Conservation of Linear Momentum and the
Types of Collisions
by SHS Encoder 3 on June 15, 2017
lesson duration of 0 minutes
under General Physics 1
generated on June 15, 2017 at 04:20 pm
Tags: Types of Collisions, Linear Momentum
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Generated: Jun 16,2017 12:20 AM
Conservation of Linear Momentum and the Types of Collisions
( 2 hours )
Written By: SHS Encoder 3 on July 8, 2016
Subjects: General Physics 1
Tags: Types of Collisions, Linear Momentum
Resources
University Physics with modern Physics (12th ed.)
Young, H. D., & Freedman, R. A. (2007). University Physics with modern Physics (12th ed.). Boston, MA: AddisonWesley.
Physics for scientists and engineers (5th ed.)
Tipler, P. A., &Mosca, G. (2008).Physics
(2008).Physics for scientists and engineers (5th ed.). New York: W. H. Freeman.
Physics (8th ed.)
Cutnell, J. D., & Johnson, K. W. (2009). Physics (8th ed.). Hoboken, NJ: Wiley.
Content Standard
The learners demonstrate an understanding of...
1. Center of mass
2. Momentum
3. Impulse
4. Impulse-momentum relation
5. Law of conservation of momentum
6. Collisions
7. Center of Mass, Impulse, Momentum, and Collision Problems
8. Energy and momentum experiments
Performance Standard
The learners are able to solve, using experimental and theoretical approaches, multiconcept, rich-context problems
involving measurement, vectors, motions in 1D, 2D, and 3D, Newton’s Laws, work, energy, center of mass,
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momentum, impulse, and collisions
Learning Competencies
The learners explain the necessary conditions for conservation of linear momentum to be valid.
The learners compare and contrast elastic and inelastic collisions
The learners apply the concept of restitution coefficient in collisions
The learners predict motion of constituent particles for different types of collisions (e.g., elastic, inelastic)
INTRODUCTION 5 mins
1. Review the learners on their mastery of the prerequisite knowledge (external and internal forces, momentum, and
motion of a system of particles) for them to fully understand the succeeding lesson.
2. Review the learners by asking them the following questions:
a. What does it mean when a physical quantity is conserved?
b. What does it mean when a system is isolated?
3. Provide real-life examples of collisions such as in sports and traffic collisions and let them analyze what happens
during collisions.
MOTIVATION
10 mins
1. Prepare a quick illustration showing an isolated system such as but not limited to the example below:
Two astronauts floating in space (labeled A and B) are moving closer, bump each other, and push themselves as a
consequence.
Ask the learners to do the following:
a. Draw the free-body diagram of the system after their collision; and
b. Label the external and internal forces acting on the system.
c. Use Newton’s Third Law of Motion to ask the learners on the nature of the impulses that act on the two astronauts.
Are they equal? Do they have the same sign?
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(Image obtained from Young & Freedman, 2007, p.253)
INSTRUCTION 60 mins
Conservation of momentum
1. Begin the discussion by going back to the previous example concerning astronauts A and B.
2. Draw the free-body diagram of both astronauts on the board.
3. Label the internal forces and show that there are no external forces acting on the system.
4. As you have now defined an isolated system, show on the board that the rate of change of momenta of the
astronauts are given by
5. Each astronaut has momentum that changes over time, however, these changes are related to each other based on
Newton’s third law of motion. These two rates of change of momentum are therefore equal but opposite in magnitude.
Hence,
Or, summing the internal forces involved would yield:
Using linear combination rule for derivatives, therefore,
6. Let the learners notice that the vector sum of the momenta
is just the total momentum
of the system of
two particles and finally, rewrite the expression
as:
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7. From the latter equation, this would now result to the following statement:
If the net external force acting on the system of particles is zero, then the total momentum of that system is constant.
