Visual Physics 218 Projectile Motion [Lab 2]

Transcription

1 In this experiment, you will be using your video equipment to evaluate two-dimensional motion. It will be necessary to plot the data in an xy-coordinate system and separate the data into x and y components. These components will be used to calculate parameters describing projectile motion, including angle, initial velocity, range, and time. You should be able to identify the random and systematic errors associated with your analysis and the cumulative effect of these errors. In addition, this lab will require analyzing video clips of actual football game footage to determine these same parameters for the projectile motion of a football kick-off. Pre-Lab Read Chapter 3 from Young & Freedman s University Physics. Study two-dimensional motion and learn how to identify and plot displacement, velocity, and acceleration on a graph, such as those graphs you created in Lab 0 and Lab 1. You need to understand how graphs of displacement, velocity, and acceleration relate to each other in two dimensions and how to incorporate different types of error into your analysis. Questions to be answered in your lab journal: 1. Search the internet and write down, in your own words, the definition of a random error and a systematic error. 2. In this experiment, you will be using the marble launcher to project the marble at different angles and speeds. List what you think the sources of random errors will be in this experiment. Also list what you think will be the sources of systematic errors in this experiment. 3. In Lab 1, Experiment 1 you were asked to plot a trendline for vertical motion. You will also be plotting trendlines in this experiment as well. What type of trendline (i.e., linear (polynomial 1storder), quadratic (polynomial 2 nd -order), exponential, logarithmic, etc.) should you use for horizontal displacement and horizontal velocity, assuming air resistance is not negligible? What type of trendline should you use for vertical displacement and vertical velocity? NOTE: During the experiments in which a marble launcher is used, all students in the lab room are required to wear safety glasses at all times. Failure to wear safety glasses will mean expulsion from the lab and a grade of zero (with no makeup available). Safety glasses are available at the front TA desk. Experiment 1 In experiment 1, you will project marbles at three different angles using the same power setting on the marble launcher. Try to get as much of the marble s parabolic trajectory in the camera s field of view as possible, though it is not necessary to get the marble s entire trajectory to calculate your results. The more of the flight parabola of the marble you can get into the movie, the more accurate your results will be that you will obtain off the movies. Even with only half of the marbles trajectory in the movie, the results you obtain inserting trendlines will be reasonably similar using different parameters. Make sure you note which angle goes with each video, in case the angle of the launcher arm cannot be determined by viewing the video. 1

2 Procedure: 1. Place the marble launcher and the white poster board on the end of the lab table as in Lab 0 and Lab 1, Experiment 3. Mount the camera on the rod and stand and place it on the lab table top. The camera should be aimed directly at the plane of the marble s trajectory. One of the lab partners should hold a meter stick in front of the poster board so that the tick marks are clearly visible in the camera image. 2. The marble launcher has different angle settings from 0 to 90. See Figure 1. This angle will be the angle θ of the initial velocity v o. The angle can be changed by un-tightening the large knob on the rear of the launcher arm of the marble launcher and rotating the launcher arm to the appropriate angle, then retightening the knob. First, set the angle to 60. Now set the power setting to the second lowest setting (#2 mark on the launcher). Perform a few test shots to make sure the camera and marble launcher are positioned properly. If possible, try to get as much of the flight parabola of the marble in the camera field of view, though this is not essential. It is imperative though that you get at least the first half and apex of the parabola. This will ensure accurate trendline equations. Capture the trajectory to video. 3. Set the angle to 75 and repeat step Finally, set the angle to 85 and repeat step After all the movies for Lab 2 have been recorded for all experiments, remove the Figure 1 memory card from the camera and insert the card into the computer. Copy the movies to your section folder under the C:\218Labs folder. Insert one of the three movies for Experiment 1 into a new LoggerPro sheet. 6. Use the Set Scale function and select two tick-marks on the meter stick as your distance. Next, use the Add Point function and select a point on the marble in every frame such that you have data points for the marble s entire trajectory that is in the movie. Now Set Origin and drag the origin of your coordinate axes to the point where the marble is resting in the launcher before the marble is launched. This should also be your 2

