# Position, Velocity and Acceleration

### Introduction

A very effective way to describe motion is to plot graphs of position with time and velocity with time. From such a graph, it is possible to determine in what direction an object is going, how fast it is moving, how far it has traveled, and whether it is speeding up or slowing down.

In this experiment, you will use a device called a 'motion sensor' to plot a real-time graph of your motion as you move across the laboratory.

The motion detector sends out a short burst of sound. The computer measures the time it takes for the sound to travel from the detector to an object and back. The computer also knows the speed of sound (about 350 m/s if you are interested). As you already may realize, knowing the speed and time allows you to calculate distance. You do not have to perform this calculation, but it is always nice to know how things work!

### Objectives

- Become proficient using the motion detector to observe and measure position, velocity, and acceleration
- Correctly ascertain an object's basic motion from graphs of its position versus time, velocity versus time, and acceleration versus time
- Practice using the LabPro and the motion detector. Become familiar with the Logger Pro software
- Practice a laboratory notebook entry

### Necessary Equipment

- Computer with LabPro and Logger Pro
- Low friction cart with track
- Motion detector
- Meter stick

### Helpful Tips and Suggestions

For more hints and technical information, visit the data sheet by clicking on the "Vernier Sensors" link in the resource menu to the upper right. Below are some common problems addressed in the resource.

- The Motion Detector does not work when objects are placed within 15 cm of the detector. This is because it takes a certain amount of time to switch between "send" and "receive" mode.
- The Motion Detector can only see objects that are directly in front of it.
- Sometimes other motion sensors, sound sources and/or irregular reflecting surfaces can cause problems.
- On most of the motions sensors, there is a switch on top that will enable you to switch back and forth between "cart" and "person walking."

### Preliminary Steps

For experiments I, II, and III you need to do the following:

- Connect the Motion Detector to your computer.
- Place the Motion Detector so that it points toward an open space at least 4 m long (if possible). Use short strips of masking tape on the floor to mark the 1 m, 2 m, 3 m, and 4 m distances from the Motion Detector.
- Open Logger Pro 3.8.4. To do this, click the Windows icon in the lower left, then go to "All Programs" and "Vernier Software."
- Prepare the computer for data collection: In Logger Pro, go to "File > Open." In the "Experiments" folder, go to "Probes and Sensors > Motion Detector > Motion Detector.cmbl." Open this file. Distance, velocity, and acceleration vs. time graphs should appear.
- Test out the equipment! Using Logger Pro, produce a graph of your motion when you walk away from the detector with constant velocity. To do this, stand about 1 m from the Motion Detector and have your lab partner click green Collect button. Walk slowly away from the Motion Detector when you hear it begin to click. Then walk toward the motion detector. There should be a difference! You need to know why later. Play around a bit until you feel comfortable, then move on to the experiment section below.

### Activity I: Graph Matching

Go to "File > Experiments > Physics with Vernier> Open," and open the "01b Graph Matching.cmbl" file. The distance vs. time graph shown below should appear.

Click the "Collect" button and then try to move in such a way that you replicate the graph on the monitor. Practice until you get it right! All members of your group should try this.

### Activity II. Constant Velocity

- Prediction: What should plots of "Position vs. Time" and "Velocity vs. Time" look like for a person moving at a constant velocity? Sketch each predicted graph in your notebook.
- Check Prediction: Now check your predictions in front of the motion sensor.
- Sketch the results next to the sketches of your predictions. Do all agree? If not, explain why your predictions were different.

### Analysis

The purpose of a curve fit is to use an equation to model real-life data. For example, if you believe that there is a direct relationship between the two variables that you are graphing, you would try applying a linear curve fit to that data. To do this using Logger Pro, select the Automatic Curve Fit option under the Analyze menu. You will get a popup curve fit menu. Under the General Equation list, you would then select the Linear Formula (click the option button by the "mx + b" equation). Then click "Try Fit". If the fit is acceptable, take a look at the coefficient area on the right side of that window. Logger Pro should tell you the best-fit values for m (the slope) and b (the y-intercept). Click OK to exit the curve fit window. A curve should appear on your graph, along with a box summarizing the coefficients in the best-fit formula.

Examine the kinematic equation:

*d = vt + d _{o}* (Eq. 1)

An examination of the variables should tell you that this is a linear equation in the form

*y = mx + b*

Therefore, if velocity is a constant, Eq. 1 is a linear equation where *d* is the vertical axis, *t* is the horizontal axis, *v* is the (constant) slope, and *d _{o}* is the y-intercept.

- In Logger Pro, determine the portion of the Position vs. Time graph in which you were moving at constant velocity.
- Highlight this portion of the graph. Under the "Analyze" menu, click "Linear Fit." This function determines the slope (m) of a best-fit line through the highlighted data. Does a straight line seem to fit the data?
- What did Logger Pro calculate the slope of your Position vs. Time graph to be? Be sure to include the units. The slope of any line is equal to the "rise" divided by the "run." In this case, the rise is distance, and the run is time. Using this idea, explain the units of the slope.
- Note the time at the beginning and ending of the highlighted section of the Position vs. Time graph. Try to highlight this same section of time on the Velocity vs. Time graph.
- Under the "Analyze" menu, click "Statistics." What did Logger Pro calculate as the average, or "mean" value of velocity over this interval?
- How does this value compare to the slope of the Position vs. Time graph? Use percent difference.
- Print this graph and staple it into your notebook.

