Download Lab 9 - Suffolk County Community College

Lab 9
CONSERVATION OF LINEAR MOMENTUM
In this experiment, you will try to verify the Law of Conservation of Linear Momentum using a
collision between two cars on the air track. Here are the important concepts to understand:
(a) THE LINEAR MOMENTUM P for a mass m is defined to be the product of the mass m and
the velocity v, i.e., P = mv. P is a vector quantity and its MKS unit is kg m/sec. A cart moving
to the right on the air track is considered to have a positive momentum.
(b) AN ELASTIC COLLISION between two objects is defined as a collision where the
mechanical energy of the system is conserved, i.e., the energy associated with the motion of the
carts is not transformed into other forms of energy. In this experiment such a collision is
produced by placing a spring between the two colliding carts.
(c) AN INELASTIC COLLISION is produced when some of the mechanical energy of the carts
is transformed into other forms of energy, e.g., heat, noise etc. An inelastic collision is produced
by using two carts that stick together after the collision. Such a collision is known as a perfectly
inelastic collision. Real collisions when the objects neither conserve their mechanical energy,
nor stick to each other after the collision lie somewhere between the above two extreme cases.
The Law of Conservation of Linear Momentum applies to ALL COLLISIONS. The purpose of
this experiment is to demonstrate this fact.
The conservation statement applies to the Total Momentum of the System P = P1 + P2
where:
P1 = momentum of the cart #1 and P2 = momentum of cart #2
Formally the conservation law states that:
In the absence of external forces, the total momentum of a system remains constant. (Total
momentum remains the same before and after the collision.) The force between the two carts at
the instant of collision is considered to be an internal force.
More specifically,
(a) AN INTERNAL FORCE is one that acts between two parts of the same system. Example:
The force of collision between two carts. Here the system is considered to be the two carts
taken together. As one cart pushes on the other the reaction force pushes back on the other
cart.
(b) AN EXTERNAL FORCE is a force between one or more parts of a given system and an
external object. Example: Friction between carts (the system) and the air track (External).
The conservation law is a direct consequence of Newton’s 2nd and 3rd laws. Consider the
collision between two carts. At the instant of collision, the cart #1 pushes cart #2 with a force
F. F is not easily measured and is usually unknown to us. Cart #2, in return, pushes cart #1
back with an equal and opposite force, namely –F. This is Newton’s 3rd Law. (Each action is
accompanied with an equal and opposite reaction.) Writing Newton’s 2nd Law for the two cars,
one arrives at:

The lab instructions for this experiment were written by R. Warasila et al. of Suffolk County Community College.
1
Cart #1
Cart #2
Force = -F
-F = m1a1 = m1v1/t
-F = m1v1/t = (m1v1)/t
-F = P1/t
Force = F
F = m2a2 = m2v2/t
F = m2v2/t = (m2v2)/t
F = P2/t
Here, the masses are written as m1 and m2 and the velocities as v1 and v2, etc. Since m1 and m2
are constants, m1v1 is written as (m1v1). F is an unknown force. However, adding the final
equations for the two carts results in:
(m1v1)
-F + F = 0 =
(m2v2)
(m1v1 + m2v2)
+
t
=
t
(P1 + P2)
=
t
Ptot
=
t
t
Therefore, the quantity Ptot = m1v1 + m2v2 is constant with time. This result is independent of
the force F which is unknown.
Procedure
PART A: Elastic Collision Between Two Equal Masses:
Choose two carts. Attach enough small masses to one to ensure equal masses for the two carts.
Leave one at rest in the middle of the air track and set the other into motion. Measure the
velocity and consequently the momentum for both carts BEFORE and AFTER the collision.
The velocity measurements are to be done as in the previous experiments that utilize the air
track. Compare the total momentum before and after the collision and give a % difference.
Repeat the experiment with at least three different initial velocities for the moving cart. A
spring at the end of the air track can be used to provide the initial velocities. Record m1 = mass
of cart #1, m2 mass of cart #2, t for each cart and calculate v1and v2, the velocities of the two
carts and the momentum before and after the collisions. There are two photocells provided in
this experiment to facilitate the measurement of the two velocities. Make sure you measure and
use the Leff appropriate to each cart.
