Motion in One Dimension

In this lab you will use a motion sensor and a LabQuest data collection device to plot and analyze the motion of a rolling cart moving along a dynamics track. You will explore the relationships between an object’s position over time and its velocity and acceleration. Your objective will be to develop a method for predicting an object’s position, given velocity and acceleration data.

Background We define displacement as the change in an object’s position: Displacement:

∆x = x f − xi

where the subscripts f and i refer to the object’s position at the end (f ) and at the beginning (i) of the time interval of interest. Time interval:

∆t = t f − t i

Note that displacement may or may not be the same as distance traveled. To measure the rate at which displacement changes, we define average velocity as: Average velocity:

v=

∆x ∆t

(1)

This is a vector quantity, having a direction as well as Δa magnitude. In one dimensional motion, there are only two directions: towards +x or towards –x along the coordinate axis. The direction is determined by the sign of Δx. We then define instantaneous velocity as the average velocity when ∆t is infinitely small. We can then think of an object as having a velocity at an instant of time: Instantaneous velocity:

v = vaverage when Δt is infinitely small.

Acceleration tells us how velocity changes. Like velocity, it has a direction. It is defined as: Average acceleration: where

a=

∆v ∆t

(2)

∆v = vf – vi.

The instantaneous acceleration a is then defined as the acceleration at an instant of time:

Instantaneous acceleration:

a = Average acceleration as ∆t

0.

As we shall see, acceleration is the key to describing motion. Knowing the acceleration allows us to determine both the velocity and the position of an object at any instant.

Constant Accelerations In general, acceleration can change with time, just as position and velocity do. But there are many examples of motion where the acceleration does not change as the object moves. We call this constant acceleration, or uniform acceleration. Most problems we will look at will be in this category. (But don’t forget that these are special cases.) There are two very useful observations about constant acceleration. For constant acceleration, •

Average acceleration and acceleration have the same value. (aaverage = a)

•

Average velocity (

∆x ) ∆t

=

Mean velocity (vf + vi)/2

(3) (4)

Experiment 1: Generating Position-Time Graphs Your first task is to generate position-time graphs for the following verbal descriptions of motion. 1. Set up the LabQuest and motion sensor to track a rolling cart moving along a dynamics track. You will roll the cart by hand. Set LabQuest to display a position graph only, and set its timing duration for 10 seconds. Set the motion detector on a block or other support at one end of the track and adjust its position so that you can compensate for the motion detector’s 3-4 cm offset: place the cart at 20 cm on the track, turn on timing, and adjust the position of the sensor until it correctly reads the position of the cart as 20 cm. Refer to “Setting up the Motion Detector and LabQuest” at the end of this document. For each of the verbal descriptions below: 2. Roll the cart to create the motion described. Once you are satisfied, make a quick sketch of the position graph in your lab notebook. Do not spend a lot of time on this – just get the character of the motion down. 3. Sketch a prediction of the corresponding velocity graph. 4. Display the velocity graph. This will be very noisy, but try to determine the average trend of the plot. Did your prediction capture its essential features? 5. During any interval where the cart was in motion, calculate its average velocity using the position data (Δx/Δt) and compare this to the velocity values in the velocity plot. Also, use the tangent tool (under “Statistics” menu) to compare the slope of the position plot to the velocity value at that time. 6. Call up the acceleration plot. This will be even more noisy than the velocity plot. At what times does the acceleration plot have its largest values (positive or negative)? Except for times when the cart is suddenly starting or stopping, what should the acceleration plot look like?

Note: Each team member should generate at least one position-time graph!

Motion A: Create a position-time graph in which the cart starts at position x = 20 cm at time t = 2 s and moves at slow but constant speed away from the motion sensor for 6 s, then abruptly halts. Stop the timer at t = 10 s. The reason for the delayed start is to give you time to position the cart 20 cm from the motion sensor. Motion B: Create a position-time graph in which the cart stays at rest at x = 20 cm for 2 s, then moves at a fast but constant speed away from the motion sensor for 6 s, then abruptly halts. Stop the timer at t = 10 s. Note: The cart should not go beyond x = 100 cm. Motion C: Create a position-time graph in which the cart starts at rest at x = 100 cm, starts moving at t = 2 s and moves at a fast but constant speed toward the motion sensor for 6 s, then abruptly halts. Stop timing at t = 10 s. Note: The cart should not go beyond x = 15 cm.

For the following, sketch the position and velocity graphs before rolling the cart. Again do not spend a lot of time on the sketches, just try to get them qualitatively correct. Motion D: The cart stays at rest at x = 20 cm for 2 s, then moves at a constant speed to x = 60 cm in 2 s, rests for 2 s, then moves at a constant speed to x = 15 cm in 2 s, then abruptly halts. Stop the timer at t = 10 s. Conclusions – Motions A, B, C and D: What do you conclude about the relation between position, velocity and acceleration at constant speeds? What is the relationship between acceleration and speed? Between acceleration and velocity?

