Physics 191 Elastic and Inelastic Collisions

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1 Physics 191 Elastic and Inelastic Collisions Introduction Momentum Energy Collisions Elastic and inelastic collisions Elastic collisions Inelastic collision Conservation laws for macroscopic bodies Kinetic Energy in Inelastic Collisions Homework Questions - Hand in before class! 4 3 Experimental setup 5 4 Measurements Data recording Operating and reading out the photogates Uncertainty estimation Inelastic collision Calculations Elastic collisions Calculations Analysis and discussion 9 1

2 1 Introduction The following experiment explores the conservation of momentum and energy in a closed physical system (ideally: no interaction of measured objects with rest of universe). As you probably know from your physics lecture course, the conservation of energy and momentum play an important role in physics and their conservation is a consequence of fundamental symmetries of nature. 1.1 Momentum For a single particle (or a very small physical object), the momentum is defined as the product of the mass of the particle and its velocity: p = m v (1) Momentum is a vector quantity, making its direction a necessary part of the data. To define the momentum in our three-dimensional space completely, one needs to specify its three components in x, y and z direction. The momentum of a system of more than one particle is the vector sum of the individual momenta: p = p 1 + p = m v 1 + m v (2) Newton s Second Law of mechanics can be written in a form which states that the rate of the change of the system s momentum with time is equal to the sum of the external forces acting on this system: d p dt = Σ F (3) From here we can immediately see that when the system is closed (which means that the net external force acting on the system is Σ F = 0), the total momentum of the system is conserved (constant). 1.2 Energy Another important quantity describing the evolution of the system is its energy. The total energy of a given system is generally the sum of several different forms of energy. Kinetic energy is the form associated with motion, and for a single particle: K = 1 2 mv2 (4) Here v without the vector symbol stands for the absolute value of the velocity, v = v 2 x + v 2 y + v 2 z. In contrast to momentum, kinetic energy is NOT a vector; for a system of more than one particle the total kinetic energy is the algebraic sum of the individual kinetic energies of each particle: K = K 1 + K Another fundamental law of physics is that the total energy of a system is always conserved. However, within a given system, one form of energy may be converted to another (such as potential energy converted to kinetic in the Pendulum experiment). Therefore, kinetic energy alone is often not conserved. 2

3 1.3 Collisions An important area of application of the conservation laws is the study of the collisions of various physical bodies. In many cases, it is hard to assess how exactly the colliding bodies interact with each other. However, in a closed system, the conservation laws often allow one to obtain the information about many important properties of the collision without going into the complicated details of the collision dynamics. In this lab, we will see in practice how the conservation of momentum and total energy relate various parameters (masses, velocities) of the system independently of the nature of the interaction between the colliding bodies. Assume we have two particles with masses m 1, m 2 and velocities v 1i, v 2i which collide, without any external force, resulting in velocities v 1 f, v 2 f after the collision (i and f stand for initial and final). Conservation of momentum then states that the total momentum before the collision P i is equal to the total momentum after the collision P f P i = m 1 v 1i + m 1 v 2i, P f = m 1 v 1 f + m 1 v 2 f, P i = P f (5) 1.4 Elastic and inelastic collisions There are two basic kinds of collisions, elastic and inelastic Elastic collisions In an elastic collision, two or more bodies come together, collide, and then move apart again with no loss in total kinetic energy. An example would be two identical superballs, colliding and then rebounding off each other with the same speeds they had before the collision. Given the above example conservation of kinetic energy then implies 1 2 m 1v 2 1i m 2v 2 2i = 1 2 m 1v 2 1 f m 2v 2 2 f (6) K i = K f Inelastic collision In an inelastic collision, the bodies collide and (possibly) come apart again, but now some kinetic energy is lost (converted to some other form of energy). An example would be the collision between a baseball and a bat. If the bodies collide and stick together, the collision is called completely inelastic. In this case, all of the kinetic energy relative to the center of mass of the whole system is lost in the collision (converted to other forms). Kinetic energy in inelastic collisions will be discussed in more detail in Section 1.6. In this experiment you will be dealing with a completely inelastic collision in which all kinetic energy relative to the center of mass of the system is lost, but momentum is still conserved, and a nearly elastic collision in which both momentum and kinetic energy are conserved to within a few percent. 3

