# 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

### Necessary Equipment

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

### Helpful Tips and Suggestions

For more hints and technical information, visit the data sheet by reading this Motion Detector resource. 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 detectors, sound sources and/or irregular reflecting surfaces can cause problems.
- On most of the motion detectors, there is a switch on top that will enable you to switch back and forth between "cart" and "person walking."

### Preliminary Steps

- Connect the motion detector to your computer.
- Place the motion detector so that it points toward an open space.
- 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: Graphical analysis of an object undergoing constant acceleration

In this lab, we will examine four graphical relationships between acceleration, velocity, and displacement vs. time.

1) The area under an acceleration vs. time curve is equal to a change in velocity.

2) The area under a velocity vs. time curve is equal to a change in displacement.

3) The slope of a displacement vs. time curve is equal to the velocity.

4) The slope of a velocity vs. time curve is equal to the acceleration.

### Prediction

Don't click the "collect" button on this activity. There is a motion detector at the high end of the track. You will use one of the low-friction carts provided. 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.

Make predictions of what the following three graphs might look like for the rolling cart:

Position vs. Time

Velocity vs. Time

Acceleration vs. Time

Show these predicted graphs to your instructor.

After you have made you predictions, perform the experiment. Quickly sketch the results of the three graphs. Do they agree with your predictions? Discuss any differences.

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

### Analysis

Examine the kinematic equation that models velocity as a function of time.

*v = at + v _{o}* (Eq. 1)

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

*y = mx + b*

Therefore, acceleration is the slope of the velocity vs. time graph. 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.

Eq. 1 can be restated thus:

*v - v_{o} = at*

This indicates that the change in velocity is equal to the area under the acceleration vs. time curve.

- In Logger Pro, determine the portion of the Velocity vs. Time graph in which the cart was moving with constant acceleration. This began when you released the cart on the way up, and ended when you caught the cart on the way down.
- 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 as the slope of your velocity vs. time graph? What are the units of the slope? (Hint: Slope is equal to the "rise" divided by the "run.")
- 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 a percent difference calculation.
- For that same highlighted time interval of the acceleration vs. time graph, under the "Analyze" menu, click "Integral." This gives you the area under the acceleration vs. time curve.
- How does this value compare to the change in velocity over the same time interval? (What is the velocity at the beginning and at the end of this time interval?) Use a percent difference calculation.

Examine the kinematic equation that models displacement as a function of time.

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

*d*(Eq. 2)

_{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. 2 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
*a/2*in Equation 2. Use this idea to determin the value of acceleration. Check to see that it is similar to the previously-determined values of acceleration. - During this same time interval, determine the integral of the velocity vs. time graph to determine the area under the curve.
- How does this value compare to the change in displacement over the same time interval? (What is the displacement at the beginning and at the end of this time interval?) Use a percent difference calculation.

Print these graphs for your report.

### Post-lab Questions

Please show all your work for credit. You will not receive credit for just writing the answer without showing all the steps and equations. Untidy work will not be graded.

1) As the cart rolled up the track, what was its acceleration at the highest point, when the cart came to a stop and reversed direction?

2) Using only the velocity vs. time graph, can you argue that the cart moves with constant acceleration? Support your answer.

Use the figure below to answer questions 3-4. Assume that each major division of the distance axis represents one meter.

3) From t = 4 s to t = 5 s, what is the velocity of the cart?

4) What is the velocity at 8 s?

5) From t = 2 s to t = 10 s, what was the displacement of the person who created this graph? Justify your answer.