Download Crumple Zone - cloudfront.net

13
Crumple Zone
Crumple Zone
Designing a Bumper to Study Impulse and Momentum
OBJECTIVE
Students will measure the impulse applied to a cart at the bottom of a ramp to bring the cart to a stop.
They will design a paper bumper to increase the time of contact and decrease the impact force acting on
the cart to bring it to a stop.
T E A C H E R
P A G E S
LEVEL
Physics
NATIONAL STANDARDS
UCP.2, UCP.3, A.1, A.2, B.4
TEKS
2(A), 2(B), 2(C), 2(D), 2(E), 2(F), 4(A), 4(B), 4(C), 4(D), 5(A), 5(B), 5(C), 5(D)
CONNECTIONS TO AP
I. Newtonian mechanics, B. Newton’s laws of motion (including friction and centripetal force),
C. Work, energy, power, 1. Work and work-energy theorem, 2. Conservative forces and potential
energy, 3. Conservation of energy, D. Systems of particles, linear momentum, 2. Impulse and
momentum
TIME FRAME
90 minutes
MATERIALS
(For a class of 28 working in groups of 4)
14 sheets of 8½" x 11" paper
7 pairs of scissors
700 cm of transparent tape
7 carts
7 meter sticks
7 25-g accelerometers
7 CBL2TM or LabPro® interface devices
7 tracks for the carts to roll down
7 4" × 4" wood blocks
412
balance to measure the mass of each cart
7 ring stands and clamps, or books to elevate one
end of the ramp
7 protractors with a string and weight attached to
the center to measure the angle of the ramps
7 large clamps to clamp the 4" × 4" wood blocks
to each table
7 computers with Logger Pro® software or
graphing calculator
Laying the Foundation in Physics
©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org
Crumple Zone
13
TEACHER NOTES
This activity allows students to work together as a team to design and engineer a cart bumper using
only a single piece of paper and some tape. Students should have the opportunity to test their designs
multiple times before the final evaluation of their product. As they test the bumpers, hopefully they
will realize that the purpose of a bumper is to reduce the force on the cart by increasing the time
interval during which the force acts. The same principle applies to the deployment of air bags in a
collision. During the conclusion students are asked to answer questions and perform calculations using
the impulse-momentum theorem.
The speed at which the cart strikes the block is not particularly important, as long as it does not vary too
much from run to run. However, you should instruct the students to choose a low enough angle for the
ramp so that the acceleration of the cart as it strikes the block is less than 25 g’s. Students need to collect
at least 1000 samples per second to get accurate data during the collision.
Note that the sample graphs given in this lesson show a “positive” force and acceleration; that is, the
spike points upward. You may prefer to have the spikes point downward to reinforce that the force
and acceleration are directed oppositely to the initial velocity of the cart. The orientation of the spikes
depend on which way the student calibrates the accelerometer probe, that is, positive upward and
negative downward, or vice-versa.
As a post-lab activity, you may want to have the students do some research on designing crash-test
bumpers. A good place to start is www.hwysafety.org.
Laying the Foundation in Physics
©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org
413
P A G E S
To evaluate student designs you may want to have one ramp at the front of your classroom that will
serve as the final test for each cart. Allowing each group to crash test their bumper in front of the rest of
the class can be fun for all the students, especially if you have a projection device for the computer or
calculator screen so that the class can see the resulting graph in real time during the crash.
T E A C H E R
Any set of ramps and carts will work for this activity, as long as the carts are the same size and mass for
each lab group. The carts used in this activity will be repeatedly crashed into a wood block; therefore
you may not want to use your best carts and tracks for this activity. (If the angle is not too high, the carts
are not usually damaged.) Inexpensive crashing carts can be made out of a block of wood and roller
skate wheels. A board serves well as a ramp, as long as you can rely on the carts to stay on the board
each time they roll down. Using a router to cut grooves in the board to guide the wheels of the carts as
they roll down will help alleviate cart mishaps.
