Collisions in One Dimension Conservation of Linear Momentum and (Sometimes) Energy

Collisions in One Dimension
Conservation of Linear Momentum and (Sometimes) Energy
Introduction and theory:
Collisions are an important way of studying how objects interact. Conservation laws
have been developed that allow one to say quite a bit about what is happening without
knowing the exact details of the interaction during a collision. In this lab we are going to
show that momentum is always conserved when there is no net external force acting on
the system and that energy is only sometimes conserved in various kinds of collisions.
These principles are important in studying automobile collisions, planetary motion, and
the collisions of subatomic particles.
Momentum is the product of mass and velocity so it has units of kg m/sec.
p = m*v
(1)
It is a vector quantity with its direction the same as the velocity. We do not have a
special name for the unit of momentum but we do commonly use the letter p to represent
the momentum vector. Conservation of momentum is derived in your textbook using
Newton’s third law and also deals with the quantity called impulse which is the Ft,
where t is the time interval over which the force F acts. Momentum is conserved
whenever the interacting objects are only interacting with each other (or at least in the
coordinate direction which we are considering). If pi is the initial momentum of the
system before the collision and pf is the final momentum of the system after the collision
then:
Δp = (pf – pi) = 0 ,
(2)
where pi = p1i + p2i and pf = p1f + p2f in case of the collision of two carts.
Another important conservation law is the conservation of energy. Energy is a scalar
quantity and not a vector. A scalar quantity just has a magnitude and no direction.
Energy is conserved depending on whether the forces between the objects are
conservative. Examples of conservative forces are gravity, electric, and magnetic forces.
There are other forces at the level of nuclear physics that are also conservative. The most
important non-conservative force we will deal with is friction. Friction is a nonconservative force because energy is converted into heat by friction. Another example of
a non-conservative force will occur when we have two bodies that collide and stick
together. This will be a special case of friction where the energy will be converted into
heat in the process of sticking together.
In this experiment we will be dealing with collisions in one dimension. The motion of
the bodies involved is constrained by a horizontal track. This means that the velocity and
momentum vectors can be only in one of two directions, +x or –x where x represents the
1
coordinate of the track. Since we will be dealing with two bodies, the conservation of
momentum law can be written as  p i   p f . This can be expressed for two bodies as
m1 v1i  m2 v 2i  m1 v1 f  m2 v 2 f .
(3)
Thus, in order to show the conservation of momentum, we must know the masses of the
two bodies and their vector velocities before and after the collision. To see if energy is
conserved we must evaluate the kinetic energy before and after the collision. There is no
change in gravitational potential energy in this case because the motion takes place on a
level surface.
We can express conservation of energy with the following equation:
? 1
1
1
1
2
2
2
2
m1i v1i  m2i v2i  m1 f v1 f  m2 f v2 f .
2
2
2
2
(4)
A question mark is placed over the equal sign because energy will not always be
conserved.
A collision in which the total kinetic energy before the collision (KEi) is the same as the
total kinetic energy after the collision (KEf) is said to be elastic.
ΔKE = (KEf – KEi) = 0
(5)
It is also useful to formulate these fundamental relations such as law of conservation of
momentum (3) and law of conservation of energy (4) in terms of velocities. Thus we have
(for elastic collisions)
(v2f – v1f)/(vi2 – vi1) = -1
(6)
Or in other words “the relative velocities after/before are equal but opposite”. This is
true for any mass values. A somewhat more restrictive case is for vi2 = 0 (stationary
“target” m2). Then it can be shown that:
v1f = (m1 – m2) vi1/(m1+m2)
(7a)
v2f = (2m1) vi1/(m1+m2)
(7b)
Note that the ratio r = v1/v2 is constant here where vi2=0.
If the total kinetic energy after the collision is different from the total kinetic energy
before the collision, then the collision is said to be inelastic.
The fundamental relations for KE and p are now modified and the velocity relations (Eq.
6, 7) do not apply. Momentum conservation holds, but simple energy conservation does
not (if we ignore work done in sticking Velcro).
2
A collision where the colliding objects stick together after the collision is said to be
perfectly inelastic. In this type of collision the kinetic energy loss is maximum, but not
necessarily 100%. Text reference: Young and Freedman §§8.3-4.
Procedure:
The two photogates will record the position of carts as a function of time. It is done by
using picket fences of known band spacing. Make sure that the photogate light beam
is level with the 1 cm spaced bans. The recording has to be started and stopped
manually. The magnitude of the cart velocity (= speed, no direction indicated) is equal to
the slope of the linear fit to the position data.
The experimental setup is depicted below:
Open the pre-set experiment file: Labs/phy122/spring 2011/collisions 1D.
Check that the photogates on the left side of track are connected to Channel 1.
Beware: the software does not know which cart is passing through a given gate, or
which direction it is going. Keep careful notes of these items in your notebook, for
example with small diagrams.
In Excel prepare two data tables: data table 1: Calibration; data table 2: Collisions. You
will use it to record the measurements.
Data table 1: Calibration; prepare 6 rows: (slow left; slow right; moderate left;
moderate right; fast left; fast right) and 2 columns: velocity photogate 1 & velocity
photogate 2.
