Lab #8: Momentum Reading Assignment: Chapter 9, Sections 9-4 through Sections 9-7 Introduction: http://www.carbuyingtips.com/disaster.htm How do insurance companies reconstruct an accident, such as a car crash, to determine exactly what happened and who was at fault? They make use of one of the most fundamental ideas in physics: conservation. During the collision, they treat the two colliding cars as two isolated particles interacting only with each other. This allows them to apply one of the conservation laws that, to the best of our knowledge, are always true: Conservation of Momentum. Conservation laws lie at the foundation of physics. Conserved quantities are fundamental to the development of both classical and quantum mechanical mo dels of systems. Conservation laws are not derived. Rather, they are empirical. That is, they result from experimental studies of the real world. In this lab, you will explore this basic idea in terms of the collisions between two carts. Christian Huygens defined the term momentum, p, to describe the motion of objects, as he realized that both mass and velocity affect the way in which objects behave. This vector quantity is easily defined: p = mv Momentum is conserved in a system only if the system is isolated from all external effects. In other words, conservation of momentum occurs when all objects that interact with each other are considered to be part of the system. For example, consider the collisions that take place on a pool table when playing billiards. If only one ball is considered to be the system, then the momentum of that ball is not conserved because the ball is being influenced by the other billiard balls. However, if all of the billiard balls together define the system, then the mo mentum lost by some of the balls is found to be gained by the others. Perfectly isolated systems rarely, if ever, occur. In the case of billiard ball collisions, the table itself is in contact with the balls and should be considered as part of the system. Some external effects are small, however, and can often be ignored. Consider a system of only two objects that are completely isolated from any outside influences. If momentum is conserved when they collide, then the momentum lost by one is the momentum gained by the other. This can be expressed mathematically as follows: p 1 = p 2 p 1f p 1i = ( p 2f p 2i ) Rearranging results in the common form of the Conservation of Momentum Equation: p 1i + p 2i = p 1f + p 2f
Notice that the left side of this equation represents the total momentum of the two-object system before the interaction and the right side represents the total momentum of the two-object system after the interaction. Even though the momentum of each object changes, the total momentum of the system remains the same. Systems involving more than two objects are more complicated. However, the Conservation of Momentum Equation can easily be generalized: p 1i + p 2i +... + p ni = p 1f + p 2f +... + p nf m 1 v 1i + m 2 v 2i +...+ m n v ni = m 1 v 1f + m 2 v 2f +... + m n v nf = p final p initial all It is important to remember that momentum is a vector quantity. Addition of momentum vectors is accomplished in the same way as the addition of any other vector quantity. When the motion is in one dimension, direction can easily be incorporated into the above equation by simply using + or - signs. Motions involving two or three dimensions can be simplified by considering components because, if momentum is conserved overall, then it is also conserved in each dimension. For the purposes of this lab, only collisions in one dimension will be analyzed. One special case of conservation of momentum occurs when the masses are equal. Notice that if all of the particles in the system have equal mass, then the above equations become a description of Conservation of Velocity: v 1i + v 2i +... + v ni = v 1f + v 2f +... + v nf all This law states that the sum of the initial velocities is equal to the sum of the final velocities for all objects in the system. One of the activities in the lab is to verify this case. Newton later defined the term impulse, I, to describe the change in momentum of an object, as he realized that an object undergoing translational motion often changes speed, direction, and/or mass. I = p = p f p i This expression shows that, when the momentum of an object changes, it is not conserved. This realization often confuses students who have been taught that the Law of Conservation of Momentum has been verified experimentally many times. Its apparent failure here is due to a lack of clarity of the conditions that must be satisfied in order for momentum to be conserved. It is important to remember that momentum is conserved only if the system being considered is isolated from all external influences.
Lab #8: Momentum Goals: Determine the impulse exerted on an object by knowing its mass and measuring its change in velocity. Verify the Law of Conservation of Momentum. Determine the conditions necessary in order for Momentum to be conserved. Analyze the data for evidence of frictional effects. Equipment List: (Note: to be shared between two adjacent groups.) Low Friction Track (2 meter) 2 collision carts with extra masses 2 ultrasonic motion detectors Science Workshop Excel Activity 1: Determine the Momentum of a Cart. 1. Set up Science Workshop to read the data collected from the motion detector located at the end of the 2-meter track nearest to your lab station. Sitting on the track should be two carts. Be aware that each motion detector will record the position of the closest cart. 2. Each cart has a mass of 500 grams. If desired, you may place one or two additional 500 gram mass blocks in your cart. Record the total mass of your cart. (You will need to enter this mass into your momentum formula in Step 3.) 3. Calculate Momentum data by using the Experiment Calculator in Science Workshop. Remember to press Enter after typing in the formula for the momentum of your cart. You may want to view the tutorial if you are unfamiliar with how to use the Experiment Calculator. 4. Using Science Workshop, open a graph window of Velocity vs. Time. Once this window is displayed, add a Momentum vs. Time graph window to it by clicking on the Add Plot Menu icon. You may want to view the tutorial if you are unfamiliar with how to use the Add Plot Menu. 5. Remove the other cart from the track so that your cart is the only one present. 6. Press Record and push your cart so that is moves freely along the length of the track. Copy and paste the graphing window into your template. 7. Compare the Velocity and Momentum graphs. How are they alike? How are they different?