4 1.5 Conservation laws for macroscopic bodies So far we were talking about the system of point-like particles. However, the conservation of the momentum is also valid for macroscopic objects. This is because the motion of any macroscopic object can be decomposed into the motion of its center of mass (which is a point in space) with a given linear momentum, and a rotation of the object around this center of mass. Then, the conservation of the linear momentum is again valid for the motion of ideal point masses located at the center of the mass of each of the objects. However, some of the linear kinetic energy can be transformed into the rotational energy of the objects, which should be accounted for in a real experiment. 1.6 Kinetic Energy in Inelastic Collisions It is possible to calculate the percentage of the kinetic energy lost in a completely inelastic collision; you will find that this percentage depends only on the masses of the carts used in the collision, if one of the carts starts from rest. After the completely inelastic collision, the carts move together, so that The initial kinetic energy is given by But, since cart 2 is initially at rest v 2i = 0 The final kinetic energy K f is given by v 1 f = v 2 f = v 3 K i = 1 2 m 1v 2 1i m 2v 2 2i. (7) K i = 1 2 m 1v 2 1i (8) From conservation of momentum, and v 2i = 0 K f = 1 2 (m 1 + m 2 )v 2 3 (9) m 1 v 1i + m 2 v 2i = (m 1 + m 2 )v 3 m 1 v 1i = (m 1 + m 2 )v 3 (10) Since the collision is inelastic, the K i is not equal to K f. You can use Eqs. (8), (9), (10) to obtain an expression for the kinetic energy fractional change D K = K f K i D K (%) = K f K i K i (11) 2 Homework Questions - Hand in before class! 1. Draw a diagram of all forces acting on each cart when they collide. 2. Equation (11) defines the the fractional kinetic energy loss in a perfectly inelastic collision. Derive an expression for D k that depends only on the masses. 3. In our experiment, can we achieve a completely elastic collision? What about a completely inelastic collision? 4

5 4. In an inelastic collision in a closed system, can some of the total momentum be lost? Some of the kinetic energy? 5. In an elastic collision in a closed system, can some of the total momentum be lost? Some of the kinetic energy? 6. If kinetic energy is lost, where does it go? Does conservation of energy apply? 7. For the inelastic collision measurements, Section 4.3, which case will change the final kinetic energy most? 8. For the elastic collision measurements, Section 4.4, write down for cart 1 (the one moving before the collision) the direction after the collision (forward or backward), the speed after the collision (faster of slower than cart 2 after the collision). 3 Experimental setup We will study the momentum and energy conservation in the following simplified situation we will look on the collision of only 2 objects, the motion of these objects will be linear and one-dimensional, so that we can choose the reference frame in such a way that only x-components of the objectsâăź momenta are nonzero; the sign of these components depends on the direction of the motion, the experimental apparatus can be set up in a way to almost completely eliminate the net external force on the system. Our objects will be two carts of different masses, with one initially at rest. The carts move on an air track, which ensures that the motion is one-dimensional and reduces the friction between the carts and the surface. The velocities of the carts can be measured with the help of the photogates, which are described in more details below. Before the beginning of the measurements, spend some time to figure out which external factors can disturb the motion of the carts on the track, and what you should do to reduce or eliminate these factors. Remember, the successful completion of this lab strongly depends on your ability to create an almost closed system. Make a few practice trials to see if you can achieve an unperturbed onedimensional collision of the carts. Adjust the level of the air track and the power of the air supply if necessary. 4 Measurements When doing measurements, make sure the carts are balanced. If you have an attachment on one side, put a dummy attachment on the other end as well. Similarly, when placing weights on the carts, distribute them evenly on both sides of the cart. If the cart is slanted, the air blowing at it from below will push it like a sail. 5

6 4.1 Data recording You will tabulate the data and calculations for this lab in two preformatted spreadsheets. Download and nscl.msu.edu/~hager/phy191/spreadsheets/elastic.xls and save to the U: drive (alternatively, the spreadsheets are on the R: drive, folder Elastic and Inelastic Collisions). The spreadsheets have some entries for fractional uncertainties (δd p (%), for example). These should be displayed in % either by multiplying the relevant fraction by 100, or by using the % formatting button in Excel. If one of the times is too long to measure (i.e. the cart is almost or completely at rest), put any very large number for the time in the spreadsheet. When you have completed the spreadsheets, save and print them Operating and reading out the photogates Photogates are used to measure the speed of the carts. The carts are equipped with fins the block the photogate; the measured time is the time it takes the fin to pass through the photogate. The speed is the calculated using this time together with the fin s length. The photogates are insensitive to the direction of the carts, so you have to pick a positive direction and remember the minus sign when the cart moves in the opposite direction. To measure the time set the photogate timer to GATE mode and the memory switch in ON position. The photogates only display one time, the first one measured after a reset. The times are added in the memory. By pushing the READ switch you can display the memory contents, which is the sum of all measurements. Example: A cart passes through the gate and the measured time is s. Afterwards, the cart bounces back and passes the gate again, which takes, let s say, s. The display will still read s, but the memory will read s. So to find that second time, you have to subtract the displayed time from the memory time. Note: if a cart keeps passing through a photogate several times, all subsequent times will be added in memory, so if a photogate measures more than two times, it is no longer possible to determine anything other than the first time, only the sum is available. So make sure your carts do not accidentally pass through a gate when they are not supposed to. 4.2 Uncertainty estimation First we test to see whether some of our assumptions are correct (the two timers give the same answer, the track is level, friction is negligible, direction does not matter). 1. Perform some trials with a single cart in which no collision occurs. Do this at different speeds, and in different directions. Record the results in appropriate tables. 2. Did you conserve momentum? Was there a bias? What time uncertainty would you deduce from these measurements? 3. What is the length of the fin and its uncertainty (justify!)? In all measurements in this lab, you may neglect the uncertainty of all masses. 4.3 Inelastic collision In the first part of the lab we make sure that after the collision the carts stick together and move with some velocity common to both masses. Thus, we have to measure the velocity of cart 1 before the collision and the common velocity of the carts 1 and 2 after the collision. For this purpose, we use 6