3 first point selected. Your second point selected should be the first frame where the marble is in flight. Your LoggerPro sheet should look similar to Figure 2. Export your data to a text file and save the text file in your section folder. Remember to either save this text file to a USB flash drive or this text file to yourself when you are finished with the lab. 7. Repeat step 6. for the remaining two movies. Questions to be answered in your Technical Memo: 1. Open up the text file in an Excel sheet. You are going to find the horizontal and vertical components of the initial velocity and acceleration by two different methods. First, plot a chart of Vx vs. t. Now add a trendline and select Linear and Display Equation on chart. The equation displayed on the chart will be in the general form y = bx + c, where b and c are the constant coefficients. For constant acceleration, recall the kinematical equation of motion v x = a x t + v ox, where a is the acceleration. Now compare these two equations and you see b = a x and c = v ox. From this you can determine v ox and the horizontal acceleration a x. The horizontal acceleration a x 0 since there is a small amount of air resistance. Next, plot a chart of Vy vs. t, and find v oy and a y using the same procedure. Do this for all three movies and Copy and Paste your charts into your TM. Figure 3 shows two example charts. Figure 2 2. Now you are going to find these same parameters using the position versus time graphs. Plot a chart of Y vs. t. Add a trendline and select Polynomial: Order 2 and Display Equation on chart. The equation displayed on the chart is in the general form y = ex 2 + fx + h, where e, f, and h are the constant coefficients. Recall for constant acceleration the vertical kinematical equation of motion y = ½a y t 2 + v oy t + y o. Compare these two equations and you see e = ½a y, f = v oy, and 3

4 h = y o. Since this is vertical motion, you know that a y = g, or g = 2e, where e is the constant coefficient of the equation displayed on the chart and g is the acceleration due to gravity. You have also determined v oy as well. Do this for all three movies and Copy and Paste your charts into your TM. Figure 4 shows an example chart. 3. Plot a chart in Excel of the horizontal position versus time, or X vs. t. Add a trendline and select Polynomial: Order 2 and Display Equation on chart. The equation displayed on the chart will be in the general form y = jx 2 + kx + n, where j, k, and n are the constant coefficients. Recall for constant acceleration the horizontal equation of motion x = ½a x t 2 + v ox t + x o. Compare these two equations and see j = ½a x, k = v ox, and n = x o. From this you have determined a x and v ox. Do this for all three movies and Copy and Paste your charts into your TM. Figure 4 shows an example chart. 4. In your TM, compare your different values of v ox, v oy, a x, and a y using the two different charting methods described above. Calculate the average v ox, v oy, a x, and a y. Figure 3 4

5 Figure 4 5. Now calculate the range and hang time of the marble. Do not obtain these values directly off the movies, but calculate them using the averages of the two values of v ox, v oy, a x, and a y you just computed. To do this, you will need the angle θ of each marble trajectory. To find the range R, use the range formula: R = v o 2 sin 2θ g Remember, this formula was derived using the condition y = y o, or Δy = 0. Thus, it is only applicable for these specific conditions. Also recall that v o in this formula is the total initial velocity calculated using the horizontal and vertical velocity components v ox and v oy : v oy v o θ v ox v o = v ox 2 + v oy 2 To calculate the hang time, recall the horizontal equation of motion x = ½a x t 2 + v ox t + x o. You just calculated the range, or total horizontal distance traveled, thus x = R. This gives the horizontal equation of motion R = ½a x t 2 + v ox t + x o. To find the hang time, solve this quadratic equation for t and substitute in your average values of a x and v ox, as well as x o, and compute t. This time t will correspond to the point of maximum horizontal distance, or the time t at R. List the values of the range and hang time you compute for all three marble shots at different angles in your TM. 5

6 6 Visual Physics 218 Projectile Motion [Lab 2] Experiment 2 In Experiment 2, you will project marbles at three different power settings using the same angle on the marble launcher. Try to get as much of the marble s parabolic trajectory in the camera s field of view as possible, though it is not necessary to get the marble s entire trajectory to calculate your results. The more of the flight parabola of the marble you can get into the movie, the more accurate your results will be that you will obtain off the movies. Even with only half of the marbles trajectory in the movie, the results you obtain inserting trendlines will be reasonably similar using different parameters. Make sure you note which power setting goes with each video, in case the power setting of the launcher arm cannot be determined by viewing the video. Procedure: 1. Set the marble launcher to the angle of 75. This angle will be the angle θ of the initial velocity v o. Leave the marble launcher at this angle θ for all of Experiment 2. The marble launcher has different power settings. Set the power setting to the third lowest setting (#3 mark on the launcher). Perform a few test shots to make sure the camera and marble launcher are positioned properly. If possible, try to get as much of the flight parabola of the marble in the camera field of view, though this is not essential. It is imperative though that you get at least the first half and apex of the parabola. This will ensure accurate trendline equations. Capture the trajectory to video. 2. Set the power setting to the second lowest setting (# 2 mark) and repeat step Finally, set the power setting to the lowest setting (# 1 mark) and repeat step After all the movies for Lab 2 have been recorded for all experiments, remove the memory card from the camera and insert the card into the computer. Copy the movies to your section folder under the C:\218Labs folder. Insert one of the three movies for Experiment 2 into a new LoggerPro sheet. 5. Use the Set Scale function and select two tick-marks on the meter stick as your distance. Next, use the Add Point function and select a point on the marble in every frame such that you have data points for the marble s entire trajectory that is in the movie. Now Set Origin and drag the origin of your coordinate axes to the point where the marble is resting in the launcher before the marble is launched. This should also be your first point selected. Your second point selected should be the first frame where the marble is in flight. Your LoggerPro sheet should look similar to Figure 2. Export your data to a text file and save the text file in your section folder. Remember to either save this text file to a USB flash drive or this text file to yourself when you are finished with the lab. 6. Repeat step 5. for the remaining two movies. Questions to be answered in your Technical Memo: 1. Open up the text file in an Excel sheet. You are going to find the horizontal and vertical components of the initial velocity and acceleration by two different methods. First, plot a chart of Vx vs. t. Now add a trendline and select Linear and Display Equation on chart. The equation displayed on the chart will be in the general form y = bx +c, where b and c are the constant