13
Crumple Zone
POSSIBLE ANSWERS TO THE CONCLUSION QUESTIONS AND SAMPLE DATA
T E A C H E R
P A G E S
DATA AND OBSERVATIONS
Impact without bumper:
414
Laying the Foundation in Physics
©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org
Crumple Zone
13
Impact with bumper:
T E A C H E R
P A G E S
Mass of cart: 1.50 kg
Angle of ramp from the horizontal: 20°
Length of ramp from the front of the cart at its initial starting position to the front of fixed block: 0.90 m
Laying the Foundation in Physics
©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org
415
13
Crumple Zone
Run
Area Under
a vs. t Graph
Mass of Cart
× Area
Under Graph
Maximum
Acceleration
Mass of Cart
× Maximum
Acceleration
Time of
Collision
∆t
Without
bumper
1.2 m/s2 × s
1.8 kg m/s2 × s
60.0 m/s2
90.0 N
0.04 s
Bumper 1
1.2 m/s2 × s
1.8 kg m/s2 × s
15.0 m/s2
22.5 N
0.13 s
Bumper 2
T E A C H E R
P A G E S
ANALYSIS
1. In this activity, we are ultimately interested in impulse, which is the product of force and time. What
are the units for
a. the area under the Acceleration vs. Time graph? Does the area represent velocity or change in
velocity?
• m/s2 × s
• This is equal to the unit m/s, and represents the change in velocity of the cart.
b. the product of the mass of the cart and the area under the Acceleration vs. Time graph?
• kg m/s2 × s, which is equal to a N·s, the unit for impulse.
2. What is the quantity that results from the product of the mass of the cart and the maximum
acceleration of the cart?
• The maximum force acting on the cart as it strikes the block.
3. Compare the area under the graph produced during the collision without the bumper to the area
produced with the bumper. Is the area under the spike greater than, less than, or equal in each case?
Explain your answer in terms of impulse and change in momentum.
• The area under the graphs are equal. The cart experiences the same impulse in each collision, but
not the same force or time interval. The students’ experimental data, however, will likely not
give exactly equal impulses due to data sampling and other limitations of the equipment.
4. In terms of the graph produced in the collision, what were you trying to accomplish by attaching the
bumper to the cart?
• The purpose of having a bumper is to extend the time of contact between the cart and the block
and thus reduce the force of impact.
416
Laying the Foundation in Physics
©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org
Crumple Zone
13
5. Using the angle of your ramp and the length of the ramp L between the cart and the block, calculate
the difference in height ∆h between the initial starting position of the cart and the point at which it
strikes the block.
• ∆h = L sin θ = ( 0.90 m ) sin 20° = 0.31 m
6. If there were no friction between the ramp and cart, the speed of the cart just before it strikes the
block could be found by the equation v = 2 g ∆h , where g = 9.8 m / s 2 . Neglecting friction,
calculate the theoretical speed of the cart just before striking the block. Show your work in the space
below.
•
v =
2 g ∆h =
m⎞
m
⎛
2 ⎜ 9.8 2 ⎟ ( 0.31 m ) = 2.5
s ⎠
s
⎝
Impulse = change in momentum
•
Impulse = m ( v f − vi ) , where v f = 0
− ( −1.8 N ⋅ s )
– Impulse
m
=
= 1.2
1.50 kg
s
m
8. Find the percent difference between the theoretical and actual speeds of the cart just before impact.
This value shows the effect of friction during the experiment. Show your work.
m
m
2.5 − 1.2
speed without friction – speed with friction
s
s × 100 = 52%
• % Difference =
×100 =
m
speed without friction
2.5
s
CONCLUSION QUESTIONS
1. Define impulse, and give its units.
• Impulse is a vector equal to the product of force and the time during which the force acts. Its
units are N · s, or kg · m/s.
2. State whether the following statement is true or false and explain your choice: The impulse applied
to an object is equal to its momentum.
• The impulse is not equal to the momentum of an object. It is equal to the change in momentum
of the object.
Laying the Foundation in Physics
©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org
417
P A G E S
vi =
T E A C H E R
7. Using the information from the graph produced in the collision without the bumper and the mass of
your cart, find the actual speed of the cart just before striking the block. Show your work, and be
sure to treat the impulse as negative, since it is opposite to the direction of the initial velocity.
