O n l i n e E xp e r i m e n t - U n i f o r m M o t i o n

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PHYS 191 - Physics I Labs

Online Experiment - Uniform Motion Line 1: Name of the experiment? (Uniform Motion)

Line 2: Your full name: (last, first)

Line 3: Lab course? (Phys 191)

Line 4: Section? (two digits code)

Line 5: Purpose of the experiment? (find the purpose after reading the theory and procedures)

Theory The study of motion, regardless of knowing or investigating what causes it, is called “kinematics” in physics. The simplest kinematic case is the uniform motion where the object moves on a straight trajectory in the same direction and takes the same amount of distance per unit time at all times.

To formulate the linear motion mathematically we can use the number line to write the position of the object as a function of time, i.e. . Then we can (t)x easily define the velocity as the derivative of the position with respect to time: . In uniform v = dt

dx motion, the distance taken per unit time is a constant, that means the velocity of the object is constant. Hence we can write as

and if we take the integral of the both sides: x dt onstant d / = v = c

. This yields where is the constant of the dt dt∫

dtdx = ∫

v (t) tx = v + c c

indefinite integral. If at time the position of the object is t = 0 x0 then which gives . (0)x = x0 c = x0

In short, in uniform motion the position of the object can be expressed as

(t) tx = x0 + v

This is a linear function of time. If we would plot the position versus time in a chart then the slope of the line fit to the data points should give us the velocity of the object.

In this lab, you will find the velocity of a toy car (a motorized dune buggy) moving on a countertop. The motor of the car runs at a steady power to ensure the uniformity of the motion.

Prepared by the lab supervisor, Dr. A�lgan (ea�lgan01@manha�an.edu) Online 191- Uniform Motion PAGE 1 / 7

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Procedure Step 1. Setup. The motion of a toy car moving on a countertop is recorded by a smart phone camera, converted into a gif format and uploaded

onto our lab website https://mc-phys-labs.weebly.com/ . The name of the movie is “ dune_buggy_25fps.gif ”. Please find it under the Downloads page and download it onto your computer: click “Download File” and the movie will be played in your browser, then right click the movie image and choose “Save image as...”.

Step 2. ImageJ. You will measure the position of the small cart in the movie by using a free image processing program from NIH named

“ImageJ”. If you do not have ImageJ, Go to https://imagej.nih.gov/ij/download.html , download, install and launch it. This control panel should show up (be patient, might take about 5 seconds to appear):

Step 3. Set the distance and the time scale. Open the movie (File>Open> select the file) and play. Let’s set the play time of the movie to the real time. Go to Image>Stacks>Animation>Animation Options and enter 25 for “speed” in the window. If the speed is already 25 just hit OK. At these settings the movie will play at the real time speed of 25 fps. This means 25 frames pass in one second.

Line 6: Calculate how much time elapses for 5 consecutive frames = ________ s.

Now let’s set the scale for the positions and distances as well. Select the line segment tool as shown (the button that is clicked). >>>


https://mc-phys-labs.weebly.com/

https://imagej.nih.gov/ij/download.html

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The wooden meter stick positioned in front of the buggy is 100 cm long. Span the line segment tool along the length of the meter stick as shown ⇑ . By zooming in and out (+ and - keys of your keyboard) you can fine tune your alignment. Take advantage of this feature often in order to align your measurement tools with better accuracy.

Go to Analyze > Set Scale. A window will open >>> Enter 100 into the “Known distance” box, and enter cm into the “Unit of length” box and click OK. This will set the scale for the positions and distances measured. All distance and position measurements will be the actual, real life values in units of cm!

<<< Let’s test the distance scale. Use the line segment tool to measure the distance between the two circles that show up on the first frame (use + and - keys of your keyboard to zoom in/out for the best alignment). You can

read the length in the “communication bar” at the bottom of the control panel ⇑. The length info appears only if you hold one of the line segment points with your mouse and move it . I recommend you hold the middle point of the segment where the program won’t change the span


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but only move the segment . You should measure the distance between the circles as 25.3 cm within less than a centimeter error. If you do not then that means perhaps you made a mistake in setting up the scale. Repeat this step again.

