Motion in 2D

3. Description of Motion in Two Dimensions* In this lab unit you will study of accelerated motion in two dimensions. Learning objectives: 1. Be able to relate features of motion of a projectile to graphs of position or velocity in the x or y direction versus time. 2. Understand separation of projectile motion into x and y motions and solving for each. 3. Learn to take and analyze video data for projectile motion. Reading Assignment: Knight, Jones and Field (161): 3.3 Coordinate Systems and Vector Components, 3.6 Motion in Two Dimensions: Projectile Motion Serway and Vuille (211): 3.1 Vectors and Their Properties, 3.2 Components of a Vector, 3.3 Displacement, Velocity, and Acceleration in Two Dimensions Serway and Jewett (251): 4.1The Position, velocity, and Acceleration Vectors, 4.2 Two-Dimensional Motion with Constant Acceleration, 4.3 Projectile Motion

PRE-LAB EXERCISES Do pre-lab problems below before you come to lab. Your TA will check that you have done the problems. You can’t enter the lab if you haven’t done the pre-lab. Before you start working on the equipment your group will discuss these and other questions. Then your recorder will write the answers on the white board and when the group is called on the recorder must present answerers. There is a time limit on this activity, so make sure you pay attention as you do the pre-lab problems. You will also hand in you’re your pre-lab and prediction questions with your lab report.

Free Fall 1. From your reading, write down formulas for position and velocity of a falling object. Use y for the vertical position coordinate and vy for the vertical velocity.

2. Sketch on the axes at the left below your prediction of a graph of position of a falling object versus time. The ball is initially at rest and y  y0 . Choose the vertical axis as upward.

velocity

position



00

t ime

t ime

 t ime 3. Sketch on the axes on the above right your prediction of a graph of the velocity of a falling object as a function of time. 4. Is the acceleration zero? Constant? Some function of time or position? *© William A Schwalm 3-1

Motion in 2D

Projectile Motion y The problems 5 through 7 below involve a ball thrown in the air with an initial velocity upward and to the right. The trajectory of the ball looks approximately like the sketch to the right. Note that the origin of the coordinate system has been chosen to be the initial position of the motion of the ball.

0 0

x

Based on your graph x vs. t, write a kinematical equation for x as a function of time.

x coordinate

 5. Sketch on the axes at the right your predictions for the x coordinate of the ball as a function of time and the y-coordinate of the ball as a function of time.

0

time



x=

y=

 y coordinate

Based on your graph y vs. t, write a kinematical equation for y as a function of time.

(a) When is the speed of the ball a maximum? (Speed is magnitude of velocity vector.) When is it a minimum?

0



(b) At the highest point in the motion, is the speed of the ball zero or not?

(c) When is the x-component of the velocity a maximum? A minimum?

(d) When is the y-component of the velocity a maximum? A minimum?

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time

Motion in 2D

Projectile Motion (continued)

6. On the axes to the right, sketch your predictions for the xcomponent of the velocity vx as a function of time and the ycomponent of the velocity vy as a function of time.

 vx

0 time

Based on your graph for vx vs. t, write an equation for vx as a function of time.



vx =

 vy

Based on your graph for vy vs. t, write an equation for vy as a function of time

vy =

0 time



Do these results agree with the formulas from your reading? Explain.

7. In the space below, draw an arrow that represents the direction of the acceleration of the ball (a) just after it is released, (b) when it reaches the highest point in the trajectory, (c) while it is on its way down. If the acceleration is zero, write ZERO above the ball.

(a)

(b)

(c)

What can you say about the acceleration in each of these cases? How is it related from case to case?

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Motion in 2D

Equipment: Drop shoot apparatus, wooden/steel ball, basket ball, stopwatch, meter stick, lab tape, video camera and a computer with Logger Pro.

Problem 1: Free fall

The fire department has requested help in studying the technique of dropping of balls filled with chemicals from helicopters to extinguish fires. (This technique was used during the Grand Forks flood and the Chernobyl nuclear power plant accident, and it is often used against forest fires.) The amount chemical in one of these balls is varied depending on the size of the fire. As first step to your study, you assume a helicopter is stationary, hovering over a fire. You are to determine if balls of same size with different amounts of chemical will fall in the same way or differently. The fire department needs actual data, not just theory.

How do the accelerations of two freely-falling objects of different mass compare?

g

Meter stick to indicate The scale

To computer USB port.

Prediction: Outline your predictions, giving explanation for each. The explanation part is important, so think carefully.

Exploration: One often assumes that acceleration of a falling body doesn't depend on its mass. Try this out. You should find two balls in the laboratory room, one wooden and the other steel. These are about the same size and shape but have different masses. Perform a simple experiment to see whether these two balls accelerate at the same rate. Do they? Did you hear one bang or two bangs? Estimate the accuracy of your conclusion. In other words, how large could the difference in acceleration be before you would detect it using the technique you plan to use? Practice using the video camera to make measurements and try to determine the size of the typical errors involved.

