LivePhoto Physics Activity 7 Name: ___________________________ Date: _________________________

Physics with Video Analysis 7 - 1

Demon Drop: A Mathematical Modeling Activity

Demon Drop is a popular ride at Cedar Point Park in Ohio. During the ride four people in a cage are in vertical freefall for a short time. Then they ride along a curved track and are finally brought to a stop while riding along on a straight level segment of track.

Newton’s Law of Gravitation tells us that the Earth exerts a constant force on an object that is close to its surface. If no other forces are present, Newton’s Second Law of Motion can be used in conjunction with data about the Earth’s mass and radius to predict that an object falling freely near its surface will undergo a constant downward acceleration of magnitude 9.8 m/s2. Is this really the acceleration of the cage full of people? To answer this question you’ll analyze a video of the of the cage’s fall.

To test the prediction of the falling rate you’ll need to (1) predict what kinematic equation will describe the vertical motion of the cage; (2) obtain height vs. time data from the video; (3) develop an analytic mathematical model, which is an equation that describes the details of the cage’s motion; and finally, (4) compare the model to the data.

Figure 1: Demon Drop Structure showing the falling cage.

You’ll begin your investigation by viewing the video clip entitled <DemonDrop.mov>. Open the movie in QuickTime Player. Play the movie or advance it frame-by-frame using the right arrow key (�) on your keyboard. The cage in the video clip looks small and there is a bit of wobble in the scene. Since the movie was taken at the amusement park with a hand-held video camera, you should appreciate that it turns out to be analyzable!

1. Preliminary Questions

Note: You will receive full credit for each prediction made in this preliminary section whether or not it matches conclusions you reach in the next section. As part of the learning process it is important to compare your predictions with your results. Do not change your predictions!

(a) What forces other than the gravitational force might act on the cage as it falls? Do you expect the effects of these forces to be measureable? Could they cause the acceleration to be non-constant or deviate from the predicted magnitude of 9.8 m/s2? Explain.

(b) Based on watching the <DemonDrop.mov> frame by frame, how does the cage move? Does it speed up, move at a constant speed, or slow down? What is the evidence for your answer?

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(c) Note that the cage’s initial position, y1, is almost 30 m above the level track. The cage is already moving downward in the first frame of the movie where t = 0.00 s. By estimating the height of the cage’s bottom in the last frame when t = 1.45 s, sketch a graph that roughly predicts the location of the cage along the vertical y-axis vs. time.

(d) Describe how you think the slope of the graph will change with time.

(e) If objects fall at a constant acceleration near the Earth’s surface, then the kinematic equations should hold. Which equation describes the height of the cage, y, in the Demon Drop as a function of time in terms of its vertical acceleration (ay), initial velocity (v0y), initial position (y0) and initial time or time offset (t0)?

2. Activity-Based Questions

Collect vertical data for the ride: To determine whether or not the gravitational acceleration of the cage in the Demon Drop is –9.8 m/s2 you’ll analyze a video clip of the ride and then use an analytic mathematical modeling technique to determine whether you can find an “analytic” equation that describes the motion of the falling cage.

Open the Logger Pro Experiment file <DemonDrop.cmbl> to obtain a video analysis setup. As you can see on the first frame of the movie, it has already been scaled based on the Cedar Point Park’s claim that the Demon Drop cage falls 30 m by the time it reaches the level track.

(a) Take y vs. t data by clicking on the Add Point tool ( ) near the top of the movie window. Then click on the same point on the cage in each frame. Logger Pro will then record the vertical position of the cage as a function of time.

After doing your video analysis, pull down the Page menu and select Auto Arrange to display the y vs. t graph. Then, sketch your data points on the graph on the right.

Hints: (1) To take data carefully, we suggest that you align the horizontal crosshair of the Add Point tool ( ) with the lower edge of the cage and place the vertical line of the crosshair at the middle of the cage; and (2) If you mess up, start over by using the Clear All Data feature in the Data menu and then returning to the beginning of the movie.

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Which equation will model your data? Call up the Model feature in the Logger Pro Analyze menu and choose the most appropriate “general equation” to model the data (Proportional, Linear, Quadratic, Cubic, and so on.). Write down the algebraic equation that you will use to develop your model and explain why you chose it. Hint: Look at the answers you gave in section 1!

(b) What do the coefficients (A, B, etc.) represent physically? The equation that you chose as your model in Logger Pro, after thinking about the kinematic equation you listed in 1(e), contains several coefficients (A, B, etc.). Please summarize how these coefficients are related to the physical quantities that describe the motion of the Demon Drop cage: its vertical acceleration (ay), initial velocity (v0y), and initial position (y0).

(c) Estimate the numerical values of the coefficients: You can use the prediction for the acceleration of the falling cage, ay, and the data for the location of the bottom of the cage, y0, shown in the first frame of the movie for t0 = 0.00 s to calculate the expected values of some of the coefficients. Another fact you can use is that the cage had already fallen about 1 m in about 0.4 s before the first frame of your movie was recorded. Show your calculations and explain the reasoning used for estimating the values of A, B, etc. Then summarize your values for these coefficients along with units in the space below.

Hints: (1) You may want to refer to the data table for initial position information; and (2) Logger Pro is set up with a conventional coordinate system where upward is the positive y-axis, so the predicted y-component of the acceleration is ay = –9.8 m/s2.

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(d) Check the model: Enter your estimated coefficients into the Logger Pro Modeling (i.e., Manual Fit) screen. Does the “model” line match the data? If not, think about the estimates you made in part (d). Start by adjusting the coefficients most likely to be off, such as the ones related to v0y and y0. Make sure you have used minus signs in appropriate places, etc. If it is impossible to get good agreement, then you may need to choose a different analytic equation for your model in question 2(b). Are the values of the coefficients (A, B, etc.) that yield the best agreement different from what you estimated in part (d)? If so, list them here with appropriate units.

(e) Go back to the Cage Height vs. Time graph in Section 2a, and sketch your model curve on it. Explain how you can determine ay from the coefficients in your model that you found in part (e) or part (d).

3. Reflections on Your Findings

(a) Comment on how well the results of this experiment agree with the prediction of Newton’s theory of Universal Gravitation, ay = –9.8 m/s2. Note: Small uncertainties in your measurements can lead to some variation in your results.

(b) Compare the value of the cage’s initial vertical velocity component v0y from part 2 (e) to the value you estimated for the cage’s initial velocity in question 2 (d). Could you have made a better estimate in 2 (d)? Explain.

(c) In some experiments, you measure a quantity and compare it to an accepted or known value. In other experiments, you make an assumption and check whether or not it is consistent with the data. What did you do in this experiment?