13
Crumple Zone
3. Two crash-test cars of equal mass are equipped with different bumpers, A and B. The cars are
initially traveling at the same speed before striking a fixed wall. The Acceleration vs. Time graphs
for each car are shown below.
P A G E S
Car A
Car B
T E A C H E R
Both graphs indicate an area under the curve of 1.2 m/s2 × s.
a. Which of the cars experiences the greater impulse? Explain your answer.
• The impulse is the product of the mass of each car and the area under the a vs. t graph. Since
the masses of the cars are the same and the areas under the curves are the same, the impulse
acting on the cars is the same.
b. Which of the cars experiences the greater maximum force? Explain your answer.
• Car B experiences the greater acceleration and therefore the greater force, as shown by the
higher peak for the acceleration vs. time graph.
c. Assuming the passengers in both cars are wearing seatbelts, which of the cars appears to have the
bumper which is safer for the passengers in the car? Explain your answer.
• Car A is the safer bumper, since there is less force applied to the passengers. In addition, the
force is applied over a larger time interval.
418
Laying the Foundation in Physics
©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org
Crumple Zone
13
4. A force acts on a cart and slows it down. The Force vs. Time graph below shows how the force
varies with time during the collision.
T E A C H E R
P A G E S
a. Estimate the magnitude of the maximum force acting on the cart.
• 150 N
b. What is the approximate time interval ∆t during which the force acts?
• 1.565 s – 1.527 s = 0.038 s
c. According to dialog box on the graph, what is another name for the area under the curve?
• The integral
d. Is the product of the maximum force and the time interval during which it acts approximately
equal to the area under the curve? Why or why not?
• No, the maximum force does not occur over the entire time interval, but only for an instant.
e. The cart is initially traveling at a speed of 2.0 m/s and is slowed by the force to a speed of
0.40 m/s during the time interval. Calculate the mass of the cart. Show your work, and be sure
to treat the impulse as negative, since it is opposite to the direction of the initial velocity.
Laying the Foundation in Physics
©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org
419
13
Crumple Zone
Impulse = change in momentum
•
Area under the curve = m ( v f − vi )
m =
•
−1.33 N ⋅ s
Area
=
= 0.83 kg
( v f − vi ) 0.40 m − 2.0 m
s
s
The impulse is negative since the force opposes the velocity.
6. The graph below represents a collision of a moving cart and a fixed block. Suppose the block at the
end of the ramp were not fixed, but could move freely when struck by the cart. On the axes below,
sketch a graph of force vs. time for the collision between the cart and free-moving block.
• The force vs. time graph would be wider and shorter, since the cart would remain in contact with
the block for a longer time interval. The block would carry away some of the total momentum.
Force vs. Time
Force (N)
T E A C H E R
P A G E S
5. What are some ways in which you could improve your bumper design?
• Answers will vary. Some improvements might include making the bumper less rigid, trying to
make the bumper more collapsible, trying a honeycomb design, etc.
Time (s)
420
Laying the Foundation in Physics
©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org
Crumple Zone
13
Crumple Zone
Designing a Bumper to Study Impulse and Momentum
Today’s cars are built with bumpers which “soften” the force of impact of a collision. A car moving
with momentum mv can be brought to rest by a force F acting during a time ∆t. The product of force and
time (F∆t) is called the impulse. The impulse applied to a mass changes the momentum of the mass. The
impulse momentum theorem can be expressed as
F∆t = m∆v = m ( v f − v i )
A car moving at 30 mph can be brought to rest over a short period of time or a long period of time. The
change in momentum is the same in each case, and therefore the impulse should be the same in each
case. However, the force acting on the car to bring it to a stop and the time during which it acts is not the
same in each case. The force is large for a short stopping time, and the force is smaller for a larger
stopping time. Thus, a bumper can reduce the impact force by crumpling to extend the time during
which the force acts on the car and its passengers during a collision.
PURPOSE
You will design a paper bumper which will soften the impact of the collision between a cart and a fixed
block of wood. Your design will be evaluated by the shape of an acceleration vs. time graph produced
during the collision.