Motion E: Read the following scenario, but before creating a position-time graph, sketch a prediction of what you think the velocity-time and acceleration-time graphs will look like (no numbers required yet): The cart stays at rest at x = 20 cm for 2 s, then starts very slowly and speeds up continuously until it reaches x = 110 cm at t = 8 s, then abruptly halts. (At the end the front of the cart will drop off the track, but that’s OK.) Stop the timer at t = 10 s. •

When you have made your prediction, replicate this motion with the cart, aiming for the following “checkpoints”: x = 20 cm at t = 2 s, x = 30 cm at t =4 s, x = 60 cm at t = 6 s, and x = 110 cm at t = 8 s.

Pointer: Start very slow! But remember to keep speeding up. Do not worry about meeting the checkpoints exactly – you just want a plot in which the speed keeps increasing. It should look something like this:

•

What is different about your position-time graph in this scenario than in Motions A – D?

•

Examine the velocity-time graphs. Does it look like you expected? Comment/discuss in your conclusion for this section.

•

Using the position-time graph, calculate the average velocity of the cart during each 2second interval between t = 2 s and t = 8 s. Compare this to the actual velocities halfway through each interval.

•

Using your velocity-time graph, calculate the average acceleration for the interval t = 2 s to t = 8 s. Compare this with the values in your acceleration graph.

Conclusion: What do you conclude about the relation between position, velocity and acceleration when speed is not constant?

Experiment 2: Analyzing a Position-Time Graph Examine the position-time graph below. Your task here is to recreate this graph, then determine the velocity-time and acceleration-time graphs that the position graph implies. This time, your graphs should be quantitatively as well as qualitatively correct. Note for all of the graphs in Experiments 2, 3, and 4: • • • •

Each plot starts at x = 30 cm. Each plot has one or more distinct segments. Each segment consists of one or more 2-second time intervals. It will be impossible to recreate the graphs perfectly; focus on the type of motion within each interval. Do your best in a reasonably short time to reproduce the characteristics of the motion: position, speed and direction at the given times.

Position (cm)

Experiment 2 Position-Time Graph

100

80

60

40

20

Time (sec) 2

6 Parabolic up here.

10 Parabolic down here.

First, deselect the velocity and acceleration displays on LabQuest. Recreate the position-time graph above. After you are satisfied, set up a table in your lab notebook like the one below and record the initial and final positions of the cart at each 2 s time interval based on YOUR position-graph. Remember that the final position of one time interval will be the initial position of the next.

Time interval

xinitial

xfinal

∆x

vinitial

vfinal

vaverge

aaverage

0 – 2 sec 2 – 4 sec 4 – 6 sec 6 – 8 sec 8 – 10 sec

•

Calculate the displacement (∆x) for each time interval. From this, find the average velocity for that interval. Use this information to find the initial and final velocities for that interval; from that you can calculate the average acceleration.

•

Draw your predicted velocity-time graph as straight-line segments for each 2-second interval. Try to get the numerical values reasonably correct as well as the general shape. NOTE: Will the velocity at each instant of time be equal to the average velocity during that time segment? If not, why not? What can you learn about the velocity-time graph based on your calculation of the acceleration? (Hint: vaverage only tells you the approximate midpoint velocity, if acceleration is not constant.)

•

Finally, draw an acceleration-time graph. Keep in mind that you may assume that the acceleration is roughly constant during each time segment.

•

Now open your velocity-time and acceleration-time graphs and compare with your prediction. Annotate your predicted velocity graph with any corrections needed.

Conclusion: Describe the motion based on the acceleration and velocities of the cart.

Experiment 3: Analyzing a Velocity-Time Graph Now try a velocity graph. Here, you want to find what motion (what position-time graph) is implied in this velocity graph.

Velocity

Experiment 3 Velocity-Time Graph

(cm/sec) 25 15

5 -5

2

4

6

8

10

Time (sec)

-15

First, make your prediction. To aid you in this, again set up a table in your lab notebook. Make sure to label with appropriate units! Time interval

vinitial

vfinal

vaverge

aaverage

∆x

xinitial

0 – 2 sec 2 – 4 sec 4 – 6 sec 6 – 8 sec 8 – 10 sec

Assume the object starts at 30 cm as before. This time, you will have to predict the acceleration and displacement during each time interval. Some hints: 

Sketch the velocity-time graph in your lab notebook.

xfinal



Record the initial and final velocities for each time interval.



Use vaverage = (vfinal + vinitial)/2. Note that this is valid only when acceleration is constant. (How can you tell from the velocity plot whether or not acceleration within a segment is constant?)



Use the average velocity to calculate the accelerations in each interval.