Step 4. Measuring the position. You will measure the position of the cart by using the “Multi-point” tool (the button that is clicked) >>> If you don’t see the Multi-point button then you see the Point tool. Right click on it and switch to the Multi-point tool.

<<< Double click the “Multi-point” tool, configure it to these settings shown.

You will notice that at the front of the buggy car there is a white spot (see image ⇓) a few pixels in size. Use this “point” to track the cart starting from the 1st to the 96th frame with 5 frame intervals. The frame index is shown on the upper left corner of the image (the very first number). In the example shown below 14/99 means the 14th frame is on display and there are 99 frames in total. You can step forward or backward buttons by using the arrow heads at the bottom corners of the movie image and embedded in the bottom sliding bar (or simply use the wheel of your mouse, if you have one). If you are a MAC user you may not see the backward/forward buttons. If so shift-right or shift-left does the same.


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Mark this white spot (the headlight of the car) on frames 1, 6, 11, … , 91, 96 without skipping any frame and do it in order. Take advantage of the zooming in/out (+/- keys) feature.

<<< This example image shows the 10th unmarked position in frame 46 and the previous 9 marked positions.

Once all 20 positions marked do CTRL-M (or go to Analyze>Measure). A window titled “Results” will open containing some information regarding the points marked including their x and y positions >>>

The positions are measured with respect to the top left corner of the picture such that right and down are the positive directions in the horizontal and the vertical axises, respectively.

Click onto any line in the Result window and do CTRL-A and CTRL-C. This will copy the whole data into your computer’s clipboard.

Step 5. Process the data. Open a google spreadsheet program and paste the data i.e. do CTRL-V. Delete all columns except the index and X column. Please note that the column titles on the Results window will not be transferred into the spreadsheet program so be careful to identify the X column correctly.

Make the index column your FIRST column in your spreadsheet program and title the column as “index”. Make the X positions your THIRD column and title the column as “x [cm]”. Since the frame speed of the movie is 25 fps and the measurements are done with 5 frame intervals, in our data between each two consecutive time points 5*(1/25)=⅕=0.2 seconds elapse. On the SECOND column create the time axis by subtracting 1 from the frame index and multiplying the result with 0.2, i.e. the code for the algorithm should be =(“ address of the index ”-1)*0.2, and title it as “t [s]”. The 20 time values should show up as the following: 0.0, 0.2, 0.4, … , 3.6, 3.8. If


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not then your coding is probably wrong. Alternatively, however, you can enter the time values “manually” as a sequence. Now plot x versus t and do a linear fit to the data. Show the equation of the fit in the chart as always.

Line 7: What is the slope? __________ cm/s. (Read it from the the equation of the fit)

Line 8: What is the velocity of the cart then? __________ cm/s.

Now by using the line segment tool measure the displacement of the car (by using the headlight to track the car we did before) from frame 25 to 50 >>>.

Line 9: What is the displacement of the car from frame 25 to 50 ? __________ cm.

Line 10: What is the time that elapses from frame 25 to 50 ? __________ s.

The average velocity of an object from an “initial” time point to another “final” time point is defined as where v̄ ti tf v̄ = ΔtΔx

and . Here is the position of the object at time and is the position of the object at time , therefore xΔ = xf − xi tΔ = tf − ti xf t = tf xi t = ti is the displacement and is the time elapsed. xΔ tΔ

Line 11: What is the average velocity of the car from frame 25 to 50? _______cm/s.

Line 12: What is the percent discrepancy between the average velocity and the velocity of the car (line 8)? ____%. Take the velocity as the reference in the calculation of the discrepancy.

Line 13: What is the y-intercept? __________ cm. (Read it from the the equation of the fit)

Line 14: What is the initial position of the car then? __________ cm. x0 =


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Now go to the first frame and measure the horizontal distance from the origin to the car. Note that origin is at the upper left corner, so simply measure the distance horizontally from the left edge of the image frame to the headlight (the white spot) >>>

Line 15: What is the measured value of the initial position of the car: _________ cm. x0* =

Line 16: What is the percent discrepancy between the measured value of the initial position of the car and the initial position of the car found x0* from the fit (Line 13)? ____%. Take (Line 15) as the reference in the calculation of the discrepancy. x0 x0*

Line 17: State your conclusion.

Your spreadsheet work should look like this >>>


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