Laboratory Exercise. Projectile Motion Energy, Momentum, & Gravity

Transcription

1 Laboratory Exercise Projectile Motion Energy, Momentum, & Gravity INTRODUCTION Humans have been using objects as projectiles from they were human. From the rock (or poo) throwing chimp to the scientists at NASA s jet propulsion laboratory we have used projectiles to extend our reach. To understand projectile motion we must also understand the energies that are required to alter motion, the momentum acquired by an object in motion, and the effect gravity has on the path of projectiles. In this set of exercises you will explore these topics separately and then in combination. Exercise 1 Momentum, Energy and Collisions Objectives Observe s between two carts, testing for the conservation of momentum. Measure energy changes during different types of s. Classify s as elastic, inelastic, or completely inelastic. computer Vernier computer interface Logger Pro two Vernier Motion Detectors Materials Dynamics track Two dynamics carts Adapted from: Physics with Vernier with permission PhysicsLabProjectileMotion2016 Page 1 of 11

2 Making predictions - your hypothesis 1. Consider a head-on between two billiard balls. One is initially at rest and the other moves toward it. Sketch the actual as well as a position vs. time graph for each ball, starting with time the and ending a short time ward. Use two different colors to distinguish the balls. As you have drawn the graph, is momentum (P=mv) conserved in this? Is kinetic energy (Ek = ½ mv 2 ) conserved? Answer and explain your reasoning on your web site. Procedure 1. Measure the masses of your carts and record them in your data table. Label the carts as cart 1 and cart Set up the track so that it is horizontal. Test this by releasing a cart on the track from rest. The cart should not move. 3. Practice creating gentle s by placing cart 2 at rest in the middle of the track, and release cart 1 so it rolls toward the first cart, magnetic bumper toward magnetic - bumper. The carts should smoothly repel one another without physically touching. 4. Place a Motion Detector at each end of the track, allowing for the 0.15 m minimum distance between detector and cart. Connect the Motion Detectors to the DIG/SONIC 1 and DIG/SONIC 2 channels of the interface. If the Motion Detectors have switches, set them to Track. 5. Open the file 18 Momentum Energy Coll from the Physics with Vernier folder. 6. Click to begin taking data. Repeat the you practiced above and use the position graphs to verify that the Motion Detectors can track each cart properly throughout Adapted from: Physics with Vernier with permission PhysicsLabProjectileMotion2016 Page 2 of 11

3 the entire range of motion. You may need to adjust the position of one or both of the Motion Detectors. 7. Place the two carts at rest in the middle of the track, with their Velcro bumpers toward one another and in contact. Keep your hands clear of the carts and click. Select both sensors and click. This procedure will establish the same coordinate system for both Motion Detectors. Verify that the zeroing was successful by clicking and allowing the still-linked carts to roll slowly across the track. The graphs for each Motion Detector should be nearly the same. If not, repeat the zeroing process. Part I: Magnetic Bumpers 8. Reposition the carts so the magnetic bumpers are facing one another. Click to begin taking data and repeat the you practiced in Step 3. Make sure you keep your hands out of the way of the Motion Detectors you push the cart. 9. From the velocity graphs you can determine an average velocity and the for each cart. To measure the average velocity during a time interval, drag the cursor across the interval. Click the Statistics button to read the average value. Measure the average velocity for each cart, and, and enter the four values in the data table. Delete the statistics box. 10. Repeat Step 9 as a second run with the magnetic bumpers, recording the velocities in the data table. Part II: Velcro Bumpers 11. Change the by turning the carts so the Velcro bumpers face one another. The carts should stick together. Practice making the new, again starting with cart 2 at rest. 12. Click to begin taking data and repeat the new. Using the procedure in Step 9, measure and record the cart velocities in your data table. 13. Repeat the previous step as a second run with the Velcro bumpers. Part III: Velcro to Magnetic Bumpers 14. Face the Velcro bumper on one cart to the magnetic bumper on the other. The carts will not stick, but they will not smoothly bounce apart either. Practice this, again starting with cart 2 at rest. 15. Click to begin data collection and repeat the new. Using the procedure in Step 9, measure and record the cart velocities in your data table. 16. Repeat the previous step as a second run with the Velcro to magnetic bumpers. Adapted from: Physics with Vernier with permission PhysicsLabProjectileMotion2016 Page 3 of 11