MATERIALS
2 sheets of 8½" × 11" paper
scissors
2 50-cm strips of transparent tape
cart
meter stick
25-g accelerometer
CBL2TM or LabPro® interface device
track
balance
ring stand and clamp (or books)
protractor with a string and weight
large clamp
computer with Logger Pro® software or graphing
calculator
4" × 4" wood block
Safety Alert
Keep your fingers out of the way when the cart strikes the block at the bottom of the ramp!
Laying the Foundation in Physics
©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org
421
13
Crumple Zone
PROCEDURE
1. Obtain two sheets of 8½" × 11" copy paper, a pair of scissors, two 50-cm strips of transparent tape,
and a cart from your teacher. Measure the mass of your cart and record the mass on your student
answer page.
2. Elevate one end of the ramp with a ring stand and clamp or several books. Clamp a wooden block at
the bottom end of the ramp so that it is securely fastened to the table. Use the protractor and hanging
weight to find the angle θ of the ramp from the horizontal. See Figure 1 below. Record this angle on
your student answer page.
L
Block of
Wood
Track
Figure 1
3. Place the cart at the top of the ramp with its back wheels at the top of the ramp. Position the cart at
the top of the ramp in its initial starting position. Measure the length of the ramp from the front of
the cart to the front of the fixed collision block. Record this length on your student answer page.
4. Connect the 25-g accelerometer to the LabPro interface, and the interface to your computer or
graphing calculator. Open the Logger Pro software on the computer, and choose the accelerometer
probe, or an experiment using the accelerometer, such as Newton’s second law. You should see an
acceleration vs. time graph appear on the screen. See Figure 2.
5. Securely attach the accelerometer to the cart so that the arrow on the accelerometer points toward the
front of the cart. To calibrate the probe using Logger Pro software, click Experiment on the toolbar,
Calibrate. Click the Calibrate tab, then Perform Now.
6. Hold the cart vertically with the arrow on the accelerometer pointing downward. In the box
containing –25, type in –9.8, to indicate the downward acceleration due to gravity. Click Keep. Turn
the cart so that the arrow on the accelerometer points upward. In the next box that appears, type 9.8,
and Keep.
7. Place the cart at the top of the ramp with the front of the cart facing the fixed block. Click Collect,
and allow the cart to roll down the ramp and strike the block. You should see an acceleration vs.
time graph such as the one in Figure 2.
422
Laying the Foundation in Physics
©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org
Crumple Zone
13
Figure 2
8. Click and drag a box around the portion of the graph which recorded the acceleration vs. time data
(the “spike”). You may want to zoom in on the area of the graph you are interested in by clicking on
on your toolbar. The area under the spike can be used to find the impulse
the “zoom in” button
which acted on the cart during the collision. Click the button on the toolbar ( ) which gives the
area under the spike you have boxed, and record the area on your student answer page. Be sure to
include the units for the area.
9. Click the Examine button (x = ) on the toolbar. Trace the curve and note the time at which the
collision began and the time at which the collision ended. Subtract the two times to find the time
interval ∆t during which the collision took place. Record the value for ∆t in the data table on your
student answer page.
10. Trace the curve to the top of the spike and record the maximum acceleration of the cart during the
collision.
11. Your teacher may want you to print, save, or sketch the acceleration vs. time graphs you produce
throughout this lab.
12. Discuss the design of your bumper with your lab partners. Your bumper should be designed to
reduce the force on the cart by increasing the time during which the force of impact acts on the cart.
In other words, you want the graph produced by the collision with a bumper to be shorter and wider
than the graph produced in a collision with no bumper. Make a sketch of several designs you think
might be effective in reducing the strike force of your cart.
Laying the Foundation in Physics
©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org
423
13
Crumple Zone
13. When you have chosen a design, use the scissors, paper, and tape to build the bumper; attach the
bumper to the front of the cart.
14. Repeat steps 5, 6, and 7 with your first bumper attached to the front of the cart.
15. Build a second bumper and attach it to the cart. Your teacher may want to observe your second run
and the resulting graph.
