Understanding Motion Objectives: Experimental objective Students will interpret motion graphs (position vs. time and velocity vs. time) and attempt to accurately replicate the motion described in the graphs. Students will also verify the acceleration of gravity on earth, g=9.81m/s 2 Learning objectives (students should learn ) To interpret physical meaning in graphs The relationships between displacement, velocity, and acceleration Equipment list: Computer, 870 interface, sonic motion sensor, photogate, 2 support stands, wood plank, picket fence scale. Apparatus: Interface Motion Sensor Photogate Yellow Black Picket Fence Theory: To describe the motion of an object, there are a few pieces of information that are critical to know; where it is, how fast it is moving and in what direction, and how its motion is changing, all relative to a reference point. We call these things, respectively, position, velocity, and acceleration. These three properties are closely related by time. The change in an object s position over some interval of time is the object s velocity, and the change in velocity over some interval of time is its acceleration. Suppose you have measured an object s starting position, xi, at an initial time, ti, and final position, xf, at a final time, tf. The change in these values can be expressed as x = x f x i and t = t f t i And the average rate of change in the position over that time, or the average velocity of the object is v = x t By replacing the Δ with d where Δ means change in, d means infinitesimal (infinitely small) change in we can write the equation for the instantaneous velocity of the object, which is the time derivative of position.
v = dx dt Graphically, this also translates to a slope. If position were graphed on the y-axis vs. time on the x-axis, the average velocity is the equivalent of finding the slope of a straight line connecting two points on the graph, while using the derivative to find velocity is equivalent to the slope of the graph at a specific point. Both of these values are the same if the graph is a straight line, that is, velocity is constant in the region being examined. The same type of relationship is true for velocity and acceleration. Acceleration describes how quickly an object is changing velocity. a = v t or a = dv dt From dimensional analysis, it is clear that if position is measured in meters (m), and time in seconds (s), velocity is expressed in the units m/s and acceleration in the units m/s 2. The inverse of these relationships can also be taken. By finding the area under the curve, of say a velocity vs. time graph, you would know the distance traveled. Mathematically, this is the same as the integral of velocity, which gives us the inverse relationships v = a dt and x = v dt Though an in-depth discussion of vectors is beyond the scope of this experiment, it is important to note that position, velocity, and acceleration are all vectors. For this experiment, they are simplified to one dimensional vectors, which means they can be either positive or negative (the scalar counterparts, i.e. distance and speed are always positive). The difference between two vectors of the same magnitude but opposite sign is the direction. For example, take the two velocities 2m/s and -2m/s; in both cases the object is moving at the same speed, but in opposite directions. It is important to recognize the distinction between positive and negative values in this experiment. As a guide for this experiment, any motion forward (toward the sensor) is considered negative (it may be useful to think of this as decreasing the distance from the sensor), while motion backward (away from the sensor) is positive (again, it may be useful to think of this as increasing the distance from the sensor). While it is, mathematically, not difficult to differentiate or integrate a simple function, in real measurement situations this is typically not feasible. The typical method for overcoming this difficulty is to take many measurements at short intervals to approximate a continuous function. This is what the motion sensor will do. It works the same as a SONAR range finder, pinging many times a second, and listening for the echo. The time delay in the echo correlates directly to the distance of the object being detected. You can hear the motion sensor doing this as a rapid clicking sound. By knowing an object s position at specific times separated by very short intervals, the computer is able to calculate and graph position or velocity vs. time. The photogate in part 3 is simply an on/off gate with a timer. It has an infrared LED on one arm and a light detector on the other. When the detector can see the LED the gate is closed or off (not triggered). When the LED/detector path is obstructed, the gate is open or on. It sends signals to the computer to mark the times of events when the gate is opened or closed.