4 DATA TABLE Mass of cart 1 (kg) Mass of cart 2 (kg) Run number Velocity of cart 1 Velocity of cart 2 Velocity of cart 1 Velocity of cart 2 (m/s) (m/s) (m/s) (m/s) Run number Momentum of cart 1 Momentum of cart 2 Momentum of cart 1 Momentum of cart 2 Total momentum Total momentum (kg m/s) (kg m/s) (kg m/s) (kg m/s) (kg m/s) (kg m/s) Ratio of total momentum / Run number KE of cart 1 KE of cart 2 KE of cart 1 KE of cart 2 Total KE Total KE Ratio of total KE / Adapted from: Physics with Vernier with permission PhysicsLabProjectileMotion2016 Page 4 of 11

5 Analysis Show all calculations on your web site. Include data tables on your web site. 1. Determine the momentum (mv) of each cart the, the, and the total momentum and the. Calculate the ratio of the total momentum the to the total momentum the. 2. Determine the kinetic energy (½ mv 2 ) for each cart and the. Calculate the ratio of the total kinetic energy the to the total kinetic energy the. 3. If the total momentum for a system is the same and the, we say that momentum is conserved. If momentum were conserved, what would be the ratio of the total momentum the to the total momentum the? 4. If the total kinetic energy for a system is the same and the, we say that kinetic energy is conserved. If kinetic were conserved, what would be the ratio of the total kinetic energy the to the total kinetic energy the? 5. Inspect the momentum ratios. Even if momentum is conserved for a given, the measured values may not be exactly the same and due to measurement uncertainty. The ratio should be close to one, however. Is momentum conserved in your s? What is your evidence? 6. Repeat the preceding question for the case of kinetic energy. Is kinetic energy conserved in the magnetic bumper s? How about the Velcro s? Is kinetic energy consumed in the third type of studies? Classify the three types as elastic, inelastic, or completely inelastic. a. Magnetic to magnetic b. Velcro to Velcro c. Magnet to Velcro Exercise 2 Momentum - Projectiles INTRODUCTION In this assignment you will be considering the catch throw process as a slow between a juggler s hand and a ball. In particular, we would like you to verify the impulse momentum theorem in a typical catch-throw process and the value of the Earth s gravitational acceleration constant. You ll be tracking the motion of the center ball that the juggler is pointing at in the photo. Each ball has a mass of 105 g. Completing this exercise will require you to (1) carefully obtain y vs. t data for the center ball for the entire video clip, which includes a catch-throw segment and another segment showing the ball s free rise and fall; (2) choose an analytic function that you think ought to fit the free rise and fall data; and (3) perform a curve fit on your data. In addition, you will be exploring the change in the velocity and the acceleration the ball undergoes during the catch-throw segment. This should allow you to verify the Impulse-Momentum Theorem. Watch the juggling video <JugglerAll.mov> through once doing any analysis in order to make your predictions. Adapted from: Physics with Vernier with permission PhysicsLabProjectileMotion2016 Page 5 of 11

