculated in completely inelastic collisions?
P = (M1+ M2)vf where vf is the same for both objects since they merged into one larger object
Where does the missing energy go?
Into the surrounding atmosphere as heat energy, heating up the colliding objects, sound, into deformation of the surfaces
Predict: What event, related to the focus topic, that may surprise students, will the students make a prediction about?
The students will use the initial momentum vectors to find the total momentum of the system for each initial condition (Head-on, one initial mass at rest, same
directions collisions, and glancing collisions, both elastic and inelastic).
They will use the principle of conservation of momentum to make predictions for the post-collision momentum vectors.
Observe: What will the students observe and/or infer during this step of the lesson? How will students communicate their
observations and inferences?
The students will compare their initial calculations with those calculated by the simulation. Then the teacher will set the initial conditions for each collision and run the
simulation. Students will manipulate the initial conditions to test each prediction and observe each collision. The teacher will calculate the final momentum vectors and
show how the results compare to the simulation. The teacher may use the following for an example:
One Dimensional Collision (One mass at initial rest, equal masses, motion along the (+) x axis)
M1= 5kg ,
= 5 m/s x
M2 = 5kg,
= 0 m/s:
M1
+ M2
=
= M1
Then the equation becomes:
+ M2
= 0 for head-on, equal mass elastic collisions with one mass at initial rest. Then:
(5kg)(5m/s) + (5kg)(0) = (5kg)(0) + (5kg)
since M1 = M2, we can divide the entire equation by M1 (or 5kg). So the equation becomes:
5m/s x =
or in the general case:
=
page 2 of 5 Note this means that in an elastic collision with the initial conditions of:
1) One Directional Motion
2) One mass at initial rest
3) Equal masses
The Conservation of Momentum predicts that the entire momentum of the first ball will be passed onto the second ball.
Explain: How will students be encouraged to develop explanations using their observations and scientific or mathematical
concepts or principles?
After the calculations have been made and the simulations have been run, the teacher will describe the implications of momentum conservation and how that helps to
construct the universe as we know it. Atoms and molecules make up mass and therefore our bodies and our surrounding environment. Idealized "collisions" are
merely representations of interactions between matter and thus understanding this concept will yield a better understanding of nature. Students will be asked
pointed questions to further their understanding of the concept.
An important concept to understand is how nature would be different if collisions didn't conserve momentum. If they didn't, then there would be no way of making
predictions about collisions in nature. Since momentum would not be conserved, there would be no laws governing collisions. Collisions in stars require momentum to
be conserved in nuclear fission reactions (inelastic collisions). This means that without Conservation of Momentum, stars wouldn't form and thus our universe as we
know it would be impossible to construct.
In addition, nature itself would be extremely different if every collision was completely inelastic (objects stick together afterward). While completely inelastic collisions
are special cases, one can imagine how different nature would be if any two colliding bodies would automatically fuse together. This result would be very problematic.
One can imagine pushing open a door. If every collision in the known universe yielded a completely inelastic collision, your fingers would automatically bond to the
door as one. One can very easily imagine extremely problematic results. So each varying type of collision in our universe must follow its their own individual rules, but
all are governed by Momentum Conservation.
The teacher should make sure the questions stay within a range of implications to ensure the discussion doesn't go off into a tangent. High School students have a
habit of not being able to picture concepts visually, so any deviation from the standard being taught will be to the detriment of the weaker students. Questions should
remain within a frame that further explains/clarifies results, and should not drift towards questions of "existential" meanings nor should they be presented to affect
students' beliefs.
Summative Assessment
The teacher should grade the lab activity to ensure the students' complete understanding of the concept as well as their mastery of the mathematics involved,
including proper vector notation. This will be done after the simulations are run and the students are given time to complete their final calculations (possibly for
homework) for the resultant vectors and given time to explain their observations, including weather or not they correctly predicted/ calculated the final results.
Formative Assessment
This lesson is designed to introduce the concept of the Conservation of Momentum. In this lesson, the instructor will guide the class in whole-group discussions about
the concept as well as its derivations and consequences in relation to collisions within a well defined, closed system in an inertial reference frame.