8. Determine the value of the Momentum of your cart. First, click and drag to highlight the region of the Momentum vs. Time graph during which the cart appears to be moving uniformly. (Note: the graphs can be resized and enlarged.) Second, using the Σ icon, calculate the mean y-value of this highlighted data, since the y-axis represents momentum. (Velocity can be found using the same method.) Activity 2: Total Momentum of the System Before and After the Collision. 1. Collaborate with the adjacent lab group to coordinate the data obtained in Activity 2. (Both carts together define the system to be analyzed in this part of the lab.) 2. Decide upon a consistent directional axis to be used by both groups. Note that the motion detectors define + to be the direction that they are facing. Since the motion detectors are facing each other, + and directions are interpreted differently by each detector. One of the groups will have to reassign directions to the Before and After momentum vectors. 3. Place both carts on the track. Add or remove extra masses so that the total masses of each cart are unequal. Record the mass of your cart. Note: Remember to update the Experiment Calculator with the new mass of your cart. 4. Press Record and push your cart so that it moves freely towards the other cart (which is also moving). Allow the carts to collide and move freely away from one another. Check to be certain that the data collected includes a substantial number of points before and after the collision. If necessary, repeat this step. Copy and paste your Momentum vs. Time graph into the template. 5. Using the same method learned in Step 8 of Activity 1, determine the momentum of each cart (1) before the collision and (2) after the collision. Record the Before and After momentum vectors of each cart according to the directional axis decided upon in Step 2. 6. What does it mean to have positive momentum? What does it mean to have negative momentum? 7. Calculate the Total Momentum Before the Collision by adding the momentum vectors of both carts before the collision. 8. Calculate the Total Momentum After the Collision by adding the momentum vectors of both carts after the collision. 9. Compare the Total Momentum before the Collision with the Total Momentum after the Collision. Determine the % difference between the two values. 10. Is momentum reasonably conserved? Why or why not? Include a discussion of error in your answer. Note: Please see your TA if you calculate an extremely large % difference. There may be several causes for this. You will either need to interpret this value or correct your mistake. Activity 3: Determine the Change in Momentum of a Cart During a Collision. 1. Calculate the change in momentum (i.e. the impulse, I) of your cart during the collision. What is the direction of this vector relative to your motion detector? 2. Is the momentum of your cart conserved? Why or why not?
Post-Lab #8: Momentum Name: Section #: In this week s lab, you investigated whether or not velocity and momentum were conserved. You accomplished this by looking at the Total Velocity and Total Momentum of a given system before and after a collision. What else could be conserved besides velocity and momentum? The search for conservation laws did not end with momentum. Gottfried Leibnitz (1646 1716), for example, considered that a quantity called vis viva might be conserved as well. He defined vis viva to be the product of mass and the square of the speed: vis viva = mv 2 Perhaps other quantities could also be conserved, such as just the square of the speed. http://www.knuten.liu.se/~bjoch509/philosophers/lei.html In this Post Lab, you will continue to investigate the concept of conservation by determining whether or not these other types of quantities are also conserved. In particular, you will consider Total Vis Viva and Total V 2. Experiments were conducted, similar to Activity 3 in the Lab, to obtain data on momentum and velocity for each of the following four situations: Experiment #1: Unequal Masses Head On Collision Experiment #2: Unequal Masses Recoil (Carts start at rest in the center of the track and are move apart via compressed spring.) Experiment #3: Equal Masses Head On Collision Experiment #4: Equal Masses Recoil (Carts start at rest in the center of the track and are moved apart via compressed spring.) Consider the following data. Note, that during each experiment a coordinate system was picked that assigned a positive value to motion originating left and moving to the right and a negative value to motion originating on the right and moving to the left.
For Experiment 1: Unequal Masses Head On Collision Mass 0.5 kg 1.5 kg P i 0.196 kg m/s -0.633 kg m/s V i 0.392 m/s -0.422 m/s P f -0.416 kg m/s -0.023 kg m/s V f -0.829 m/s -0.015 m/s For Experiment 2: Unequal Masses Recoil Mass 0.5 kg 1.0 kg P i 0 0 V i 0 0 P f -0.329 kg m/s 0.328 kg m/s V f -0.664 m/s 0.328 m/s For Experiment 3: Equal Masses Head On Collision Mass 0.5 kg 0.5 kg P i 0.478 kg m/s -0.302 kg m/s V i 0.956 m/s -0.603 m/s P f -0.301 kg m/s 0.478 kg m/s V f -0.602 m/s 0.956 m/s For Experiment 4: Equal Masses Recoil Mass 0.5 kg 0.5 kg P i 0 0 V i 0 0 P f -0.292 kg m/s 0.290 kg m/s V f -0.584 m/s 0.580 m/s
Analysis of Experimental Results 1. For each of the four experiments above, state whether or not velocity was conserved. Show your calculation for each experiment. 2. For each of the four experiments above, state whether or not momentum was conserved. Show your calculation for each experiment. 3. Determine whether or not the quantity vis viva was conserved in each of the four lab experiments. Show your calculation for each experiment. 4. Determine whether or not the quantity speed-squared was conserved in each of the four lab experiments. Show your calculation for each experiment. 5. Summarize your conclusions by placing an X in the appropriate box if the quantity was reasonably conserved within the two-cart system. Experiment? v? p?(mv 2 )?(v 2 ) # 1 Unequal - Head On # 2 Unequal - Recoil # 3 Equal Head On # 4 Equal Recoil 6. A car crash is an example of a collision. A car crash rarely constitutes a system that is isolated from external effects. Therefore, how can conservation of momentum be applied to this situation?