Data table 2: Collisions: Each collision contains 4 runs. The table should include the
values for the following quantities:
 mass of the cart 1 – m1;
 mass of the cart 2 – m2;
 velocities of each cart v1i and v2i before the collision;
 velocities of each cart v1f and v2f after the collision;
 the momentum of each cart before the collision;
 the momentum of each cart after the collision;
3










the total momentum of the two cart system before the collision;
the total momentum of the two cart system after the collision;
the relative percentage change in momentum of the system
the kinetic energy of each cart before the collision;
the kinetic energy of each cart after the collision;
the total kinetic energy of the two cart system before the collision;
the total kinetic energy of the two cart system after the collision;
the relative percentage change in kinetic energy of the system.
The ratio of relative velocities for elastic collisions only (eq. 6);
The ratio of velocities related to the ratio of masses (eq. 7 b) for collision 4 only.
Look in the Introduction and Theory section for the useful equations of momentum and
kinetic energy to enter them in the Excel.
Part 1 – Calibration
How good is your equipment? Let’s test this using a trivial known result – continuous
motion (no collision). Send a single cart (with no extra weights) though both gates at
slow speed, first from the left, then repeat from the right. Repeat this pair, but now at a
moderate speed. Repeat this pair now at fast speed.
Record the value of speed through both gates for each trial in the prepared data table 1
v
(don’t forget signs!). For each pass calculate 2 . You can add the column in Excel for
v1
these calculations. The ratio of speeds through the two detectors should be 1. Compare
your data with this “theory”. Is there any systematic effect of left vs. right, slow vs. fast,
etc. Pick the speed that had the least systematic error (such as tilted track, friction, etc) to
be used in the second part of the lab to launch the cart. In data analysis show the sample
of ratio calculations for the speed picked to use in the second part of the lab.
In the discussion, explain the physical origin of the systematic and/or statistical data.
Propose a way to correct the experimental data for this effect.
4
Fig. 1
Part 2. Collisions
Collision 1: Perfectly inelastic collision (carts of about equal masses stick together after
the collision)
a) Measure the mass of each cart.
b) Set the carts on the track with Velcro ends facing each other.
c) In all collisions: the stationary cart must be positioned between the gates as
close as possible to the second photogate. The first cart must pass through
photogate 1 before it collides with the second cart.
d) Make a collision with m1 incident on stationary m2 (v 2i =0) at chosen speed,
with m1 approaching from the left. Apply linear fit to the position vs time
graphs to find the velocities of the carts before and after collision. Because the
track as well as the carts is not exactly frictionless it is recommended to fit
only a small part of the recorded position closest to the instant of collision.
(Fig. 1)
e) Complete the row in the prepared data table 2. Repeat the above steps b), c)
and e) 3 more times.
Collision 2: Perfectly inelastic collision (carts stick together after the collision), where m1
≠ m2 and v2i=0.
5
a) Add the heavy rectangular block to the second cart, m2= M. Set the carts
on the track with Velcro ends facing each other. Set up the carts as
directed in steps b) and c) in the collision 1.
b) Make a collision of m1 on stationary M, with m1 approaching from the
left.
c) Apply linear fit to the position vs time graphs to find the velocities of the
carts before and after collision. Because the track as well as the carts is not
exactly frictionless it is recommended to fit only a small part of the
recorded position closest to the instant of collision. (Fig. 1).
d) Complete the row in the prepared data table 2. Repeat the above steps b),
c) and d) 3 more times.
Collision 3: Elastic collision (carts of about equal masses)
Remove additional mass from the cart 2.
a) Set them on the track with magnet bumpers facing each other.
b) Make a collision with m1 incident on m2 at chosen speed, with m1
approaching from the left.
c) Apply linear fit to the position vs time graphs to find the velocities of the
carts before and after collision. Because the track as well as the carts is not
exactly frictionless it is recommended to fit only a small part of the
recorded position closest to the instant of collision. (Fig. 1).
d) Complete the row in the prepared data table 2. Repeat the above steps b),
c) and d) 3 more times.
Collision 4: Elastic collision (carts stay separate after the collision), where m1 < m2 and
v2i=0.
a) Add the heavy rectangular block to the second cart, m2= M;
b) Make a collision of m1 on stationary M, with m approaching from the left.
c) Apply linear fit to the position vs time graphs to find the velocities of the
carts before and after collision. Because the track as well as the carts is not
exactly frictionless it is recommended to fit only a small part of the
recorded position closest to the instant of collision. (Fig. 1).
d) Complete the row in the prepared data table 2. Repeat the above steps b),
c) and d) 3 more times.
In Data Analysis derive equation (6); based on the data provided in the Excel’s data table
p
2, show sample calculation of relative percentage change in momentum
and relative
pi
KE
percentage change in kinetic energy
for the best run in each collision. For collision
KE i
# 3 and #4 show calculation of the relative velocity for one of the best runs. For run # 4
6
show one sample calculations of the experimental ratio r 
v1 f
v2 f
and compare it with the
mM
.
2m
Report the major results of the lab in the result table.
expected value rexp ect 
In the discussion explain your results for the velocity ratios in part 1. Comment on your
experimental results on the relative velocity ratio compare to the theoretical value. For
p
KE
the momentum and kinetic energy to be conserved
 0 and
 0 , but why do we
pi
KEi
p
KE
allow the values of
to be less than 10% and
to be less than 20% to say whether
pi
KE i
or not momentum or kinetic energy is conserved?
In the conclusion state if the objective of the lab has been met. Do the results of the
relative percentage change in momentum and relative percentage change in kinetic
energy agree with the theory?
7