The Law of Conservation of Momentum can be derived from Newton's Second Law of Motion, which states that an unbalanced force acting on an object will create an
acceleration on that object, or
F = ma where F is the force vector, m is the object's mass, and a is the resultant acceleration vector, in the same direction of the force applied.
A common rewrite can be shown as follows:
F = (m* v)/t or
F = (mvf-mvi)/t where F is the force vector, m is the object's mass,
v is the change in velocity vector of the mass' motion, vf is the object's velocity vector after
the force acts on it, vi the velocity vector before the force acts on it, and t is the overall time that the force acts on the mass.
If both sides are then multiplied by the change in time (t), then the equation can take another form:
Ft = m( v) =
p where Ft is called an impulse acting on the mass and m v is the mass' change in momentum (which can be expressed as
p.) Note that this
means that F= p/t.
Using Newton's Third Law of Motion (For any action there is an equal and opposite magnitude reaction) if two objects collide, they create an equal and opposite force
on each other. This can be expressed as:
F12 = -F21 where F12 is the force exerted on the first object by the second, and F21 is the force exerted on the second object by the first. Note that the negative in
the equation states that F12 and F21 act in opposite directions from each other. Let us now define:
F12 =
p12/t = (m1vf1-m1vi1)/t and
F21 =
p21/t = (m2vf2- m2vi2)/t
Using Newton's third law, we know that:
In the presence of no outside forces, the balls will only experience/exert one force on each other, equal in magnitude and opposite in direction. This can be expressed
as follows:
F12 = -F21, or F12+F21 =0 (The first clue to the conservation)
Since:
F12= p12/t and F21= p21/t, Then
p12/t +
p21/t =
ptotal/t or
pt/t
page 3 of 5 This then means that:
(m1vf1-m1vi1)/t + (m2vf2- m2vi2)/t = 0 =
m1vf1-m1vi1 + m2vf2-m2vi2 = 0 =
pt/t or if we multiply the 3 sides by t
pt which we can then rearrange to separate the initial conditions from the final conditions as follows:
m1vi1 + m2vi2 = m1vf1 + m2vf2 = 0 = p_initial = p_final
Note that in this case both the initial sum of the the momenta as well as the final sum of momenta equal the constant zero, meaning that total momentum of the
system is conserved. This is based on deriving the equation purely from Newton's Laws of Motion. In practice, using a well-defined reference frame in which to
describe the motion (ie. straight line collisions in one-dimensional space) will yield a unique nonzero momentum constant, which in this lesson we will call "p"
An Online Simulation titled "2D Elastic Collision Simulation of Two Hard Spheres" will be used to model collisions. In addition, students will be able to apply the concept
by manipulating the equation:
M1V1I + M2V2I = M1V1F + M2V2F = p
Where: M1 is the First Mass
M2 is the Second Mass
V1I, V2I, V1F, V2F are the velocity vectors of the first and second masses in the instant just before and after the collision (respectively)
p is a constant throughout the collision
Students will be able to make accurate predictions for the final velocity vectors of both masses. In addition, the students will use varying sets of initial momentum
information to compare their predictions to actual results modeled by the simulation. Possible sets to include are:
1) M2 is at initial rest. M1 collides perfectly head on. Three scenarios are possible:
a. M1 > M2 (M2 will begin to move in the direction of M1 just after the collision. M1 will continue along its path with a greatly reduced speed).
b. M1 < M2 (M2 will begin to move in the direction of M1. M1 will bounce back in the opposite direction with a smaller velocity vector.)
c. M1 = M2 (M2 will take ALL of the initial momentum of M1, moving in the same direction of motion as M1 with its same speed. M1 will come to complete rest).
2) M2 and M1 are both in straight line (one-directional) motion. This set allows for a much wider range of situations.
a. Both masses move directly towards each other.
b. Both masses move in the same direction, but V1i>v2i, meaning that M1 will eventually "catch up to" (ie. collide with) M2.
If V2i>V1i, Mass 1 will never catch up to Mass 2 and so no collision is possible.
