Physics with a Rocket-Borne Video Camera
Model rocketry has long been a source of inspiration to physics students and teachers alike[1],[2]. A popular project is to determine the peak height of the rocket by using “sighting protractors” to determine the angular height of the rocket from a vantage point of known distance from the launch pad using basic trigonometry[3]. Desbien showed how to take side-view, high-speed video of a rocket launch and analyze the rocket’s velocity and acceleration[4]. Other projects have used still cameras aboard the rocket to take photographs of the ground, which students can use to learn remote sensing techniques. Studying videos from on-board cameras dates back to the 1970’s. An article by Gaede and Weller described how to process 8 mm movies taken with a rocket-borne camera to expedite their in-class kinematic analysis[5]. More recently, lightweight digital video cameras have become available in commercially available model rockets. Videos are a powerful learning tool for introductory physics students since the frame rate of the movie provides a natural Dt for calculating velocity and acceleration from distances measured on successive video frames[6]. In this article, we describe a methodology for measuring the height, velocity, and acceleration of a rocket as a function of time from a rocket-borne video camera, and illustrate it with examples from a particular video sequence. We tested various versions of this project in physics classes at two high schools and in a summer program for incoming college freshmen majoring in STEM education. Insights and suggestions learned from this testing are included throughout the article.
We used an Oracle model rocket from Estes Inc. that provides a moderate resolution video with a frame rate that is well suited to measuring the rocket’s kinematics. The video camera is mounted in the nosecone and a flat mirror provides a view angled backwards, so the rocket “sees” the ground below it during its ascent. Figure 1 shows an image extracted from our flight video, which can be viewed at TPT Online[7].
Figure 1: An image from our video taken 2.5 sec into the flight. The red region on the left is the rocket body, with tail fins and smoke trail leading back to the launch pad (yellow “X”). The “Y”-shaped pattern of white marks on the ground include three markings 3.0 m from the launch pad and three more at 10.0 m. We used the vertical dimension of the frame to define the camera’s field of view and to determine scale via Eqn. 2.
The basic concept in measuring the rocket’s height above the ground, h, on any frame of the movie is the trigonometric relation
(1)
where q is the angular size (in degrees) of an object on the ground that has a known linear size, s (in meters)[8]. The high school students in our physics classes were familiar with this relation and most were able to utilize it. More novel, and hence challenging to our students, was the scaling equation that relates q and s to the size of objects on the computer screen[9], x (in mm), which is used to display the video:
(2)
The subscript “o” refers to the object in the video with a known size, so, while the subscript “f” refers to the field of view of the camera. We have had success in describing the ratio Eqn. 2 using the language an English teacher might use in describing an analogy question, “xo is to xf as qo is to qf.” It may also be helpful to emphasize that, though the objects on the ground appear smaller as the rocket ascends (that is, their apparent or angular size decreases), their linear size does not change.
Students can measure xo and xf from a given frame of the video using a ruler, and thus define the proportion in Eqn. 2. If we can determine qf, the angular size of the camera’s field of view, students can calculate qo from the right side of Eqn. 2, and then use h = so / tan qo to determine the height of the rocket when that frame was taken. Fortunately, qf is a fixed value since it is determined by the camera optics. Our solution was to take a video of a meter scale drawn on a chalkboard while holding the camera a known distance away, d. Using the video, students can view the number of meters, s, on the scale subtended by the camera’s field of view and calculate qf = tan-1 (s/d). For our rocket, we found qf = 33.9°. A frame from one such calibration video is shown in Figure 2. Our calibration videos are available on our webpage[10]; experimenters using a different rocket should create their own calibration videos since qf and Dt may vary from one rocket to the next.
Figure 2: Frame from a video used to determine the angular size of the camera’s field of view. Sighting along the red rocket tube to a meter scale drawn on a chalkboard 3.0 m from the rocket.
Using this method, one can in principle determine qo and h for every frame in the video. In practice, one must have in each frame an object on the ground of known size, so. To provide a suitable sized object on every frame as the rocket ascends, we have learned to provide a number of objects on the ground having a wide range of sizes. In the accompanying video[8], we have a stick under the launch pad with contrasting bars 0.10 m long. We also have white markers on the ground laid out in a “Y” pattern at 3.0 and 10.0 m from the launch pad (another set at 30 m is recommended), which is marked with an “X” (see Figure 1). These markers are suitable for most frames in the video, but in some frames the rocket has rotated so no markers are visible. For these frames, one can use another feature whose size has been “bootstrapped” from a nearby frame. For example, in frame 19 of the included video, we used the length of the launch pad leg as the size standard. In frame 18, both the leg and the stick are visible, and we used the ratios xo / x’ = so / s’ to compute the length of the leg, s’ = 0.30 m from the known 0.10 m bars and the apparent sizes (in mm) of the bar and leg measured from the image.
We successfully measured the height of the rocket in frames 18 through 56 of our video. The results are shown in Figure 3. In order to convert from frame numbers to time since launch in seconds, we need to know the time between successive frames in the video. The literature from Estes indicates the frame rate is 9 frames per second. We confirmed this using a video taken of a clock with an analog second hand (see webpage resources10). By counting the number of frames elapsed over 30 seconds, we determined a frame rate of 9.13 frames/sec, or Dt = 1/9.13 = 0.110 sec between frames.