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Motion in 2D

The image could be quite blurred because the balls are moving too fast for our video camera. You can reduce the blurriness by reducing the shutter speed. To do this: Click on the options button then from Video Capture Options tab and select your camera’s setting: Camera Setting  Driver Setting  Advanced Then un-click Automatic Gain Control and adjust the exposure to a shorter time. (say 1/60 to 1/120 s). But, if you select an exposure time that is too short the image gets dark. So you may have to adjust the brightness as well. To do this select Camera Settings  Device Settings then adjust the brightness.

Take videos of dropping ball with two different shutter speed setting and see the difference. Plan: Let’s suppose you were to make a graph of position versus time. Write a measurement plan here. What do you expect to see? Include an outline of your predictions with your plan.

Execute: Cary out your plan. Collect video data, including length calibration etc. Analysis: Plot position versus time and fit the graph to an appropriate function. (What does appropriate mean here?) Find an experimental value for acceleration directly from the parameters of the curve you fitted to the data. Compare the result with the expected value. Usually you have to figure out some way to estimate the error in your measurements. In this particular case you do something a bit different, though. A simple way to get the percent difference between the expected value and the measured value is to calculate

% difference =

experimental  expected

x 100%

expected This is not the same as figuring out the measurement error, since in reality you don’t know whether either the expected or the measured result is actually correct. It just shows how much the two differ. Now plot velocity versus time and fit the graph to an appropriate function. Find an experimental value for acceleration again, this time using velocity versus time. Compare the result you get by this method with the expected value and with one you obtained from position versus time graph.

Always complete your data processing (analysis) before you take your next data.

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Motion in 2D

Conclusions: From your reading and from the laboratory activities, what do you know about the acceleration of freely falling object? To what extent does mass of the object affect its motion in free fall? What else affects the acceleration and to what extent? Does the video analysis method give a reasonably good value for the acceleration of a falling object? How can you estimate the errors from the various sources?

Problem 2: Projectile motion In medieval warfare they used to use a device called a trebuchet, which hurls horse-sized rocks into castles to break down the walls. (Truly awesome.) You are asked to study the motion of such projectile for a group of local enthusiasts planning an ancient war reenactment. Your team should realize that no one has seen this particular design in action for nearly a thousand years. The final full-scale construction will be expensive and probably dangerous. You have to build a desk-top model before the actual fullscale one is constructed in order to make sure the group really understands how it’s supposed to work. For a thrown object: 1. How should the horizontal component of its velocity change with time according to the usual theory described in your text? To what extent is this actually true? 2. How does the vertical component of its velocity actually change with time? 3. Of all the factors that might affect the motion, which ones actually play a role and which ones are negligible? How accurately can you determine this using equipment available in the lab? Prediction: Based on the pre-lab exercises and what you learned from the text summarize your answers (predictions) to above problems.

Exploration: Practice throwing the ball until you can get the ball's motion to fill the video screen after it leaves your hand. Determine how much time it takes for the ball to travel and estimate the number of video data points you can get in that time. Is that enough points to make the measurement?

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Motion in 2D

Suppose you are planning to plot the trajectory of the ball and choose the horizontal axis pointing on the screen to right. Which way should you throw the ball, relative to the camera’s view? If this is hard to arrange because of the orientation of the equipment in your lab room there is a way to invert the video image. To get a mirror image select Options  Camera Setting  Advanced. This will take you to the Properties tab. From there select mirror horizontal. Note the effect of this setting. Perform some measurements of total the distance the ball travels and the total time to determine the maximum and minimum value for each axis before taking data, so that you can see the range of values involved.

Plan and Implementation: Outline your measurement plan. Which graphs do you need for the purpose? On the Logger Pro screen, choose a function to represent the horizontal position-versus-time graph and another for the vertical position-versus-time graph.

These data points depend on the initial conditions such as the position at t = 0 (initial position) or velocity at t = 0 (initial velocity) or acceleration (it may not depend on time). Your data points carry the information about these constants. In another words if the initial conditions or other constant (acceleration) were different from one you have the shape of the graph (may belong to the same family of curves) would be different. So a part of your job is to find these constants for a particular case when the video was taken. How can you estimate these values of the constants? As you have done in the Lab 1 (Motion in 1D) you will use a curve fit. You know how each component of position or velocity of projectile motions changes as a function of time. These equations contain some constants in symbols. Using such an algebraic form of these equations fit your data. Curve fitting process reveals these constants you are after in numbers so that you know these constants. Question: Suppose you are fitting y-component of velocity versus time data to an equation and found that a cubic equation or a linear equation fit better than a quadratic. Which result will you choose for your analysis? Justify your answer. Explain why you would make this choice.

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Motion in 2D

Analysis: Now do a similar analysis for horizontal velocity-versus-time graph and another for the vertical velocity-versus-time graph.

Determine the launch velocity of the ball from your graph. Can you think of a way to decide whether or not this result is reasonable? Determine the velocity of the ball at its highest point. What can you say in general about the horizontal velocity during the entire motion?