Procedure: Part 1: in this part of the experiment, you will be the object, and the computer will graph your motion. You will be presented with a graph of position vs. time, and you will try to accurately reproduce the motion as the computer graphs your motion real time. 1. Make sure the computer sensor interface is connected to the computer and turned on before turning on the computer. 2. Make sure the motion sensor is correctly connected to the computer sensor interface. The yellow plug should be connected to channel 1 and the black to channel 2. 3. Open the Physics Lab folder on the desktop and open the P01_MOT1 file. 4. Study the graph and answer question 1 before beginning the experiment. 5. Mount the motion sensor on the stand so it is about elbow height, and make sure the face is pointing parallel to the ground in a direction where you can move at least 2m without obstruction. 6. Position the computer monitor so that the person moving in front of the motion sensor and the person operating the computer can both see it clearly. Make the graph as large as possible. 7. When you are ready, stand in front of the motion sensor with the plank (to reflect the pings better). Note: make sure the area behind you is clear you will be walking backward. 8. The person operating the computer will click start and the motion sensor will begin clicking, The timer will show a few second countdown in orange, and the y-axis will show your position. Use this brief time to adjust your position to the proper distance. 9. When the timer turns green, your position vs. time will be recorded on the graph. 10. Perform the data collection at least twice per group member. 11. After performing all of the data collection runs, select the best run, and draw a box encompassing your data between 2 and 6 seconds. 12. Click Fit and select Linear Fit. A linear fit line is added to your graph with an equation. The m- term, or slope from that equation is the average velocity over the selected time range. 13. Take a screenshot of the graph for later reference. Part 2: in this part of the experiment, you will again be the object. This is much like the first part, except now the graph you are presented with depicts velocity vs. time. 1. The computer and sensor should be set up the same as in part 1. 2. Make sure the file from part 1 is closed, and open the MOT2_PLT file. 3. Study the graph and answer question 4 before beginning. Also, it is helpful to go through each segment and discuss with your group which direction you should be moving and if you should be speeding up, slowing down, or at a constant speed. 4. When you are ready, stand in front of the motion sensor. When the group member operating the computer clicks start recording will immediately begin. There is no delay. 5. Do your best to copy the graph. Each group member will attempt this at least twice. Note: if the graph of your motion suddenly jumps, it is likely that you moved the plank to high or too low, and the sensor lost it. The graph is probably rescaled and difficult to read. Click stop, delete the run, and click the auto-scale button to rescale the graph. Also, a tip: in this part of the experiment, you shouldn t need to move your entire body it is often easier to do this part by leaning and moving your arms only. 6. After performing all of the data collection runs, select the best run, and draw a box encompassing all of your data between 3.5 and 4.5 seconds. 7. Click Fit and select Linear Fit. As in part 1, a linear fit line is added, but this time the slope of that line is your average acceleration over the time range selected.
Prelab: 8. Take a screenshot of the graph for later reference. Part 3: in this part of the experiment, you will drop a scale through a photogate to confirm the acceleration of gravity on earth (9.81 m/s 2 ). 1. Replace the motion sensor with the photogate, plugging it in to channel 1. 2. Make sure the previous experiment file is closed and open the PO6_FALL file in the Physics Labs folder on the desktop. 3. Click start and drop the scale so that it falls straight through the photogate detection area. Warning: the scale can shatter if it hits hard ground. Place a backpack or something else soft to break its fall. 4. Click stop. The computer will display a graph of position vs. time and velocity vs. time with a fit line. 5. Rescale both graphs so that they fill the entire screen and any curves can be seen. 6. The slope of the linear fit is the average acceleration due to gravity. 7. Save a screenshot of this graph for future reference. 8. There is also a window with three tables. Select that window. 9. Click the dropdown arrow next to the statistics ( Σ ) button, and make sure minimum, maximum, mean, and standard deviation are selected. 10. Copy the far right (acceleration) table into your notebook. 1. What is the difference between uniform motion and uniformly accelerated motion? 2. Suppose an object starts at time t=0s at a position defined as x=0m. At t=0.5s, the object is at x=1.5m, and at t=1.0s, the object is at x=2.0m. What is the average velocity for the first interval (approximated at t=0.25s) and for the second interval (approximated at t=0.75s)? 3. In the example above, what is the average acceleration of the object? 4. The accepted value for the acceleration of gravity is a constant 9.81m/s. knowing that, what should the graph of velocity vs. time for a falling object look like (assume down is the positive direction). What should the graph of position vs. time look like? Report: You should complete these tables in your report (don t forget units). Note: do not write directly in these tables all notes and records must be kept in the notebook section of this book. These are for your reference only. Time Table 1 Acceleration Minimum Maximum Mean Std Deviation
Table 2 (results) Experiment Measured Value Expected Value %error Position vs. Time Velocity vs. Time Freefall (graph) Freefall (table) Questions: Part 1: 1. Determine the following from the position vs. time graph: a. How close should you be to the motion sensor at the beginning? b. How far away should you move? c. For how long should you be moving? d. How fast should you move (during the motion portion)? 2. For your best attempt in the graph, what is the slope of your best fit line for the motion portion? How well does this match the given graph (use %error and briefly discuss how good your result is)? 3. Describe your motion. Start at time = 0 and end when the timer stopped (example: Constant motion at 1.3m/s for 2 seconds followed by no motion for 3 seconds etc.). Part 2: 4. Determine the following from the velocity vs. time graph: a. Which direction should you move in the first motion? b. What is the maximum speed (positive or negative) that you should reach? c. How long (total) should the positive velocity portions of your motion last? d. What is the total acceleration during the positive velocity portions of your motion? e. Should you end up closer to or further from the sensor? 5. How well did your motion in the specified area fit the given graph (use %error and briefly describe how good your result was)? Part 3: 6. How closely does the slope of velocity vs. time for the scale match the expected acceleration of gravity? How well does the mean value from the table compare to the expected acceleration? Discuss in terms of %error. 7. What is the uncertainty in your mean acceleration from the table? 8. What may have caused the experimental value to be different from the expected value? Hint: things like human error and calculation error are never acceptable answers be specific. 9. What is the computer measuring? What information does it already have? How does it come up with a graph and table for velocity and acceleration? Hint: measure the width of each of the transparent and opaque sections of the scale.