6 Making predictions - your hypothesis Draw the path of a single ball as it is rising and falling freely in the air as it is juggled. 1. Think about a typical toss the juggler is making. What kind of analytic function might be used to fit the graph of y vs. t while the ball is rising and falling freely in the air? Explain the reasons for your answer. Do you predict a constant acceleration? If so, can you guess its magnitude and direction? 2. What kind of acceleration do you think the ball experiences as the juggler makes contact with the ball throughout the catch and throw process? Do you think it will be constant or vary? What direction is it, upward or downward. Explain the reasons for your predictions in terms of what you see in the movie. 3. What is the impulse-momentum theorem ( try;< Impulse-Connection>)? In other words, if you know the momentum change during a ball s with the juggler s hand, how should the impulse the ball experiences be related to the momentum change? Include the formulation for the impulse momentum theorem on your web site. Define what force should be used to determine the impulse the gravitational force, the juggler s hand force, or the net force? Procedure You will be working with a short video clip entitled <JugglerClip4.mov>. It shows three ball juggling with only two distinct segments on it (1) a catch-throw segment (frames 3 13); and (2) a free rise and fall (Frames 15 33). Do a careful curve fit on each of the segments of the video clip with special attention to the catch-throw shown in segment Collect vertical data for both segments: Open the Logger Pro file <Juggler.cmbl>. The video clip has been scaled. Skip the title frame (frame 0) by advancing the movie by one frame and then record y vs. t data in meters for the remaining frames. First, click the Add Point tool ( ) and select the same point on top of the middle ball for frames 1 (where t = s) through 33 (where t = s). Logger Pro will then record the vertical position of the ball as a function of time. Sketch your data points on the graph to the right. Work carefully. If you mess up, start over by using the Clear All Data feature in the Data menu. Hint: The grey band corresponds to the catch-throw segment when the juggler is touching the ball. Adapted from: Physics with Vernier with permission PhysicsLabProjectileMotion2016 Page 6 of 11

7 2. Do a curve fit for the y vs. t data in the free fall segment: Select the free fall y vs. t data (frames 15 through 33). Call up the Curve Fit feature in the Analyze menu and choose the type of equation you think will best match the data (Proportional, Linear, Quadratic, or Cubic, and so on.). Write down the equation that you obtained from your Curve Fit and list the value of the coefficient A with appropriate units. Also report on the time offset checking the time offset in the lower left corner of the curve fit dialog box. Note: Recall that t 1 and t 0 are alternate notations for the initial time or Time Offset for the toss, which should be given by t 0 = s. 3. Does the function you chose for your curve fit in segment 2 seem appropriate? In other words, does the fit line match your data well? Yes or no. Explain your basis for deciding whether or not the fit is satisfactory. You may want to look at the reported RMSE. * Note: If the curve fit isn t satisfactory perhaps you are trying to use the wrong analytic function and should repeat the analysis in step 2 as needed using another function. 4. Use the value of A to determine the ball s y-component of acceleration during the free fall segment. Is it changing or constant? How does it compare to what you predicted in Part 1(a). Cite evidence for your answer. Hint: You may either refer to the y vs. t graph or the a y vs. t graph in your answer. 5. Find the catch and throw velocities: Use the data table to determine the instantaneous vertical velocity ** of the ball just as it s being caught (frame 3 at t = s). Also determine the instantaneous vertical velocity of the ball just the juggler releases it (frame 13 at t = s). List initial and final velocity components with appropriate units and signs to 4-significant figures. 6. Find the momentum change for the catch and throw process: Recall that the ball s mass is 105 g. Calculate the vertical momentum of the ball when it first falls into the juggler s hand (as in frame 3). Also find the vertical momentum of the ball when it is just about to leave the juggler s hand (as in frame 13). What are the magnitude and direction of the momentum change, p y, in the vertical direction that the ball undergoes during this time period? Show your calculations. Beware: Momentum is a vector quantity! Do not fall into the trap of simply subtracting the magnitude of one quantity from the magnitude of the other. 7. Determine the catch-throw impulse: Using the definition of net force as that which causes acceleration, and the fact that net y F ma, you can use Logger Pro to find the integral (as an area y under the curve) in the a y vs. t graph. You should be able to determine the impulse between frames 3 and 13. Explain what you did and show your results. 8. Determine the juggler s force: If the maximum net force on the ball is 2.85 N, using the information provided, can you calculate the maximum vertical force the juggler exerts on the ball during the throw? Draw a free-body diagram showing the magnitudes and directions of the forces on the ball * RMSE stands for Root Mean Square Error. It is a measure of how far away, on average, the data points are from the fitted curve. RMSE is in the units of the y-axis, in this case is meters. ** The numerical derivative we chose here is the weighted average of the slope of 5 points around each point. Adapted from: Physics with Vernier with permission PhysicsLabProjectileMotion2016 Page 7 of 11