16. Repeat steps 5, 6, and 7 with your second bumper attached to the front of the cart.
424
Laying the Foundation in Physics
©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org
Crumple Zone
13
Name _____________________________________
Period ____________________________________
Crumple Zone
Designing a Bumper to Study Impulse and Momentum
DATA AND OBSERVATIONS
Mass of cart: ________________________ kg
Angle of ramp from the horizontal: ______________
Length of ramp from the front of the cart at its initial starting position to the front of fixed block: ______
Run
Area Under
a vs. t Graph
Mass of Cart
× Area
Under Graph
Maximum
Acceleration
Mass of Cart
× Maximum
Acceleration
Time of
Collision
∆t
Without
bumper
Bumper 1
Bumper 2
ANALYSIS
1. In this activity, we are ultimately interested in impulse, which is the product of force and time. What
are the units for
a. the area under the acceleration vs. time graph? ______________ Does the area represent velocity
or change in velocity?
b. the product of the mass of the cart and the area under the acceleration vs. time graph? _________
2. What is the quantity that results from the product of the mass of the cart and the maximum
acceleration of the cart?
Laying the Foundation in Physics
©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org
425
13
Crumple Zone
3. Compare the area under the graph produced during the collision without the bumper to the area
produced with the bumper. Is the area under the spike greater than, less than, or equal in each case?
Explain your answer in terms of impulse and change in momentum.
4. In terms of the graph produced in the collision, what were you trying to accomplish by attaching the
bumper to the cart?
5. Using the angle of your ramp and the length of the ramp L between the cart and the block, calculate
the difference in height ∆h between the initial starting position of the cart and the point at which it
strikes the block.
6. If there were no friction between the ramp and cart, the speed of the cart just before it strikes the
block could be found by the equation v = 2 g ∆h , where g = 9.8 m / s 2 . Neglecting friction,
calculate the theoretical speed of the cart just before striking the block. Show your work in the space
below.
7. Using the information from the graph produced in the collision without the bumper and the mass of
your cart, find the actual speed of the cart just before striking the block. Show your work, and be
sure to treat the impulse as negative, since it is opposite to the direction of the initial velocity.
8. Find the percent difference between the theoretical and actual speeds of the cart just before impact.
This value shows the effect of friction during the experiment. Show your work.
426
Laying the Foundation in Physics
©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org
Crumple Zone
13
CONCLUSION QUESTIONS
1. Define impulse, and give its units.
2. State whether the following statement is true or false and explain your choice: The impulse applied
to an object is equal to its momentum.
3. Two crash-test cars of equal mass are equipped with different bumpers, A and B. The cars are
initially traveling at the same speed before striking a fixed wall. The acceleration vs. time graphs for
each car are shown below.
Car A
Car B
Both graphs indicate an area under the curve of 1.2 m/s2 × s.
a. Which of the cars experiences the greater impulse? Explain your answer.
b. Which of the cars experiences the greater maximum force? Explain your answer.
Laying the Foundation in Physics
©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org
427
13
Crumple Zone
c. Assuming the passengers in both cars are wearing seatbelts, which of the cars appears to have the
bumper which is safer for the passengers in the car? Explain your answer.
4. A force acts on a cart and slows it down. The force vs. time graph below shows how the force varies
with time during the collision.
a. Estimate the maximum force acting on the cart.
b. What is the approximate time interval ∆t during which the force acts?
c. According to the graph, what is another name for the area under the curve?
428
Laying the Foundation in Physics
©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org
Crumple Zone
13
d. Is the product of the maximum force and the time interval during which it acts approximately
equal to the area under the curve? Why or why not?
e. The cart is initially traveling at a speed of 2.0 m/s and is slowed by the force to a speed of
0.40 m/s during the time interval. Calculate the mass of the cart. Show your work, and be sure
to treat the impulse as negative, since it is opposite to the direction of the initial velocity.
5. What are some ways in which you could improve your bumper design?
6. The graph below represents a collision of a moving cart and a fixed block. Suppose the block at the
end of the ramp were not fixed, but could move freely when struck by the cart. On the axes below,
sketch a graph of force vs. time for the collision between the cart and free-moving block.
Force (N)
Time (s)
Laying the Foundation in Physics
©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org
429