PHYS111Q : Labs

2 Kinematics II

Introduction

In addition to reading this assignment, review Appendix B.

In class and homework we are studying a model describing the position and velocity of an object moving in one dimension with constant acceleration. Our model for the velocity of such an object as a function of elapsed time $\Delta t = t_2 - t_1$ is

$\begin{displaymath} v_2 = v_1 + a \Delta t, \end{displaymath}$

(2)

where

is the initial velocity of the object (the velocity of the object at time

), and

is the (constant!) acceleration of the object. The position of the object, according to the model, is given by

$\begin{displaymath} x_2 = x_1 + v_1 \Delta t + \frac{1}{2} a \Delta t^2, \end{displaymath}$

(3)

where

is initial position of the object (the position of the object at time

). You will test this model by studying a ball bouncing vertically while it is in flight.

Experiment

For the remainder of the semester, it is fine for one member of each lab group to collect data for the whole group. However, each student is responsible for getting the cmbl files and analyzing the data.

Mount the motion detector on the stand provided at a height of about 1.0 m above the lab table, facing down.
Run Logger Pro and use File -> Open ... to open the file Probes & Sensors -> Motion Detector -> Motion Detector.
Use the Data Collection ... button $\scalebox{0.7}{\includegraphics{LoggerPro3-data.eps}}$ . to set the sampling rate to 30 points per second. Also shorten the experiment length to the time it takes the ball to stop bouncing.
Start collecting data, and drop the ball directly below the detector, letting it bounce until it comes to rest. You will need to release the ball about 50 cm below the detector, because it is unable to ``see'' objects within about 50 cm of its front face. With repeated attempts, you should be able to collect five runs in which the ball bounces several times below the detector without leaving its field of view. The distance vs. time graphs should have a form that is familiar to you.
Trouble shooting : If the distance calibration seems to be way off or jumping around, adjust the sampling rate within the 20-50 points per second range. The detector may be mixing up reflected pulses, and this is a rate-dependent effect. Also be sure that there are no objects other than the floor and the ball within the ``field of view'' of the detector.

Analysis

It is far easier to tell whether data is linear than whether it follows a parabola or some other curve. We also have a linear fitting procedure which gives us uncertainties in slope and intercept. For these reasons, we always try to find a way to analyze data in a linear form.

According to our model (Eq. 2), if an object has constant acceleration, we expect its velocity vs. time graph to be linear with slope equal to its acceleration. We will focus our analysis on the velocity vs. time data, rather than trying to fit a parabola (Eq. 3) to the distance vs. time data itself. The following steps apply to the analysis of both experiments.

When used with the motion detector, Logger Pro should automatically show distance vs. time, velocity vs. time, and acceleration vs. time graphs.
If your velocity vs. time data appear to be compatible with a linear model, use theCurve Fit ... button $\scalebox{0.7}{\includegraphics{LoggerPro3-curvefit.eps}}$ to fit straight lines (mx + b) to linear segments of your data between bounces. If not, ask for help. Choose the best five to ten segments available. Record your fit results with uncertainties.
Save the entire Lab Pro section as a cmbl file. (You will need a representative graph with fits to hand in.)
Use a spreadsheet to calculate a ``best value'' of the slope of your velocity vs. time graphs and the corresponding uncertainty.

Before You Leave Lab

Discuss preliminary answers to questions 1-4 below with your instructor. (This means you'll need to complete your spreadsheet analysis in lab.)

Group Assignment

Hand in a representative velocity vs. time graph with fits, your spreadsheet, and answers to the following questions.

Sketch a representative set of distance, velocity, and acceleration vs. time graphs stacked vertically so that the time scales of the graphs line up. Include only three or four bounces in your sketch so that the details of a single bounce are easily discernable. Label the regions in which the ball is in contact with the table (the bounces) and regions between bounces during which the ball is rising and falling.
Which direction (negative or positive) corresponds to ``up'' on your graphs? Why?
Imagine that you missed this week's lab, and you are trying to make sense of the Logger Pro graphs collected by your lab group in your absence. Describe strategies (if possible) for identifying upward motion and downward motion on (a) distance vs. time, (b) velocity vs. time, and (c) acceleration vs. time graphs? Come up with strategies (if possible) that do not depend on knowing which direction corresponds to ``up'' (see question 2).
Is your ``best value'' of the slope of your velocity vs. time graphs consistent with the value given in your text?
Where would you see evidence in your data that the free-fall acceleration of the ball is not constant? Do you see any such evidence?
The distances plotted on your distance vs. time graphs are directly proportional to the raw data from the motion sensor. Logger Pro calculates velocity and acceleration values using the distance vs. time data for you. How would you do those calculations yourself?