PHY 221 Lab 2
Acceleration and Uniform Motion
Name:
Partner:
Partner:
Introduction:
Recall the previous lab
During Lab 1, you were introduced to computer aided data acquisition. You used
a device called a Motion Detector that measured the time delays of reflected ultrasonic
signals, which the computer used to calculate displacements. Very quickly, the computer
graphed the displacement-time data. The shape of the graph immediately gave you
information about the nature of the motion. During those activities, we looked at
constant speed motion. The displacement-time graph was a straight line with a positive,
negative, or zero slope. The velocity was correspondingly either positive, negative, or
zero with approximately zero slope. Do you remember what a negative velocity meant?
The Motion Detector establishes a coordinate axis
The displacement measured by the Motion Detector was always a positive
number. Walking away from the Motion Detector resulted in an increase in the
displacement. This is the positive direction. Similarly, walking toward the Motion
Detector (that is, walking in the negative direction) resulted in a decrease in the measured
displacement, though the position was never negative because you were always on the
positive side of the Motion Detector. Remember, Δs = xf – xi.
Any vector with a direction pointing away from the Motion Detector will result in
a positive number, and any vector with a direction pointing toward the Motion Detector
will result in a negative number. For example, when you walk toward the Motion
Detector, your velocity (a vector) is toward the Motion Detector and will be a negative
number. On the other hand if you walk away from the Motion Detector, your velocity
will be pointing away from the Motion Detector and hence will be a positive number.
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Materials:
Metal angle
PC with LabQuest interface for measuring instruments
Motion Sensor (also known as a "sonic ranger")
PASCO cart on aluminum tracks
Activity:
1. Average acceleration
Recall the previous laboratory. You used a sonic ranger to detect the motion of a cart
rolling on an aluminum track. The refresh your memory, open Logger Pro 3.8.3. Set up
the readout program as in Lab 1 with two graphs, position versus time and velocity versus
time. Set the collection time to 5 sec if it is not already set to 5 sec. Collect data and
ensure that the sonic ranger detects only the cart all along the track and not part of you or
another object along the track.
Start moving the cart gently and smoothly back and forth along the track driving it with
your fingers, keeping parts of your body that might be picked up the motion detector out
of the way. Cover the range of about one meter in this back and forth motion. Don’t get
too close to the motion detector since it has some dead space right in front of it (allow at
least 15 cm separation). Collect the data. You should make 2 full cycles in the back and
forth motion during the data collection period. You should see a wave-like graph on your
screen. Try to make it smooth without any sudden steps or spikes in the entire time range.
Once you are happy with the recorded result, use PRINT to copy your graph to the
printer. Do not forget to improve your ability to read the graph by “Autoscale”ing the
graphs before you print.
As in the last lab, find an instant in time on your computer copied graph (we will refer to
it as “A”) when the cart was at the middle point between the closest and the farthest
distance from the motion detector. Since the cart was at this position several times,
choose one when the cart was moving away from the motion detector. You also want to
see the cart come back to this position, at least once, at a later recorded time. Indicate the
point A on your printed graph. Now go to the next time moment when the cart came back
to the same position, point “B” and indicate that on your graph as well. Now find the
instant in time between A and B when the cart was furthest from the motion detector.
This will be time moment “C” and indicate it on your graph.
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Now analyze the v(t) vs. t graph in a similar way to that we used to analyze x(t) vs. t in
the last laboratory. Using the “v=?” cursor applied to the graph of velocity calculate
average accelerations in the time intervals AC and CB.
aAC= ∆vAC / ∆tAC = (vC- vA)/(tC- tA)
aAC=
aCB=
As in the previous lab calculate the average velocities for the same time intervals.
vAC=
vCB=
Q1. Did the sign of average velocity change from AC to CB? Why?
Does the sign of velocity indicate the direction of motion?
Did the sign of the average acceleration change from AC to CB?
Does the sign of acceleration indicate the direction of motion?
2.
