Download 3.5 Represent and Reason Consider the experiments from

3.5 Represent and Reason
Consider the experiments from activity 2.1 again. Construct a bar chart to represent how the momentum
changes for the system with both carts.
a) Cart A (200 g) moving left at a constant 0.70 m/s speed hits identical cart B (200 g) that is stationary.
Cart A stops and cart B starts moving at speed 0.70 m/s to the left.
b) Cart A loaded with blocks (total mass of the cart with blocks is 400 g) moving left at 0.70 m/s hits
stationary cart B (mass 200 g). After the collision, both carts move left, cart B at speed 0.86 m/s and cart
A at speed 0.27 m/s.
c) Cart A (200 g) with a piece of modeling clay attached to the front moves left at 0.70 m/s. Identical cart
B (200 g) moves right at constant speed 0.70 m/s. The carts collide, stick together thanks to the clay, and
stop.
d) Repeat experiment (c) but this time cart A is loaded (total mass of the cart with blocks is 400 g). After
the collision both carts stick together and travel left at speed 0.23 m/s.
3.6 Represent and Reason
Now consider your system to be cart A only. Create a conservation bar chart to represent the momentum before
and after the collision for the system with only cart A. Use the center column (Δ) to show how the momentum
changes. Then write a mathematical expression showing how the momentum before is related to the momentum
after.
Experiment
a) Cart A (200 g) moving left at a constant 0.70 m/s speed
hits identical cart B (200 g) that is stationary. Cart A stops
and cart B starts moving at speed 0.70 m/s to the left.
Math statement:
Bar Chart
before
Δ
after
before
Δ
after
+
0
b) Cart A loaded with blocks (total mass of the cart with
blocks is 400 g) moving left at 0.70 m/s hits stationary
cart B (mass 200 g). After the collision, both carts move
left, cart B at speed 0.86 m/s and cart A at speed 0.27 m/s.
Math statement:
+
0
-
c) Cart A (200 g) with a piece of modeling clay attached to
the front moves left at 0.70 m/s. Identical cart B (200 g)
moves right at constant speed 0.70 m/s. The carts collide,
stick together thanks to the clay, and stop.
Math statement:
before
Δ
after
before
Δ
after
+
0
-
d) Repeat experiment (c) but this time cart A is loaded (total
mass of the cart with blocks is 400 g). After the collision
both carts stick together and travel left at speed 0.23 m/s.
Math statement:
+
0
-
e) Look at the definition for system and the types of interactions. What does the center column (Δ) represent?
Explain.
Impulse to the Rescue
4.1 Hypothesize
In activity 3.1, we saw that the momentum of cart A changed during each process (i.e., from the initial state to
the final state). Why does the momentum of the system object change? Use Newton’s laws to come up with a
hypothesis that can explain the mechanism through which the momentum of a system changed.
4.2 Derive an Expression
r
Cart A of mass m is moving at velocity vA1 on a horizontal, frictionless track.
r
At clock reading t1 your hand begins to push on the cart exerting a constant force Fh on A .
r
At clock reading t2 the cart is moving at faster velocity vA2 and your hand stops pushing.
a) Use the cart as your choice of system. Let the initial state be before you push on the cart and the final state
after the push. Sketch the system in the initial and final state, and label all given quantities.
b) Has the total momentum in the system changed from before the push to after? If so, identify the external
object that caused the change. Sketch a conservation bar chart.
c) Let's try
to calculate the quantity of change. Use Newton’s second law and the definition of acceleration to
r
r
r
show Fh on A (t2 ! t1 ) = mvA2 ! mvA1 .
Did You Know?
r
r
The term on the left Fh on c (t2 ! t1 ) is called the impulse due to the external force Fh on c during that time interval.
r
r
The term on the right mvc2 ! mvc1 is called the change in the momentum of the cart.
d) Why is it important to have a constant force when we are deriving an expression for the impulse?
e) Suppose the cart’s forward velocity decreased due to the force exerted by a hand pushing lightly back on it.
How would you write the expression for the impulse due to that force?
Here’s An Idea!
Impulse-momentum bar charts provide a concrete way to represent impulse-momentum processes. Notice
that each bar of an impulse-momentum bar chart represents the component of momentum along an axis
for an object in the system (an up bar represents momentum in the positive direction and a down bar
represents momentum in the negative direction). A shaded impulse bar in the middle of the graph
indicates that an external object interacted with the system for a non-zero time interval causing a change
in the momentum of the system object(s). In this particular example, the external impulse is zero. Nonzero impulses also have direction (up + and down –).
1. Make initial-final sketches.
Initial state
r
v i1
Final state
r
v i2
r
vf
2. Draw
velocity
arrows.
3. Choose
a system.
!
1
2
!
1
!
x
2
x
Bar chart
pi1
+
0
p i2
I
pf1 pf2
4. Draw initial and
final momentum
bars (watch signs)
for each object in
the system.
_
In the above impulse-momentum process, we chose our system to be both cars. However, we could choose to
have just car 1 as the system object. The following bar chart represents the impulse-momentum process for this
system choice.
We don’t know the exact amount of momentum or impulse usually but we can still make estimates based
on the situation. The column for the impulse bar (I) is shaded to indicate that it is not a type of momentum
but is instead a process involving an interaction between a system object and an object outside the system;
in the scenario above, the interaction is between car 1 (the system) and car 2 (not in the system).
3.9 Reason
Read each of the scenarios below. For each scenario, identify two systems, one where the quantity is constant
and one where the quantity is conserved. Represent each scenario with a conservation bar chart. Write a
mathematical expression to show the conservation of this quantity.
a) In the morning, you leave the house with $5.00 in your pocket. At lunchtime, you pay the cashier $2.75
for a tray of pasta, garlic bread, and a bottle of water.
b) You have seven Famous Scientists trading cards (from the latest issue of Scientific American). Your
friend has a larger collection, with 15 Famous Scientists trading cards. Your friend trades you Madame
Curie, Albert Einstein, and Isaac Newton for the much-coveted Lise Meitner card. In addition, she gives
you her duplicate of Niels Bohr.
c) James (75 kg) is sitting in a canoe (15 kg) on a flat lake moving forward at 0.3 m/s. A large inflatable
raft with 3 passengers (mass 180 kg) crashes into the front of James’ canoe with a speed of 0.7 m/s.
After the crash, the canoe and raft are moving together with a speed of 0.37 m/s.
Reflect: How do bar charts help you to understand or solve the problem? What does
the center column of the bar chart graphs signify? Explain how you can use a bar
chart to write a mathematical statement for a process.