Figure 3: Blue squares represent the height of the rocket as measured on successive frames of the video. The red line is a least-squares fit to the data over the time interval it spans.
Figure 3 shows that the rocket accelerated upward for about 0.7 sec, then continued at a nearly constant velocity over the remaining part of the measured flight. This surprised us at first, since the video shows the Estes C12-5 engine clearly providing thrust for the first 2.3 sec of the flight. However, inspection of the specifications sheet provided with the engine revealed that the engine provides a strong initial thrust, presumably designed to get the rocket clear of the launch pad and moving fast enough to be aerodynamically stable, followed by a lower period of thrust intended to counter the downward force of gravity without producing excessive speed and drag-induced damage.
The difference in height between successive frames, Dh, allows us to measure the average velocity between the frames, v = Dh/Dt. Figure 4 shows the plot of v versus time for our flight. The time difference between successive v measurements is still Dt (a fact that was not obvious to many students), and so the average acceleration of the rocket can be calculated from a = Dv/Dt as shown in Figure 5.
Figure 4: Blue squares represent the velocity of the rocket as calculated from the height on successive frame pairs. The red line is a least-squares fit to the data over the time interval it spans.
This project offers students an opportunity to see how measuring errors propagate through a series of calculations. For example, the early part of the height vs. time curve in Figure 3 appears smooth, but as the height increases, the curve becomes more jagged. This results from uncertainties in measuring xo from the images. At lower heights, our markers subtended a larger angle, and the change in their apparent size from one frame to the next was substantial. At higher h, the apparent size of the markers was smaller, and the change from one frame to the next was comparable to the measuring uncertainty, causing more random variation from one data point to the next in the time series. Because velocity is computed directly from pairs of h, the variance in v is larger. Similarly, the variance in a is larger still, since a is computed from pairs of v values. We encourage students to describe how small measuring errors present in Figure 3 compound and grow as these data are used to calculate velocity and acceleration.
Figure 5: The acceleration of the rocket as calculated from successive pairs of velocities. Error propagation results in significant scatter in this figure.
A number of statistical techniques such as data smoothing could be used to overcome these observational uncertainties. Here, we use linear regression, available in most spreadsheet applications. In Figure 3, we fit a line to the points between t = 0.77 and 3.29 sec, where the rocket appears to be traveling at a constant speed. The slope gives the average speed over this interval, 27 m/s. Similarly, we fit a line to the points in Figure 4 between t = 0.27 to 0.82 sec, where the rocket appears to be accelerating at a constant rate. We found the average acceleration to be 51 m/s2, over five times the acceleration due to gravity! The standard deviation of the points around each of these fits gives the size of a typical measuring uncertainty in h (1.3 m) and v (3.2 m/s). Clearly, this project offers an opportunity for students to employ statistical methods to describe data from a real life situation.
This project combines several topics covered in most first-year physics classes including trigonometry, kinematics, and spreadsheet analysis, and can provide an introduction to basic statistics. It works well as a capstone project in the second semester, and can re-energize a class suffering from spring fever. If time is short, students can simply use our video, with different groups analyzing different segments of the video. We find it helpful if two or more groups analyze the same segment to compare results and check methods and measurements. Once consensus is reached, results from the different segments is pooled to obtain a record of the complete flight. However, students will be more invested in the project if they fly their own rocket. Students can also measure the altitude of the flight using “sighting protractors”3 or a barometric altimeter2 and compare the result with the height measured from the onboard video. They can also use a regular camera to record and analyze a side-view video of the launch4 to measure the height, velocity and acceleration early in the flight (we used a $100 point-and-shoot digital camera in video mode at 13 frames/sec), and compare these results to those of the onboard video. Both side-projects provide an opportunity to demonstrate a principle essential to all sciences: the need to observe a phenomenon in multiple, independent ways to confirm and refine one’s result.
As teachers, we enjoyed flying the rocket, analyzing the data, and developing lesson plans that were appropriate to our respective students. Our students, while challenged by the project, also enjoyed it and some students were inspired to carry the analysis well beyond our expectations. We encourage readers to develop their own versions of this project and share them with the community on our webpage10. A spreadsheet analysis of the video described in this article is available upon request from the first author.
[1] Robert A. Nelson and Mark E. Wilson, “Mathematical Analysis of a Model Rocket Trajectory, Part I: The powered phase,” Phys. Teach. 14, 150-161, (Mar. 1976).
[2] Ken Horst, “Model Rocketry in the 21st-Century Physics Classroom,” Phys. Teach. 42, 394–397 (Oct. 2004).
[3] For example, see lesson plans from Estes Inc., like “Mathematics: Altitraking” at http://www2.estesrockets.com/pdf/ERL_M_9-12_2.pdf .
[4] Dwain M. Desbien, “High-Speed Video Analysis in a Conceptual Physics Class,” Phys. Teach. 49, 332-333, (Sep. 2011).
[5] Owen F. Gaede and Charles M. Weller, “Processing your own super-8 rocket films for classroom use,” Phys. Teach. 13, 167-168, (Mar. 1975).