7 coefficients. For constant acceleration, recall the kinematical equation of motion v x = a x t + v ox, where a is the acceleration. Now compare these two equations and you see b = a x and c = v ox. From this you can determine v ox and the horizontal acceleration a x. The horizontal acceleration a x 0 since there is a small amount of air resistance. Next, plot a chart of Vy vs. t, and find v oy and a y using the same procedure. Do this for all three movies and Copy and Paste your charts into your TM. Figure 3 shows two example charts. 2. Now you are going to find these same parameters using the position versus time graphs. Plot a chart of Y vs. t. Add a trendline and select Polynomial: Order 2 and Display Equation on chart. The equation displayed on the chart is in the general form y = ex 2 + fx + h, where e, f, and h are the constant coefficients. Recall for constant acceleration the vertical kinematical equation of motion y = ½a y t 2 + v oy t + y o. Compare these two equations and you see e = ½a y, f = v oy, and h = y o. Since this is vertical motion, you know that a y = g, or g = 2e, where e is the constant coefficient of the equation displayed on the chart and g is the acceleration due to gravity. You have also determined v oy as well. Do this for all three movies and Copy and Paste your charts into your TM. Figure 4 shows an example chart. 3. Plot a chart in Excel of the horizontal position versus time, or X vs. t. Add a trendline and select Polynomial: Order 2 and Display Equation on chart. The equation displayed on the chart will be in the general form y = jx 2 + kx + n, where j, k, and n are the constant coefficients. Recall for constant acceleration the horizontal equation of motion x = ½a x t 2 + v ox t + x o. Compare these two equations and see j = ½a x, k = v ox, and n = x o. From this you have determined a x and v ox. Do this for all three movies and Copy and Paste your charts into your TM. Figure 4 shows an example chart. 4. In your TM, compare your different values of v ox, v oy, a x, and a y using the two different charting methods described above. Calculate the average v ox, v oy, a x, and a y. 5. Now calculate the range and hang time of the marble. Do not obtain these values directly off the movies, but calculate them using the averages of the two values of v ox, v oy, a x, and a y you just computed. To do this, you will need the angle θ of the marble trajectory. To find the range R, use the range formula: R = v o 2 sin 2θ g Remember, this formula was derived using the condition y = y o, or Δy = 0. Thus, it is only applicable for these specific conditions. Also recall that v o in this formula is the total initial velocity calculated using the horizontal and vertical velocity components v ox and v oy : v oy v o θ v ox v o = v ox 2 + v oy 2 To calculate the hang time, recall the horizontal equation of motion x = ½a x t 2 + v ox t + x o. You just calculated the range, or total horizontal distance traveled, thus x = R. This gives the horizontal 7