8 as scaled vectors. In your solution be sure to include both an equation for the net forces on the ball and the magnitude and direction of the gravitational force on the ball. Analysis 1. Does the Impulse-Momentum Theorem hold for the catch-throw process? Explain why or why not. Hint: In order to say yes, your momentum change and impulse values need to be within at least 10% of each other. 2. What force is relevant in the impulse calculation the gravitational force, the juggler s hand force, or the net force? 3. How did the acceleration you determined during the catch-throw process compare to prediction question 2. Describe what the a y vs. t graph tells you about the motion of the ball when it is in contact with the juggler s hand? Exercise 3 Net Work Kinetic Energy Theorem INTRODUCTION The Net Work Kinetic Energy Theorem can be derived from Newton s Second Law. If the velocity of an object is measured, this theorem can be used to calculate the net work on an object when the force on it is not constant. In this assignment you will view a movie of a lowfriction cart with a force sensor mounted on it. The cart and force sensor are being pulled with a variable force. You can use the position and force data recorded by sensors to verify the theorem. Motion Detector Force Sensor Making predictions - your hypothesis 1. State the Net Work Kinetic Energy Theorem using both equations and words. 2. Double click on the file <NetW_KE.mov> to open the movie in QuickTime Player. Play the movie or advance it frame-by-frame using the right arrow key ( ) on your keyboard. If friction is present and the cart is being pulled, there are four forces on the cart. Make a drawing of the system being studied and provide vector labels for the horizontal and vertical forces present. Adapted from: Physics with Vernier with permission PhysicsLabProjectileMotion2016 Page 8 of 11

9 3. If the frictional forces experienced by the cart are negligible, explain why the net force on the cart at each moment is the same as the pulling force measured by the force sensor at each moment. 4. Suppose a cart is pulled from left to right along the positive x-axis from position xa to position xb by unsteady horizontal forces applied to a force probe attached to the cart. Let s denote the x-component of the force as F pull (x) where the x indicates that the force is a function of position. What equation can be used to calculate the work, W pull, done by this force? 5. Observe the cart motion and plan your analysis: Open the Logger Pro experiment file <NetW-KE.cmbl>. This file opens the movie <NetW_KE.mov>. The movie has been synchronized with data recorded by the motion detector and the force sensor. Since the mass of the cart and its affixed force sensor is known to be kg, the Logger Pro new column feature has already been set to calculate vx (the x-component of the cart s velocity) and K (the kinetic energy of the cart) in each frame. a. Watch the events in slow motion by clicking on the Start button in the Replay box. b. Near the left end of the Toolbar switch from Page 1: Data to Page 2: Analysis. Then examine the F pull vs. x graph and explain how you can use this graph and the Integral tool in the Logger Pro Analyze menu to determine the net work done on the cart as it moves from a position xa to a position xb. 6. Still on the analysis page (denoted as "2: Analysis") of the Logger Pro file, indicate the range of positions for which the net work on the cart is positive. Then indicate the range of positions for which the net work on the cart is negative. Round your answers to two decimal places. 7. Note that the information in the Logger Pro data table includes calculated columns for the velocity (vx) and kinetic energy (K), corresponding to each value of x. Explain how the value of K at each location can calculated from the mass, m, of the cart and from the velocity of the cart corresponding to a given value of x. 8. What equation can be used to calculate the kinetic energy change ( K a b ) as the cart moves from positions xa to xb? You can denote the corresponding horizontal velocity components va and vb. Procedure Does the Work Energy Theorem hold from one arbitrary position to another? Let s start your verification of the Net Work Kinetic Energy Theorem by assuming that friction forces are negligible and see what happens. Continue to use the Logger Pro experiment file <NetW- KE.cmbl>. Start by considering what happens from xa = m to xb =0.607 m. 1. Find the value the cart s kinetic energy change in Joules for the chosen range of motion. Perform your calculations using three decimal places. Hint: You can use the fact that the new column feature has already been used to calculate the kinetic energy corresponding to each value of x. 2. Next use the definition of work you reported in your answer to 1(d) to find the value of the work on the cart in Joules done by the pulling force between positions xa and xb. Also, use the technique you described in answer 1(e). Show your equations and calculations. Note: Once again bring up page 2: Analysis and select all the data from x = m to x = m. 3. Compare your answers, and then calculate the percentage difference between the net work and the change in kinetic energy. Adapted from: Physics with Vernier with permission PhysicsLabProjectileMotion2016 Page 9 of 11