Determine the acceleration of the ball independently for each component of the motion as a function of time. Is the acceleration constant from just after launch to just before the ball is caught? Determine the magnitude of the ball's acceleration at its highest point. Does this agree with the simplified description of projectile motion found in your text or lecture notes?

Plot vertical position versus horizontal position. To do this take the cursor on the graph and from Options  Graph Options  Axes Options and for Y-axis Columns select Y then for X-axis Columns select X. Now you should be able to see a graph of Y versus X position. Is the shape of the graph as you expected? Why is the shape of y-x graph similar to y-t graph?

Does the percent difference for the acceleration fall within the limits of accuracy in measurement, or not? Explain. Do you think part of it might represent an actual physical effect?

Conclusion: What was the main point of this exercise? How did the results compare to your initial predictions? Was there any surprise? Did you have to change you predictions? If so, why? What are the sources of errors? Using the video tools you used how accurately can you determine the constants?

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Motion in 2D

Problem 3: Drop and shoot You have seen in Lab I that if an object is thrown horizontally on a smooth surface such as an air track it will move with constant velocity. In this lab unit you have examined the motion of a freely falling object with initial velocity zero. You should also know that if you throw a ball horizontally in free space, subject only to gravity, the ball follows a parabolic trajectory. How do the vertical and horizontal components respectively of the motion in two dimensions of a freely falling object relate to accelerated and non-accelerated motion in one dimension? In solving the following problem you are going to examine the relation between these three types of motion using a DROP SHOOT apparatus and video camera.

Prediction:

The drop and shoot apparatus is a spring loaded device that fires two steel balls simultaneously. One ball is released with zero initial x velocity and the other ball is shot with an initial x velocity. Neither ball is given an initial y velocity.

Steel Ball

Cocking pipe

Does the second ball reach to the ground at the same time as the first ball? Does the horizontal speed of the first ball influence its time of flight? If it does, how? And, how can you tell? If not, how can you tell it doesn’t, and why doesn’t it?

1. Draw a frame-by-frame sketch of the position along the x-axis of an object moving with constant horizontal velocity on smooth horizontal surface as it would appear in successive frames of a video, or a stroboscopic picture.

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Motion in 2D

2. Superimpose on the first sketch a frame-by-frame sketch of position along the y-axis (vertical) of freely falling object. 3. Draw a frame-by-frame sketch of the vector sum of position vectors of the ball in the question 1 and question 2. You may have to think a moment about what this means. Discuss this with your team members. How does the combined trajectory of the ball look? Describe the result.

Exploration: Everyone in the group should operate the Drop and Shoot apparatus at least once. Experiment with this device to see how it works and how to create the initial conditions you want to study. How do you line up the two balls? Do this by positioning the first ball carefully before loading the second ball. (There is a really good way to do this.) Be careful not to get shot by the steel ball.



Safety warning: Do not look into the spring launcher when there is a ball Is loaded inside and the spring is cocked!!

Practice shooting the drop-shoot mechanism until you have the process under control. Practice taking data until you can control the pattern of data points on the resulting video. You want to take several frames before the balls hit the floor, and you want the images to arrange themselves so that you get the best accuracy. It’s up to you to figure out how. Make sure to place a meter stick aligned perpendicular to the line of sight from the camera for a length scale reference. Get an experimental estimate of the accuracy of the measurements. You need to be able to explain how you arrived at this estimate, so discuss it with your team.

Plan: Outline the measurement plan for taking the video.

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Motion in 2D

Analysis: Because the motion is two dimensional we want to display y-x graphs. But before doing so first we display position versus time graphs. Trace the second ball (with the non-zero initial velocity) as usual. Notice that the y-position (height) of the first (with zero initial velocity) ball moves about the same as the second ball. We can trace both balls at the same time. Place the cursor on the video then selecting options  Movie Options  then check Video Analysis: Allow Multiple Points per Frame. On the graph you will see two sets of red points (horizontal position-time) and two sets of blue points (vertical position-time). If you see the time column on the data table there are two same-time values. Display only x-time graphs. Can you tell which set corresponds to which ball? Next display only y-time graphs. Again can you tell which set corresponds to which ball? Unfortunately we can not analyze the data in this mode. If you try it then the program try to fit positions of both balls. However we can display y-x graph with this mode. To do this, take the cursor on the graph and from Options  Graph Options  Axes Options  and for Y Axis Columns select Y then for X-Axis Columns select X. Now you should be able to see a graph of Y versus X position for both balls. Is the shape of the graph as you expected? (Neither the x or y axis is a time axis, so how does time enter here?)

Conclusion Is the sketch you made as part of your planning activity similar to the graph you created on Logger Pro? Are there any surprises? What was the point of this exercise? Why did you make the composite graph in the planning exercise? Using your on words explain what are the points of this activity, or in other words: What is one supposed to learn here?

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Motion in 2D

Reference 1. Pre-lab problems are adapted from David R. Sokoloff and Ronald K. Thornton, Interactive lecture Demonstrations: Active Learning in Introductory Physics (John Wiley & Sons Inc. 2004).

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