Designing a Bumper to Study Impulse and Momentum · · 2014-03-05Designing a Bumper to Study Impulse and Momentum ... power, 1. Work and work-energy theorem, 2. ... Designing a

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412 Laying the Foundation in Physics

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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.

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

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

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©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org

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Laying the Foundation in Physics 413

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.

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.

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.

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.

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.

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POSSIBLE ANSWERS TO THE CONCLUSION QUESTIONS AND SAMPLE DATA

DATA AND OBSERVATIONS Impact without bumper:

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Impact with bumper:

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

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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.

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

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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. • ( )= sin = 0.90 m sin 20 = 0.31 mh L θ∆ °

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 = 2v g h∆ , where 2= 9.8 m / sg . Neglecting friction, calculate the theoretical speed of the cart just before striking the block. Show your work in the space below.

• ( )2

m m= 2 = 2 9.8 0.31 m = 2.5s s

v g h ⎛ ⎞∆ ⎜ ⎟⎝ ⎠

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.

• ( )( )

Impulse = change in momentum

Impulse = , where = 0

1.8 N s – Impulse m= = = 1.21.50 kg s

f i f

i

m v v v

vm

−

− − ⋅

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 m2.5 1.2speed without friction – speed with friction s s% Difference = 100 = 100 = 52%mspeed without friction 2.5s

−× ×

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.

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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.

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.

Car A Car B

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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 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.

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• ( )

( )

Impulse = change in momentum

Area under the curve =

Area 1.33 N s= = = 0.83 kgm m0.40 2.0s s

f i

f i

m v v

mv v

−

− ⋅− −

• The impulse is negative since the force opposes the velocity.

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.

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)

Time (s)

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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 it m m∆ ∆ −F v v v

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 balance ring stand and clamp (or books) protractor with a string and weight large clamp

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

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!


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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.

Block of

Wood

L

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.


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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 the “zoom in” button on your toolbar. The area under the spike can be used to find the impulse 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.


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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.


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Name _____________________________________

Period ____________________________________



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: ______

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?

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


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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 = 2v g h∆ , where 2= 9.8 m / sg . 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.


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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.

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.

Car A Car B


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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?


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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)


Documents

Designing a Bumper to Study Impulse and Momentum · · 2014-03-05Designing a Bumper to Study Impulse and Momentum ... power, 1. Work and work-energy theorem, 2. ... Designing a