Lab 2



Lab 2

Velocity

Overview

In this lab, we will explore the relationship between the position function and the velocity function of a moving object. In the lectures, we have seen that the velocity function v(t) is the derivative of the position function s(t) (and therefore s(t) is an antiderivative of v(t)). We will be examining these statements in this lab by using a motion detector to measure the velocity and position of a person walking.

Vocabulary used in this lab

• Distance: This is the distance an object is from the motion detector. The words distance and position will be used interchangeably in this lab.

• Displacement: This is “change of position,” that is, the difference between the initial and terminal coordinates the object. Displacement can be either positive or negative. (Think about what the difference between positive and negative displacement is.)

• Velocity and speed: The important thing to note here is that velocity can be either positive or negative, depending on whether motion is in the positive or negative direction. But speed = |v(t)| is never negative: it is the absolute value of the velocity. For example, a car’s speedometer is measuring speed and always reads positive, regardless of the direction you’re traveling.

• Total distance traveled: Perhaps this is best illustrated by an example. If you walk 2 meters forward and then 2 meters backward then your displacement will zero (the initial and terminal positions are the same so therefore the difference of coordinates is zero). But, you actually traveled 4 meters. In this case, the total distance traveled would be 4 meters. The “total distance traveled” is never negative.

Mathematics for this lab

The mathematics that we will need is the relationship between displacement and velocity:

• Displacement is equal to the definite integral of velocity v(t) over a time interval.

• Total distance traveled is equal to the definite integral of speed |v(t)| over a time interval.

These ideas are discussed in the textbook on pages 374-375.

Materials

• Computer with Vernier computer interface.

• Vernier motion detector. This will plug into the LoggerPro computer interface.

• Separate “Lab Report 2” sheet. This is what you will turn in. It is due one week after your scheduled lab day, by 4:30 pm, in the Mathematics Office (Cupples I, room 100).

Comments on the Motion Detector

The motion detector works by emitting short bursts of ultrasonic sound waves. You will hear a clicking sound when the detector is operating. The detector “listens” for the echo of these ultrasonic waves returning to it. The motion detector measures the time it takes for the sound waves to make the trip from the detector to the object and back to the detector. Knowing the speed of sound, the detector is then able to calculate the distance to the object.

The specifications on the detector state that it has a minimum range of 0.4 m. and a maximum range of 6.0 m. The detector seems to work best if there is a smooth flat surface for the sound waves to bounce off of.

Practically speaking, this means that as you collect data

• Your motion needs to be parallel to the sensor and not perpendicular to it (i.e., you will need to walk either away from or towards the sensor, or both).

• You will probably want to be holding a book or piece of cardboard in the sensor’s “line of sight” while you walk.

• If neighboring lab teams are moving while you are collecting data, your motion detector might pick up their motion as well. You will have to cooperate with your neighboring teams so that your motion detector is only picking up data from your walking.

• It is very important that the motion detector does not move. (What would the motion detector measure if it was moving?)

Procedure

The first thing to do is get the computer and motion detector ready.

Connect the motion detector to the Dig/Sonic 2 port of the Lab Pro. Make sure the Lab Pro is connected to the USB port of the computer. (It is likely that these have already been done for you.)

• Set the detector upright on the edge of the desk, facing out in a direction in which you have a clear path at least 2 m. long.

• Open the LoggerPro program. You will want to select “Lab Pro USB” for the port in the Setup Interface. It is possible (hopefully not likely) that LoggerPro will not recognize the hardware. If this happens, unplugging the Lab Pro and plugging it back in should take care of the problem.

• Open the experiment file for this lab. (Click on the “Open” button and open the file labeled “lab2_velocity.” (It is possible that LoggerPro will complain about this file – just ignore any complaint and click “OK.”)

• At this point, you should be able to collect distance and velocity data. Try playing around with the motion detector by pressing the “collect button” on the toolbar. (For example: turn the sensor to face the nearest wall and collect data—what happens? Turn the sensor around and collect data by moving your hand back-and-forth in front of it.) When you start collecting data, notice that velocity and distance data is graphed and the actual data appears in a table to the right of the graphs. Notice also that each time you start collecting new data, the data from the previous run is cleared. (Another way to clear recently collected data is to select Data, Delete Run from the menu.)

You are now ready to start collecting data that you will analyze.

• Decide who will monitor the computer (and hit the collect button) and who will be the “mover”—the mover will create distance and velocity data by walking back and forth in front of the motion detector.

• Your goal is to generate a data set that exhibits both positive and negative velocity during the five seconds of movement.

• There is a limited amount of free space in the lab for straight-line movement, so you will have to coordinate your data collection with your neighbors. Make sure you have a clear path before you start collecting your data.

• The mover should take an initial position standing in front of the motion detector, holding a book or a piece of cardboard in the detector’s “line of sight.” This will help to make a smoother velocity graph as you collect the data.

Before beginning, note the mover’s initial position. When the data collection stops, the mover should remain in position while his/her partner notes the terminal position. (An acceptable method to note these positions is to put a marker, like a pen, on the floor where the mover’s feet start.)

Measure and write down these positions (number of meters in front of the sensor); you’ll need this information later. (There will be a tape measure in the lab for you to measure the distances. You’re not looking for great accuracy—just something in the right ballpark. Remember that 1 meter is approximately 3.28 feet.).

• Click on the “Collect” button and gather the data. When the mover hears the motion detector making the clicking sound, s/he should start moving along a straight line in front of the motion detector. Remember your goal to create data with positive and negative velocity.

After collecting your distance and velocity data, you should make sure your data looks “reasonable”: for example,

• Is the graph of the velocity reasonably smooth?

• Does the distance data in the table look like what you were expecting? Do the distances more-or-less match up with what you noted for the initial and final positions?

• Is the velocity positive and negative when you were expecting it to be positive and negative?

• If your data looks questionable, collect another set of data. (The good news is that it doesn’t take much time to collect another data set if necessary.)

• If your data looks good, save it so that you can finish your report after the lab period if necessary. (If you didn’t bring a diskette, you could e-mail the data to yourself.)

At this point, you are ready to analyze your data.

• You should have two graphs – position (labeled “Distance vs. Time”) and velocity (labeled “Velocity vs. Time). You should print both of these graphs and also the table: activate each graph or table window before you print (so that you get a nice printout of each one separately).

• Click on the “Examine” button (Analyze – Examine in the menu). This should allow you to more easily read both the graph and the data (watch what happens when you move the mouse to the graphs and the data table). Using this, answer #1-3 on Lab Report 2.

Displacement

We want to calculate the displacement of the mover after 5 seconds. We can do this in three different ways:

a) by having LoggerPro estimate the integral of the velocity

function v(t) for 0 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download