8 equation of motion R = ½a x t 2 + v ox t + x o. To find the hang time, solve this quadratic equation for t and substitute in your average values of a x and v ox, as well as x o, and compute t. This time t will correspond to the point of maximum horizontal distance, or the time t at R. List the values of the range and hang time you compute for all three marble shots at different power settings in your TM. 8 Experiment 3 In this experiment, you will be analyzing actual game video of a football game and plotting the trajectory of a football during a kick-off. You do not need to see the entire flight of the football during the kickoff, just enough to plot an accurate trendline of the parabola. There is no equipment to set up for this experiment. The movies are already saved on your computers. Procedure: 1. The three kick-off movies are in the C:\218Labs\Kickoff folder. Start LoggerPro and insert one of the kick-off movies from this folder into a new LoggerPro sheet. 2. Use the Set Scale function and select two yard line markers on the football field as your distance. These yard line markers are ten yards apart. If you want the data you will calculate to be in meters, then you must convert yards to meters and enter the scale in meters. Next, use the Add Point function and select a point on the football in every frame such that you have data points for the football s entire trajectory that is in the movie. Now Set Origin and drag the origin of your coordinate axes to the point where the football is resting on the tee before the football is kicked. This should also be your first point selected. Your second point selected should be the first frame where the football is in flight. Export your data to a text file and save the text file in your section folder. Remember to either save this text file to a USB flash drive or this text file to yourself when you are finished with the lab. 3. Repeat steps 1. and 2. for the remaining two kick-off movies. Questions to be answered in your Technical Memo: 1. Open up the text file in an Excel sheet. You are going to find the horizontal and vertical components of the initial velocity and acceleration of the football by two different methods. First, plot a chart of Vx vs. t. Now add a trendline and select Linear and Display Equation on chart. The equation displayed on the chart will be in the general form y = bx +c, where b and c are the constant coefficients. For constant acceleration, recall the kinematical equation of motion v x = a x t + v ox, where a is the acceleration. Now compare these two equations and you see b = a x and c = v ox. From this you can determine v ox and the horizontal acceleration a x. The horizontal acceleration a x 0 since there is a large amount of air resistance on the football. Next, plot a chart of Vy vs. t, and find v oy and a y using the same procedure. Do this for all three kick-off movies and Copy and Paste your charts into your TM. 2. Now you are going to find these same parameters using the position versus time graphs. Plot a chart of Y vs. t. Add a trendline and select Polynomial: Order 2 and Display Equation on chart. The equation displayed on the chart is in the general form y = ex 2 + fx + h, where e, f, and h are the constant coefficients. Recall for constant acceleration the vertical kinematical equation of motion y = ½a y t 2 + v oy t + y o. Compare these two equations and you see e = ½a y, f = v oy, and

9 h = y o. Since this is vertical motion, you know that a y = g, or g = 2e, where e is the constant coefficient of the equation displayed on the chart and g is the acceleration due to gravity. You have also determined v oy as well. Do this for all three kick-off movies and Copy and Paste your charts into your TM. 3. Plot a chart in Excel of the horizontal position versus time, or X vs. t. Add a trendline and select Polynomial: Order 2 and Display Equation on chart. The equation displayed on the chart will be in the general form y = jx 2 + kx + n, where j, k, and n are the constant coefficients. Recall for constant acceleration the horizontal equation of motion x = ½a x t 2 + v ox t + x o. Compare these two equations and see j = ½a x, k = v ox, and n = x o. From this you have determined a x and v ox. Do this for all three movies and Copy and Paste your charts into your TM. 4. In your TM, compare your different values of v ox, v oy, a x, and a y using the two different charting methods described above. Calculate the average v ox, v oy, a x, and a y. 5. Now calculate the range and hang time of the football. Do not obtain these values directly off the kick-off movies, but calculate them using the averages of the two values of v ox, v oy, a x, and a y you just computed. To do this, you will need the angle θ of the football trajectory. To find the angle, you can use the average initial velocity components you calculated. From the vector diagram of the initial velocity, recall that the relation between the initial velocity and angle is: v oy v o θ v ox tan θ = v oy v ox Using this expression, you can solve for θ and calculate the angle of the initial velocity of the football. To find the range R of the football, use the range formula: R = v o 2 sin 2θ g Remember, this formula was derived using the condition y = y o, or Δy = 0. Thus, it is only applicable for these specific conditions. Also recall that v o in this formula is the total initial velocity calculated using the horizontal and vertical velocity components v ox and v oy : v o = v ox 2 + v oy 2 To calculate the hang time of the football for the entire kick-off, recall the horizontal equation of motion x = ½a x t 2 + v ox t + x o. You just calculated the range, or total horizontal distance traveled, thus x = R. This gives the horizontal equation of motion R = ½a x t 2 + v ox t + x o. To find the hang time, solve this quadratic equation for t and substitute in your average values of a x, v ox, and x o and compute t. This time t will correspond to the point of maximum horizontal distance, or the time t at R. List the values of the range and hang time you compute for all three kick-off movies in your TM. 9

10 EQUIPMENT LIST FOR LAB 2: Safety Goggles Camera (memory card and AC charger) Heavy duty stand with finger clamp Launcher and marble Poster board Meter stick THINGS TO DO AT THE END OF THE LAB SESSION: 1. Replace the heavy stand, rod, clamp on the lab table. 2. Replace the camera (with the LCD view finder window closed) on the shelf of the lab table. 3. The meter stick should be put on the shelf of the lab table. 4. The memory card should be left in the slot in the computer monitor. 5. Leave the other items in neat order on the lab table. Be sure to leave the marble in the launcher. 6. Put your safety goggles back in the box at the front TA table. 10