### III: Constant Acceleration

**SITUATION 1: ROLLING DOWN A RAMP**

Close Logger Pro and reopen it. Do not load a graph matching file for this activity. Logger Pro should auto-detect the attached motion sensor. You will have to add a third graph, "acceleration vs. time," manually. To do this, go to Insert > Graph. The new graph should default to acceleration vs. time. Resize the graphs so that they are stacked vertically and will fit on one page. To do this go to Page > Auto Arrange.

Elevate one end of the cart track 8 to 10 cm. Keep the motion detector at the high end of the track. You will use one of the low friction carts provided. In your notebook, predict the motion of the cart AFTER you release it from near the top of the track. Make sure to indicate direction. Sketch your predictions for Position vs. Time, Velocity vs. Time, and Acceleration vs. Time graphs.

After you have made you predictions, perform the experiment. Sketch the results next to your predictions. Do all agree? If not, explain why your predictions were different.

**SITUATION 2: ROLLING UP AND DOWN A RAMP**

Repeat situation 2, except this time push the cart from the bottom of the track, towards the motion detector. Not so hard that it hits the detector at the top of the track! Be sure to catch the cart before it returns to the low end. Before collecting data, first make predictions as in Situation One.

After you have made you predictions, perform the experiment. Sketch the results next to your predictions. Do all agree? If not, explain why your predictions were different.

Don't delete these graphs. You will use them in the next activity.

### Analysis

Examine the kinematic equation:

*v = at + v _{o}* (Eq. 2)

An examination of the variables should tell you that this is a linear equation in the form

*y = mx + b*

Therefore, if acceleration is a constant, Eq. 2 is a linear equation where *v* is the vertical axis, *t* is the horizontal axis, *a* is the (constant) slope, and *v _{o}* is the y-intercept.

- In Logger Pro, determine the portion of the Velocity vs. Time graph in which the cart was moving with constant acceleration.
- Highlight this portion of the graph. Under the "Analyze" menu, click "Linear Fit." This function determines the slope (m) of a best-fit line through the highlighted data. Does a straight line seem to fit the data?
- What did Logger Pro calculate the slope of your velocity vs. time graph to be? Be sure to include the units. The slope of any line is equal to the "rise" divided by the "run." In this case, the rise is velocity, and the run is time. Using this idea, explain the units of the slope.
- Note the time at the beginning and ending of the highlighted section of the Velocity vs. Time graph. Try to highlight this same section of time on the Acceleration vs. Time graph.
- Under the "Analyze" menu, click "Statistics." What did Logger Pro calculate as the average, or "mean" value of Acceleration over this interval?
- How does this value compare to the slope of the Velocity vs. Time graph? Use percent difference.
- Print this graph and staple it into your notebook.

Examine the kinematic equation:

*d = 1/2at ^{2} + v_{o}*t +

*d*(Eq. 3)

_{o}An examination of the variables should tell you that this is a quadratic equation in the form

*y = Ax^{2} + Bx + C*

Therefore, if acceleration is a constant, Eq. 3 is a quadratic equation where *d* is the vertical axis, *t* is the horizontal axis, *1/2a *is the quadratic coefficient A, *v _{o}* is the linear coefficient B, and

*d*is the constant C.

_{o}- In Logger Pro, determine the portion of the Position vs. Time graph in which the cart was moving with constant acceleration.
- Highlight this portion of the graph. Under the "Analyze" menu, click "Quadratic Fit." This function determines the coefficients A, B, and C of a best-fit quadratic through the highlighted data. Does a quadratic function seem to fit the data?
- What did Logger Pro calculate the quadratic coefficient A to be? Be sure to include the units.
- Remember that the quadratic coefficient A is equal to
*1/2a*in Equation 3. Use this idea to determin the value of acceleration. - Note the time at the beginning and ending of the highlighted section of the Position vs. time graph. Try to highlight this same section of time on the acceleration vs. time graph.
- Under the "Analyze" menu, click "Statistics." What did Logger Pro calculate as the average, or "mean" value of Acceleration over this interval?
- How does this value compare to the value of acceleration you determined in #4 above? Use percent difference.
- Print this graph and staple it into your notebook.

Using percent difference, compare the acceleration found using the velocity vs. time graph to the acceleration found using the acceleration vs. time graph.

### Suggested items for consideration in your discussion

- From your analysis of the first situation of "Acceleration vs. time", can you argue that the cart moves with constant velocity? Support your answer by using information from all three graphs.
- From your analysis of the third situation of "Acceleration vs. time", can you argue that the cart moves with constant acceleration? Support your answer by using information from all three graphs.

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