Note - Both sets of possible initial conditions in this group can be further divided. Initial mass conditions can be divided into 3 groups like before (M1 > M2, M1 < M2,
M1 = M2). Also, different velocity values (both away and towards each other) will greatly affect the results. Such varying possibilities can provide a great range of
practice problems and calculations.
3) M1 and M2 collide at angles (ie. two dimensional motion). The theory of conservation of momentum implies that momentum is also conserved in two dimensional
motion (using vectors).
Such sets can be further broken up into the same groups as before, such as one object at initial rest; motion towards each other; motion away from each other; one
mass greater than the other, one velocity greater than the other; and so on. Such scenarios represent a great representation of nature, as it allows for infinite initial
scenarios. Examples of this kind should be introduced by the teacher to the whole group and should be limited to teacher calculations in front of the whole class, or
presented to students with a strong mastery of vector addition (eg. AP level students and some honors level students). Calculations of this type among weaker groups
will bog down progress and take away valuable time from the whole group discussions.
A determination may be made to simplify models if students struggle with the mathematics involved. These students will be given initial conditions from within the first
set of collisions (one object at initial rest). For students with exceptional understanding of the concepts involved, the teacher may include collisions that occur when
two objects collide at angles.
Feedback to Students
This lesson is an application based activity that will foster students' understanding of the concept of Conservation of Momentum. The instructor will present the
equation and explain how to calculate an object's momentum vector within a well-defined reference frame. After the teacher explains the process of calculating
momentum vectors, students will be given specific varying initial conditions and calculate the overall momentum vector (p) of the system before the collision. The
teacher will then monitor the progress of the students and check individual student's calculations.
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
Students who struggle with the concepts and/or mathematical manipulations may be given simple scenarios to calculate, such as head on collisions or one mass at
initial rest. These situations offer one dimensional motion only and will thus be easier to understand.
Students who display exceptional understanding of the simpler situations can be given models of more complex systems (such as glancing collisions and/or collisions
with different masses). These scenarios require a more developed understanding of vector addition as well as an ability to conceptualize each resultant scenario.
page 4 of 5 Extensions:
An easy extension to this lab is to construct a grid for air puck collisions. Using an overhead camera with stop motion abilities, students can accurately calculate, to a
great degree of certainty, the initial and final momentum of each puck. Mass can be added to each puck to further diversify options for investigations.
Suggested Technology: Computer for Presenter, Computers for Students, Internet Connection, Interactive Whiteboard, Basic Calculators, Adobe Flash Player
Special Materials Needed:
Computer with internet access and a smart board for presentation
Further Recommendations:
If more than one computer is available, students can be broken up into groups based on similar levels of understanding. Each group may be given multiple similar
initial conditions with slight changes to each. For example, a strong group may be given the task of calculating momentum constants for masses in glancing collisions
(which require vector addition). This group may then be asked to calculate momentum for the same angular collisions only this time varying masses (ie. one mass
greater than the other and vice-versa) and make predictions. The situations may be modeled by the simulation to show the results. The group will learn that there are
infinitely many types of collisions in the universe, all with their own final conditions that are uniquely ordered by the initial conditions under the conservation of
momentum
For a group that is struggling with understanding the concept altogether, the teacher may provide a situation where one mass is initially at rest and the other mass will
come to a complete stop after the collision (ie. a pool ball striking another in a perfect straight line). The students will come to understand that in a perfect collision, all
of the initial momentum of the first ball will be given to the second ball. This provides a basic explanation of the concept and may create a foothold for the struggling
students to build on.
Additional Information/Instructions
By Author/Submitter
The Online Simulation can be found on Geogebra.org and is titled "2D Elastic Collisions of Two Hard Spheres"
SOURCE AND ACCESS INFORMATION
Contributed by: Michael Bravo
Name of Author/Source: Michael Bravo
District/Organization of Contributor(s): Miami-Dade
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
Description
Apply the law of conservation of linear momentum to interactions, such as collisions between objects.
SC.912.P.12.5:
Remarks/Examples:
(e.g. elastic and completely inelastic collisions).
page 5 of 5