Instantaneous acceleration
To move on to the concept of instantaneous acceleration, activate the “tangent” cursor on
the graph of velocity. Go to the point of maximal velocity and record the slope of the line
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tangent to the velocity graph. That will give you the instantaneous acceleration at that
instant in time:
a(vmax)=
Now go to the point where the velocity is zero (in between the maximum and minimum
velocity) and read out the instantaneous acceleration:
a(v=0) =
Finally go to the point where the velocity is the most negative:
a(vneg)=
Q2. When the cart is moving fast, does it mean that the acceleration is large?
When object momentarily stops (zero velocity), is the acceleration zero?
3.
Graph of instantaneous acceleration a(t)
Add the graph of acceleration versus time to your screen (this will be the third graph on
your screen). To do this: Go to ‘Insert’ on Menu. Click on Graph which will add the
graph. Go to Page and click ‘Autoarrange’(or Ctrl r). Click on labels of graphs to switch
between position, velocity and acceleration as needed. The best would be top to bottom
Position, Velocity and Acceleration. PRINT the graphs.
Q3. Compare, using “tangent”, the slopes of the velocity curve with the “a=”s on the
acceleration curve at the same times. Are they the same? Why?
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We could continue our procedures and analyze this graph the same way we did
previously, that is, use the “tangent” cursor to find the derivative of acceleration with
respect to time, da/dt, at different moments in time. It turns out that such activity is not
very useful from the dynamical point of view, and we have no need to go beyond the
second derivative of position over time (a=dv/dt=d(dx/dt)/dt=d2x/dt2). While
kinematics deals with description how objects move, dynamics deals with explanation
why they move. We will explore these relationships during an upcoming laboratory.
4.
Constant position
An object at rest doesn’t move. Yet all kinematical variables: position, velocity and
acceleration can still be measured. Thus, being at rest is just a very special type of
motion. To investigate motion with constant position, use the three graphs; position vs.
time, velocity vs. time and acceleration vs. time that you have set up. Make sure that the
sonic ranger detects the cart all along the track. Collect data for the cart at rest. PRINT the
three graphs.
A measurement process always carries some inaccuracies called measurement errors.
They may show up as oscillations of the measured value around the true value,
fluctuating with time.
Q4. Taking into account possible measurement errors answer the following questions:
Is the position constant with time?
Is the velocity constant with time?
Is the acceleration constant with time?
Let us now quantify what the values of position, velocity and acceleration are. Instead of
reading off values of these quantities at one particular moment in time, we can average
the measurements for different times (as long as we believe that the quantities are
constant). The averaging procedure results in much smaller measurement errors. This can
be easily accomplished in our program by using the “STAT” button. It is useful to
exclude the initial and final time moments from your graph while averaging, since they
tend to have larger measurement errors.
To measure position within selected time ranges, left-click on the position vs. time graph
with the pointer positioned at the lowest time you want to include in the average (the
pointer should be within the rectangular area defined by the graph axes), hold the mouse
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button down and stretch the selection range to the largest time to be included in the
average. Vertical bars superimposed on the graph, connected by horizontal dashed lines,
will indicate the selected range. If you are not happy with the selection just repeat the
procedure. To average the position measurements within the selected range click on the
“STAT” button. The average value (called also the “mean”) will be displayed in a small
box superimposed on the graph. What is the average position value?
x=
Now determine the average velocity for the same time interval (you will need to stretch
the selection bars on the velocity graph before using the “STAT” button again). Do not
use scientific notation to write down the value; just put the right number of zeros after the
decimal point. Since velocity is obviously zero for any object at rest, small but non-zero
measured values just gives you an idea about the measurement error on the average
velocity.
v=
Repeat the averaging process for the acceleration:
a=
5.
Constant velocity
Now try to realize motion in which position changes but velocity remains constant. Give
the cart a strong push and let it go. Be sure someone is catching the cart. PRINT the
graphs. Identify and mark parts of your graph that correspond to this type of motion.
Indicate the range of values of each kinematical variable on the vertical axis.
Q5. How does the position depend on time in this case?
What can you say about the acceleration?
