Objectives Conservation of Energy and Momentum You will test the extent to which conservation of momentum and conservation of energy apply to real-world elastic and inelastic collisions. Equipment air track air track carts Various masses Laboratory balance Level Meter stick EITHER Stopwatch Masking Tape OR (if available) Photogates LabPro Data Acquisition Interface LabPro Digital Adapter for Photogates Introduction During a simple collision, two bodies, initially free of any external forces, interact with each other so as to change the motion of both. Under the right conditions reasonably accurate predictions of the final velocities and directions can be made using the principles of conservation of momentum and energy alone. In this experiment you will use these principles to analyze collisions of the simplest kind, those involving two bodies confined to straight-line (one-dimensional) motion. The basic principles, however, are essential to the understanding of a wide variety of phenomena ranging from the interaction of subatomic particles to rocket motion. The collisions you will analyze are to be considered to be perfectly elastic or inelastic (sticking collision). You will push carts on an air track. Since air does supply a small amount of friction, some kinetic energy is lost. But the loss is usually minimal. With proper experimental technique, the discrepancy between theoretically expected and experimental values should be small. One of the requirements of this lab is to quantify the extent to which the theory (which assumes conservation of energy and momentum and no friction) holds. Note that this lab can be completed using either a simple stopwatch for timing the cart motion or a more sophisticated timing method involving two photogates and a computer data acquisition
system (LabPro). The notation used in the following discussion is as follows: before collision, bodies of masses m and m have speeds v and v respectively. After collision, the speeds are v and v. Conservation of Momentum Requires m v + m v = m v + m v Conservation of Energy Requires m v + mv = mv + mv These relationships can be combined to give expressions that are useful in analyzing the physics of the collision experiments in this exercise. We will examine both elastic and inelastic collisions. The approach in both experiments is to collide a cart possessing a known speed and mass (v, m ) with a stationary cart of known mass (v =0, m ). After the carts collide, the final speeds of the carts will depend on whether or not the collision is elastic or inelastic and whether or energy and/or momentum are conserved. If energy and momentum are conserved, we can obtain expressions for the final speeds of the two carts from the above conservation equations. Solving the equations simultaneously for the final velocities after an elastic collision where the second cart is at rest before the collision (v = 0) yields v m m For the inelastic case (v = v ) and v = 0: Procedure = v v = v m + m m + m v m = v = v m + m. With the air blower turned on, check the level of the track by placing a cart on it. Adjust the leveling screw at one end of the track to raise or lower the track so that the cart does not move. Some tracks have a slight curvature so you may never get the cart to remain perfectly stationary at every point along the track. The level of the track is important because you do not want to complicate the system by adding a component of gravitational acceleration. A diagram of the track and carts is shown below (with the optional photogates). m
D J C = J A D J C = J A E? A J. A? A + = H J + = H J. If you are using the optional photogates and LabPro. (If you are using the stopwatch method, you may skip this part):.. attach a picket fence on the top of each cart. The relative positions of the photogates is unique to each part of the exercise. Refer to the descriptions of the individual parts of the exercise... Adjust the height of the photogates so the.5 cm bar on the picket fence blocks the beam. See Figure..3. Plug the photogates into the # and # Dig/Sonic jacks of the Lab Pro..4. Open the computer program by clicking on the Logger Pro icon..5. Select Open in the File menu. Then open the folder Probes & Sensors by doubleclicking on that folder. Open the folder Photogates. Then double-click on Collision Timer. You should now see blank Time Measurements and Velocity Measurements graphs as well as a blank data table..6. To tell the computer what type of picket fence we are using choose Show Sensors from the Experiment menu. Then click on Dig/Sonic (or ). Choose Set Distance or Length. Enter 0.05 for each sensor..7. Set formula for v and v :.8. At the top of the table, double-click on v. Set equation as 0.05 / "Gate Time ". Double-click on v. Set equation as 0.05 / "Gate Time ". Data collection For each part of the experiment, use the following method for data collection. Perform at least five trials for each part of the experiments. i) If using the photogates and LabPro: a. Position the two photogates approximately 40 cm apart in the middle portion of the track. b. Initially, the first cart should be at the far left of the track and the second cart should be just to the left of the right-hand photogate. c. Practice rolling the first cart (at the left end of the track) at a medium pace to towards the second cart. Use either the stopwatch or LabPro to measure the velocity of the carts. When
you are confident of your technique, make enough runs so that you have enough data to plot a decent graph of v ' vs. v, i.e. the v 's should be as variable as possible. d. When you are ready to collect data, click on Collect. e. Give the first cart an initial velocity. After the gate flags pass the photogates, click on Stop. f. The monitor will display a table of times associated with the carts passing the photogates. You need to record the velocities v and v in your lab notebook. v corresponds to the photogate plugged into the # Lab Pro jack. v corresponds to the photogate plugged into the # Lab Pro jack. ii) If using the stopwatch method: a. With tape or some other removable method, mark four positions along the air-track. Place the first two marks at 30 cm to the left and right of the center of the track. Place the second two marks at 60 cm to the left and right of the center of the track. These are marks against which you will time the carts to measure their velocities. b. Initially, the first cart should be at the far right of the track and the second cart should be in approximately the center of the track. c. Give the first cart an initial velocity and have your lab-partner use the stop-watch to time how long it takes to pass between the two right hand marks. This is the velocity v. d. After the first cart hits the second (previously stationary) cart, measure the second cart s velocity by timing its passage between the two left hand marks. This is the velocity v. e. When you are confident of your technique, you are ready to start taking data. During the data taking, make enough runs so that you have data to plot a decent graph of v vs. v, i.e. the v 's should be as variable as possible. Record your values in a table similar to Table shown below. For each part of the exercise you will need at least good run for 5 different v s to draw a decent graph. Example of Table. For elastic and inelastic collisions (Make tables like this in your lab notebook.) Trial # m (kg) m (kg) v (m/s) v (m/s) 3 4 5 Part : Elastic collisions with m = m Arrange the two carts so that they repel each other If the carts do not repel each other ask the instructor for assistance. Label the carts and and find each of their masses using a balance. Make a graph of v vs. v.
Using the equations developed in the lab preparation, find the theoretical slope of the line and record it in your notebook. (Note: In this experiment the equation of the line would take the form, v = av, where a is the slope.) Find the extent to which this relationship holds by comparing the experimental value of the slope with the theoretical value. Part : Elastic collisions with m m Repeat the above experiment with different masses. Compare slope with experimental value. Part 3: Inelastic collisions with m = m This part is performed almost exactly as the first part of the experiment except that when the carts collide they stick together. There are two parts for the inelastic collision experiment as well. In the first part the masses of the cart are nearly equal. For the second part the masses are different. Be sure your two carts are oriented so that they will stick together when they collide. Repeat the sequence of runs as in Parts and. Make a graph of v vs. v. Calculate the theoretical slope of this line. Check to see if this relationship holds by comparing the experimental value of the slope with the expected value. Inelastic collisions: m m Repeat the above experiment with different masses. Compare the slope with the experimental value. Questions. The slope a did most likely not agree perfectly with the theoretical value in any of the four cases. In any of the four cases was the deviation of the actual value farther than two error bars away from the theoretical value? If so, to what is the discrepancy due?. Does your experiment support the application of the laws of conservation of energy and momentum in everyday contexts? Explain your answer carefully!