Collisions in one dimension
1. Start by differentiating elastic collisions from inelastic ones and the conditions needed for a specific type of collision
to happen.
a. State what happens to the system’s:
i. Total kinetic energy
ii. Total momentum
iii. Observations on the physical state of the bodies involved before and after collision
2. After briefly discussing each type of collision, by cold calling, ask the learners the following:
a. Site examples of elastic and inelastic collisions (one each).
b. In inelastic collisions, what other forms of energy can the lost kinetic energy be transformed into?
c. What does it mean when we say a collision is “completely” inelastic?
3. You may limit the number of particles involved to two.
Completely inelastic
1. Derive the final velocity equation for a completely inelastic collision. Assign initial velocities for the two particles to be
and
with masses
and
, respectively.
2. Ask the learners that if the total kinetic energy of the system is lost, and if the kinetic energy depends on the speed
of the particles, why is it that the entangled objects still move with the same final velocity? (Answer:
Consider relative
(Answer:Consider
velocities.)
3. You can also tell the learners to modify the variables involved to obtain a specific ratio for velocity (e.g. setting both
masses to be equal and one particle initially at rest, you can easily verify that
).
4. Write this problem on the board:
A ballistic pendulum can be used to measure the speed of a projectile, such as a bullet. The ballistic pendulum shown
in Figure 7.13 a consists of a stationary 2.50-kg block of wood suspended by a wire of negligible mass. A 0.0100-kg
bullet is fired into the block, and the block (with the bullet in it) swings to a maximum height of 0.650 m above the initial
position (see part b of the drawing). Find the speed with which the bullet is fired, assuming that air resistance is
negligible.
Elastic
1. Consequently for elastic collisions with one particle initially at rest, derive the expression for the final velocity of each
particle.
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2. Again, encourage them to alter the initial conditions to obtain a specific final velocity value.
3. Tell the learners that during the course of your derivation for the final velocity, they may have encountered a relation
between the relative velocities of the particles.
4. Tell them that their observation is a condition needed to be satisfied in order for the kinetic energy of the system to
be conserved.
Elastic collisions and relative velocities.
ENRICHMENT 30 mins
1. Extend the discussion to collisions in two dimensions. Site further examples and practice problems such as the one
below:
Two objects slide over a frictionless horizontal surface. The first object, mass m1 = 5 kg, is propelled with speed v1 =
4.5 m/s toward the second object, mass m2 = 2.5 kg, which is initially at rest. After a collision, both objects have
velocities that are directed ? = 30o on either side of the original line of motion of the first object. What are the final
speeds of the two objects? Is the collision elastic or inelastic?
2. Discuss the coefficient of restitution e as a measure of the elasticity or “bounciness” of a collision.
Show the equation and discuss the variables involved.
Relate it with the kinetic energies before and after impact. From here, you can show the simple derivation.
Discuss the values of e for completely inelastic and elastic collisions.
Provide examples such as in sports equipment.
3. Ask the learners if it is possible for the kinetic energy to be greater after collision (superelastic collision). After they
have answered, give examples of cases involving superelastic collisions.
4. Discuss briefly the mechanics of rocket propulsion along one dimension.
EVALUATION 15 mins
1. You may present the table below as a summary of the discussion:
2. Instruct the learners to answer the following questions in a sheet of paper:
a. State the law of conservation of momentum and the conditions for it to be valid.
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b. Problem 1. Two gliders move toward each other on a frictionless linear air track
.
After they collide, glider b moves away with a final velocity of 2 m/s. What is the final velocity of glider a? How do the
changes in momentum and in velocity compare for the two gliders?
c. Problem 2. A marksman holds a rifle of mass mR = 3 kg loosely in his hands, so as to let it recoil freely when fired.
He fires a bullet of mass mB = 5 kg horizontally with a velocity relative to the ground of vrx = 300 m/s. What is the recoil
velocity of the rifle? What are the final momentum and kinetic energy of the bullet? Of the rifle?
3. As an assignment to be submitted next meeting, refer to the problem below: Derive the general expression for the
final velocity of two particles A and B colliding elastically along the x-axis. Particle A with mass mA starts with velocity
vAwhereas particle B with mass mB initially moves with velocity vB in a head-on collision.
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