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Determine average value of acceleration from the a(t) vs. t graph. You are likely to
observe much larger measurement errors than in the previous activity. Select a time
interval in which acceleration fluctuates around the same value.
a=
Determine the average value of the velocity. Use the same time interval as selected for
the acceleration.
v=
Since the position is not constant, an average value of position is not very interesting. For
motion with constant velocity, position depends on time via the following formula: x(t) =
x0 + v t , where v is the velocity and x0 is the position at t = 0. To verify this formula you
“fit” a line to your graph. To “fit” means to adjust values of free parameters, in this case
x0 and v, so the formula gives the best match to the collected data. The program can do
the fit for you! Select the time range on the position graph to be used in the fit (the same
way you used to select it for averaging). Use the same interval as used to calculate the
average values of a and v listed above. Go to “Analyze” menu and select “Linear Fit”.
The fitted line will be superimposed on your data extending also beyond the selected time
range.
Q6. Does the fitted line match your data well within the selected range?
The fit results are displayed in the small window which says: “y = mx+b”. In our case
“y” (vertical axis) is the position denoted by us as x, and “x” (horizontal axis) is the time
denoted by us as t (this may be confusing!). The y-intercept “b” is our x0 , and “m” is the
slope of the line i.e. our v. What is the value of velocity determined by the fit?
vfit =
Q7. Is it close to the value you obtained by averaging the velocity data? What criteria
did you use to decide whether vfit is close to vavg?
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During the time interval between when you let go of the cart and stopped the cart, fit the
velocity curve with a linear fit and find the slope corresponding to the acceleration of the
cart and compare to the average acceleration over the same period.
From the slope, m, of the linear fit of the velocity curve
afit =
From the acceleration curve using STAT function of the time the cart was moving
uniformly,
aavg =
Q8. Are these values similar?
6.
Constant acceleration
Finally we will investigate motion in which position and velocity vary but acceleration
remains constant.
To produce constant acceleration, we will take advantage of gravity, which produces a
constant force on an object. According to Newton’s Second Law, which we will study
later, acceleration is proportional to the force, so if the force is constant, the acceleration
remains constant.
Tilt the track by putting a metal bracket under the end of the track where the sonic ranger
is attached. Make the longer side of the bracket vertical to obtain a larger tilt. Position the
cart close to the sonic ranger (but not too close, because of the dead space in the sonic
ranger readout). Collect the data while you allow the cart to roll down freely. Please
catch the cart. Be sure the cart does not roll off the table. PRINT the graphs.
Identify and mark parts on your graph that correspond to this type of motion. Indicate the
range of values of each kinematical variable in the vertical axis.
Q9. How does the velocity depend on time in this case?
Q10. Any ideas what kind of function would describe the dependence of position on
time?
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Determine the average value of acceleration from the a(t) vs. t graph. Select a time
interval in which acceleration fluctuates up and down around the same value.
a=
In this motion, the velocity should be changing linearly with time: v(t) = v0 + a t. Perform
a linear fit to the graph of velocity in the same time interval as used in the above average.
Record the fitted slope “m” giving you the acceleration and the fitted intercept “b” for
future reference:
afit =
v0 =
Q11. Is the fitted value of the acceleration close to the average value determined
above? How are you deciding what is ‘close’?
The position should have a quadratic dependence on time: x(t) = x0 + v0 t + ½ a t2 . To
verify this formula select the same time range as on the graphs of acceleration and
velocity. Go to “Analyze” menu and select “Curve Fit…” In the function selection
window choose: “At^2+ Bt + C” and click on the “Try Fit” button. “C” is our x0, “B” is
our v0 and “A” is equal to ½ a. Click OK to close the fit control window. The fit results
should be superimposed on your graph. Does the quadratic function describe your data
well?
From the displayed fit results, read off the fitted acceleration (A=½ a) and initial
velocity:
afit =
v0 =
Q12. Do they agree with the values obtained by the fit to the graph of velocity?
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Let’s learn about some other tools available to us in LoggerPro. Looking at your v-t
graph, find a region where there are no obvious glitches in your data, i.e. no sudden or
sharp spikes. Select and shade a region by clicking and dragging the cursor across the
graph. This will highlight a portion of your v-t graph. Under Analyze, choose the Integral
tool. A box displaying the area under the shaded v-t curve should appear in your graph.