Use vaverage = ∆x / ∆t to determine the displacement during each time interval. From this, determine the initial and final positions for each interval.



Draw your predicted position-time graph in your lab notebook. For intervals in which velocity is changing (a not zero), consider carefully! What does this mean for the shape (slope) of the position graph during those times? You may want to break such intervals into sub-intervals of one second to get a better prediction of how the cart should be moving in those intervals.

Reproduce your predicted position-time graph using LabQuest. Again don’t expect to be able to move the object exactly as indicated by the graph. Just get the basic type of motion down: is it speeding up, slowing down, going forward, going backward, and roughly how fast in each case. Call up the velocity graph and compare this with the given velocity graph. How well did you do? At what time during any interval is the actual velocity (instantaneous velocity) equal to the average velocity? This is a useful observation to remember! Finally, call up the acceleration window and compare your calculated values with the measured values. (Ignore the spikes that occur because of the abrupt transition between some of the time segments.) Conclusion: Describe the motion based on the acceleration and velocities of the cart.

Experiment 4: Analyzing an Acceleration-Time Graph Now try an acceleration graph. Your goal is to predict the motion (velocity and position) produced by this acceleration. Graph 6 Accel

(cm/sec2 ) 5 3

1 -1

2

4

6

8

10

Time (sec)

-3

-5

1. Predict the velocity graph. Again, set up a table for the data for each interval. This time, you need to predict vinitial and vfinal for each interval. Work backwards from the given acceleration to find the velocities. Draw the velocity graph in your notebook.

Graph

Time interval

7 or 8

0 – 2 sec 2 – 4 sec 4 – 6 sec 6 – 8 sec 8 – 10 sec

vinitial

vfinal

∆v

aaverge

2. On the basis of your velocity table and graph, you can predict the position table and graph. Assume

the object starts from rest at x=30 cm. Work backwards from the initial and final velocities in each interval to the average velocity for that interval to the displacement for that interval. From this, you can find xfinal for each interval. Draw your predicted position graph in your lab notebook. Graph

Time interval

7 or 8

0 – 2 sec

xinitial

xfinal

∆x

vaverge

2 – 4 sec 4 – 6 sec 6 – 8 sec 8 – 10 sec

3. Call up the position graph on LabQuest and reproduce your position graph. 4. Call up the velocity and acceleration windows and confirm (or revise) your results by comparing these graphs with your predictions and calculations.

Conclusion: Describe the motion based on the acceleration and velocities of the cart.

Motion Sensor and Vernier LabQuest Set Up •

Plug the motion sensor cable into the motion sensor and into a digital socket on the side of the LabQuest module. Turn on the LabQuest. It may take a full minute to boot up. Once booted and if necessary, press the home button on the front face of the LabQuest, then tap the icon for “Labquest App”.

•

To select features on the LabQuest, always use the stylus or the plastic cap of a pen. Never touch the screen with ink, pencil lead or pencil eraser.

•

Set the duration for data taking: At the Sensor screen (this has a meter icon), tap the duration field on the right side of the screen and set the data-taking duration to 10 seconds.

•

Call up the graph screen by tapping the graph icon.

•

Set the motion sensor at one end of a dynamics track and tilt its screen so that it points parallel to the track. The sensor sends out a narrow cone of ultrasound pulses and determines the distance to an object from the time it takes the echoes to return. It should key on any object that is directly in front of its screen, but can sometimes pick up spurious echoes from objects to one side or the other. Clear all objects from the sides of the track to avoid spurious signals.

•

Run a test trial to confirm the system is working and to calibrate the graph screen. Set the rolling cart about 15 cm in front of the sensor, tap the green “go” button, and move the cart by hand the full length of the track. You should see the position and velocity data being plotted as you move the cart. Once the timing is finished, LabQuest should set the scale of the graphs so that the data fills the screen.

Some notes on the sensor and LabQuest:

•

The motion sensor collects data on position, velocity, and acceleration. You can select which data set LabQuest will plot by tapping the label on a graph axis and selecting the data set from the menu.

•

You can also select how many graphs to display at once by tapping on the “Graph” menu and the “Show Graph” option. You can change graph parameters at the “Graph Options” screen.

•

The motion sensor will not detect objects closer to its screen than 15 cm, so arrange your set-up so that the sensor is at least 15 cm behind your starting point.

•

The motion sensor also has an automatic offset of 3-4 cm. This means an object 20 cm will be plotted as being 23 or 24 cm away. If this is a problem, set the sensor at 3 or 4 cm on the track so that an object at 20 cm on the track will be correctly plotted.

•

It is also possible to set the zero point at a desired position. Go to the Sensor screen, tap the red sensor field and select “zero”.

•

LabQuest always graphs data in units of meters and seconds. For our purposes, it will be more convenient to work in centimeter units, so record the data in centimeters: 22.7 cm rather than 0.227 m.