10 4. You should have found that the Net Work Kinetic Energy theorem holds for our arbitrarily chosen range of motion to within about 5%, but that the work done by the pulling force was greater than the change in kinetic energy that it caused. Note: You should check your calculations if you didn t get this result. Explain why the presence of a small amount of friction in the cart s wheel bearings might explain this discrepancy. Analysis Net Work-Kinetic Energy with Friction: You should have concluded that the Net Work-Kinetic Energy Theorem holds reasonably well for the relatively friction-free situation of the cart on the track. If we assume some friction is present, then the net work-kinetic energy theorem should still hold! For example, let s do a thought experiment involving a special case. Suppose the cart is replaced by a block that is pulled at a constant velocity with a constant force in the presence of sliding friction. 1. What is the net force on the block while it is moving at a constant velocity from point a to point b? Explain how you arrived at your answer. 2. What is the net work done on the block while it is moving from point a to point b? Explain how you arrived at your answer. 3. What is the change in the kinetic energy of the block as it moves from point a to point b at a constant velocity? Explain how you arrived at your answer. 4. Does the Net Work-Kinetic Energy Theorem hold for this situation where friction is present? Exercise 4 Projectiles INTRODUCTION The projectile you will be studying for the last exercise in this series is an impulse rocket. NASA uses thrust rockets which provide a force throughout the early flight stages of the rocket. An impulse rocket provides a short initial force (impulse). The impulse rocket is an easier model to use when studying trajectory since the mass of the projectile is constant and only the initial impulse force need be considered. Materials Computer Air Burst Rocket Vernier computer interface Logger Pro Vernier Probes as required Other Instrumentation as required Making predictions - your hypothesis Derive the range formula for an impulse style projectile. Show all of your work. Your final equation must be in the form dx =. The initial equations you will use are the; Adapted from: Physics with Vernier with permission PhysicsLabProjectileMotion2016 Page 10 of 11

11 distance: dx = v*t acceleration: a = v/t falling bodies: d = ½ gt 2 Check your derived formula using the trajectory simulator found at; Procedure 1. Determine which variables you need to measure in order to predict the landing point of the AirBurst rocket when launched at a known angle. 2. Determine the instrumentation required to measure values for those variables given a constant initial impulse. 3. Set up instrumentation and fire rocket vertically. You may do this as many times as you feel are necessary. 4. Calculate the landing point of the rocket for a given angle of firing. 5. Fire the rocket to test your calculations. Analysis 1. Calculate the percent error between your calculated landing and the actual landing site. 2. What variables were difficult to control or measure? 3. Calculate the Ek of the rocket at point of launch. 4. Calculate the Ep at the rockets highest point of flight. 5. Calculate the momentum of the rocket at point of launch and at the highest point of flight? Was momentum conserved throughout the flight of the rocket? Explain. 6. Make free body diagrams showing forces, velocities, acceleration, and energy of the system. 7. Using a watt meter measure the electrical energy used by the compressor to launch the rocket. (1 watt = 1 joule/second). Calculate the efficiency of the Compressor/AirBurst rocket system. Adapted from: Physics with Vernier with permission PhysicsLabProjectileMotion2016 Page 11 of 11