You can click and drag this box to a convenient location out of your way.
Q13. What is the physical meaning of the value of this integral?
Q14. Using the data on your screen, how can you verify that the integral is correct?
Q15. Please show your calculations below.
Q16. You need not do this, but what would an integral of some region under you’re a-t
graph give? How could you verify this?
7.
Different initial conditions for motion with
constant acceleration (optional if time permits)
Without changing the tilted track inclination used for Activity 6, place the cart near the
lower end of the track. Give the cart a push upwards on the track, until it stops without
getting too close to the sonic ranger, and then rolls back down. You may need to practice
until you manage to obtain the desired trajectory. PRINT the graphs and mark them as
before after you have finished the rest of the fitting described on this page. Repeat the
averaging and fit procedures as in Activity 5.
Find the average acceleration from the a(t) vs. t graph: (keep the statistics fitting box on
the screen).
a=
Linear fit to the v(t) vs. t graph: (keep the fitting box on the screen)
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afit =
v0 =
Quadratic fit to the x(t) vs. t graph: (keep the fitting box on the screen)
afit =
v0 =
Q17. Do the acceleration and initial velocity values determined in different ways
agree?
Is the initial velocity close to the value in Activity 6? Please explain how you determine
the meaning of ‘close’.
Is the value of acceleration close to the value in Activity 6?
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Pre-Lab Questions
Print Your Name
______________________________________
Read the Introduction to this handout, and answer the following questions before you come to
General Physics Lab. Write your answers directly on this page. When you enter the lab, tear off this page
and hand it in.
1. You stand in front of the Motion Detector and let Logger Pro record your
displacement. Is the displacement Logger Pro assigns you positive or negative?
2. If you walk toward the Motion Detector, does the displacement that Logger Pro
records get larger or smaller?
3. Suppose you walk towards the Motion Detector and let Logger Pro record your
velocity. Is the velocity that Logger Pro calculates positive or negative?
4. The same as the previous question, but you are walking away from the Motion
Detector instead of towards it.
5. Suppose the Motion Detector is mounted on the ceiling and pointed so that it can
record the displacement of objects beneath it. In the resulting Logger Pro graphs
(with displacement on the vertical axis and time on the horizontal axis), does the
positive direction represent up toward the sky or down toward the center of the earth?
6. Write the kinematic equation that gives displacement as a function of time for an
object that is experiencing motion with constant acceleration. (Hint See page 9.)
7. Suppose you used Logger Pro and the Motion Detector to graph Displacement versus
Time for an object thrown vertically, and, further, you had Logger Pro do a fit to the
graph to determine the equation that describes Displacement as a function of Time.
The general form of the equation that Logger Pro gets from doing the fit to the graph
is x = C + Bt + At2, and Logger Pro tells you the values of the constants A, B, and C.
Suppose Logger Pro says that C = 1.2, B = 0.3, and A = 4.82. Use this information
along with your answer to pre-lab question 6 to determine the acceleration of gravity,
g, in m/s2. (The answer to this question is not 9.8 m/s2.)
Which way was the object thrown? Up or down? How can you tell?
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8. In the blank graph immediately below, take the origin of the coordinate system to lie
on the floor of the lab, take up to be the positive direction, and draw the shape of a
Displacement versus Time graph for a ball that is thrown straight upward but then
falls back to the ground. Label the axes. (This is not discussed in the Introduction to
this handout. Refer to your general physics text, if necessary.)
2.0
1.5
Displacement
(meters)
1.0
0.5
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.7
0.6
0.8
0.9
1.0
Time (seconds)
Graph for Pre-Lab Question 8
9. The same as the previous question, but this time take the origin of the coordinate
system to be on the ceiling in the lab, and take down to be the positive direction.
2.0
1.5
Displacement
(meters)
1.0
0.5
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Graph for Pre-Lab Question 9
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0.7
0.8
0.9
1.0
Time (seconds)