PART B: Inelastic Collision Between Two Carts:
Now choose two carts that stick together after the collision. Masses are not important. Equal or
unequal masses are equally suitable but they must be measured. Repeat the experiment as
before. Here the initial momentum comes from the moving cart and is given as m1v1 (the
stationary cart carries no momentum). The final momentum after the collision is given by:
Pafter = m1v + m2v = (m1 + m2)v
Where v is the velocity of the two carts connected together. This part should also be done with
three different initial velocities.
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PART C: Elastic Collision Between Two Unequal Masses:
In this trial send the smaller mass cart into the larger mass cart. The only difference between this
part and Part A is that now the two masses are not equal. You will find that both carts will be in
motion after the collision. You will therefore have to measure the final momentum from both carts.
There are three velocities to measure (v1 initially and v1 and v2 after the collision) and two timers
to use. The same photocell is to be used for two measurements. Some quick reading of the timers
and resetting is necessary here. This part of the experiment should also be done, using three
different initial velocities.
PART D: Elastic Collision Between Two Unequal Masses:
In this trial send the larger mass cart into the smaller mass cart. Otherwise the experiment is the
same as part C, so be ready to reset the timers as necessary.
Summarize your results for the Law of Conservation of Linear Momentum in your report and verify
that it applies to ALL COLLISIONS, perfectly elastic, perfectly inelastic and any collision in
between these two extreme cases.
PRELIMINARY QUESTIONS:
Define momentum P and give its MKS unit here (in your words):
Compute the momenta for each of the following particles and compute the total momentum of the
three mass system:
m1 = 50 gm
v1 = 150 cm/s
m2 = 100 gm
v2 = 200 cm/s
m3 = 1 kg
v3 = 10 cm/s
State the law of conservation of total linear momentum for a system COMPLETELY:
For the system of a cart plus a small mass hanging in the air, as was done in the experiment on
Newton’s second law, draw all the forces on the system of two masses and identify which ones are
internal and which ones may be considered to be external. Do this work below (or in a separate
page to present in your report):
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DATA AND PRESENTATION
Use the data sheets for the 4 different collision setups.
Elastic Collision of Two Equal Masses
Inelastic Collision of Two Masses
Elastic Collision of Two Unequal Masses - large cart hits small cart
Elastic Collision of Two Unequal Masses - small cart hits large cart
Conservation of the Total Momentum
Error Table
Trial
Ptot before
Ptot after
% difference
Elastic Collision Equal Masses
Inelastic Collision Equal Masses
Elastic Collisions Unequal Masses
small into large
Elastic Collisions Unequal Masses
large into small
Discuss how reasonable the % differences are in terms of the various sources of errors in the
experiment.
Include a conclusion regarding the conservation of momentum in elastic and inelastic collisions.
Finally, answer the questions:
1) What was your initial expectation: Ptot before > or < Ptot after? WHY?
2) If Ptot before > or < Ptot after, make a list of the experimental effects that might contribute to this
difference. Explain how each experimental effect can change the results (does it increase Ptot after?
Decreases it? Etc.)
3) What was/were the collision(s) with larger % difference? Suggest an explanation.
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Elastic Collision of Two Carts of Equal Mass
m1 = _________________________
m2 = _________________________
Leff = _________________________
Leff = _________________________
Before
t
Cart #1
v1
After
Cart #2
P1
t
v2
P2
-
0
0
-
0
0
-
0
0
P1 + P2
5
t
Cart #1
v1
Cart #2
P1
t
v2
P2
P1 + P2
Inelastic Collision of Two Carts of Equal Mass
m1 = _______________________
m2 = _________________________
Leff = _______________________
Leff = ________________________
Before
t
Cart #1
v1
P1
t
After
Cart #2
v2
P2
-
0
0
-
0
0
-
0
0
P1 + P2
6
t
Cart #1
v1
Cart #2
P1
t
v2
P2
P1 + P2
Elastic Collision of Two Carts of Unequal Mass - light impacts heavy
m1 = _______________________
m2 = _______________________
Leff = _______________________
Leff = ______________________
Before
t
Cart #1
v1
After
Cart #2
P1
t
v2
-
0
-
0
-
0
P2
P1 + P2
7
t
Cart #1
v1
Cart #2
P1
t
v2
P2
P1 + P2
Elastic Collision of Two Carts of Unequal Mass - heavy impacts light
m1 = _______________________
m2 = _________________________
Leff = _______________________
Leff = ________________________
Before
t
Cart #1
v1
P1
t
After
Cart #2
v2
P2
-
0
-
0
-
0
P1 + P2
8
t
Cart #1
v1
Cart #2
P1
t
v2
P2
P1 + P2