7 two photogates (see Fig. 1). In these measurements, use the needle and putty bumper attachments. Make sure there is enough putty; if you can hear the two attachments hit, add more putty. Otherwise the sudden impact and the resulting shaking of the carts will lead to wrong results. Position cart 2 Figure 1: Drawing of the carts on the airtrack before and after the inelastic collision. Note: the carts in these drawings are not balanced. Always make sure your carts are! close to the gate 2. You will do 3 sets of inelastic collisions consisting of 2 trials each. For each set vary the masses of the carts by adding the masses (small metal disks) to them and measure the initial and final velocities. The sets are as following: 1. cart 1: no mass disks, cart 2: 4 mass disks 2. cart 1: 2 mass disks, cart 2: 2 mass disks 3. cart 1: 2 mass disks, cart 2: no mass disks In each measurement, you need to find all the initial and final masses and velocities Calculations Now that you have collected the data, you can fill in the rest of the inelastic spreadsheet. 1. Calculate the initial and final momenta for each measurement. 2. Calculate the initial and final kinetic energies for each measurement. 3. Calculate δd P (%). 4. Calculate δd K (%). Note on uncertainties of δd: These uncertainties are a bit tricky to calculate exactly. However, we do not need them exactly, a good approximation in this case is δd P (%) δd P /P i 7

8 4.4 Elastic collisions In an almost elastic collision, the main difference from the previous part of the lab is that after the collision the carts move separately. The rubber band bumpers allow carts to collide with almost no conversion of the kinetic energy into the other forms of energy. As before, cart 2 initially stays at rest, and before the collision we have to measure only the velocity of the cart 1 v 1i (Figure 2 top). However, after the collision we have to measure the velocities of both carts, v 1 f and v 2 f (Figure 2 bottom). Thus, all in all we have to measure three times (t 1i, t 1 f, t 2 f ), while the photogate system can simultaneously measure only two of them. In order to overcome this, Figure 2: Drawing of the carts on the airtrack before and after the elastic collision. Note: the carts in these drawings are not balanced. Always make sure your carts are! you have to be fairly quick and coordinate the measurement with your partner. You have to reset the timer after the measurement of the initial time t 1i but before the collision. You have to memorize and write down the time before the reset. Perform a few practice trials to quickly remember and reset the contents of the timer before the carts collide. Measure again three sets of 2 trials each with the following masses 1. cart 1: no mass disks, cart 2: 4 mass disks 2. cart 1: 2 mass disks, cart 2: 2 mass disks 3. cart 1: 4 mass disks, cart 2: no mass disks In each measurement, you need to find all the initial and final masses and velocities. Note: Pay attention to the sign of the velocities, which depends on the direction of motion of the cart Calculations Now that you have collected the data, you can fill in the rest of the inelastic spreadsheet. 1. Calculate the initial and final momenta for each measurement. 2. Calculate the initial and final kinetic energies for each measurement. 8

9 3. Calculate δd P (%). 4. Calculate δd K (%). If the percentage change in momentum or kinetic energy before and after the collision is greater than 10%, talk to your instructor about possible mistakes and redoing some or all of the measurements. 5 Analysis and discussion Remember to add error bars whenever you plot a quantity with an uncertainty! 1. For the inelastic collisions create a graph of D P (%) vs. the number of the measurement (1 through 6). Indicate also the theoretically predicted value of D p (%). 2. Create the same plot for the elastic collisions. 3. Was the total momentum conserved in the collisions? (See Reference Guide for how to check if a measurement is compatible with a hypothesis.) 4. For the inelastic collisions create a graph of D K (%) vs. m 1 m 1 +m Perform a linear fit to the D K (%) graph above. What are the slope and intercept of the fit? 6. The above fit line intercepts the y axis and the x axis; give the (x,y) coordinates of both intercepts. What are the masses m 1 and m 2 at those points? 7. For the elastic collisions create a graph of D K (%) vs. the number of the measurement (1 through 6). Indicate also the theoretically predicted value of D K (%). 8. What is the average loss of kinetic energy in your elastic collision data? Is the deviation from the theoretically expected value statistically significant? 9