Uniform Motion Lab Report Answers

Uniform Motion Lab Report Answers

Uniform Motion
Department of Chemistry and Physics
PHYS 1201
Lab 4: Uniform Motion

Preparation: Read Physics: Third Custom Edition for Mount Royal University, Sections 1.2,
1.3, 1.4, 2.1, 2.2, and 2.3. Read the Lab Standards Manual section titled Estimating Uncertainties
of Slopes Using a Graphical Method
Equipment: Spark Table Data printout sheet, ruler, PhET simulation (Warm-Up Questions).
Learning Goals: Students will be able to explain the characteristics of uniform motion and
recognize uniform motion by looking at a position-versus-time graph.
Experimental Skills: Students will learn to graph coordinates by hand, include error bars, create
a best fit line, and calculate slope in order to determine average velocity.
Introduction
Position, Displacement, and Average Velocity
A change of position is called displacement. The displacement Δx of an object as it moves from
an initial position xi
to a final position xf
is
Δx = xf – xi
Graphically, Δx is a vector arrow down from position xi
to position xf
. The time interval
Δt = tf
– ti
is the elapsed time for an object to move from one position xi
at time ti
to another
position xf
at time tf
.
The average velocity of an object during the time interval Δt, in which the object undergoes a
displacement Δx, is the vector:
Vave
= Δx / Δt (Equation 1)
The average velocity vector points in the same direction as the displacement vector. This is the
direction of motion.
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Lab 4: Uniform Motion
Experiment
In this lab, we will simulate uniform motion using a puck launched on a leveled air table. A
paper is placed underneath the puck, on top of carbon paper. A spark timer leaves a series of dots
on the paper as the puck moves at regular time intervals. These dots are used to investigate the
motion of the puck. In today’s lab, the spark timer was set to 100 ms.
Question 1:
1) Use the printout of the spark timer data, titled Lab 4 Spark Table Uniform Motion Data. You
can find this in the Week 4 folder as well as in the package of essential printables. The
beginning of the puck’s path is marked with a “O”, so that you know which way to orient the
page. Choose 7 consecutive dots that best represent the uniform motion phase of the puck
trajectory, while reducing uncertainties. Be sure not to choose the very first point on the
paper, as you don’t know whether the puck was stationary for some time before being
released. If it was, then the puck was not in motion for the entire 100 ms interval between
creating this point and its closest neighbour. Label these points 0 to 6 on the dotted paper.
2) The position of dot 0 will be defined as . Measure the positions using a ruler as shown in
Figure 2. The positions of all dots are measured with respect to dot 0:
i. For each dot, align the 0 marker of the ruler with dot 0
ii. Place the edge of the ruler directly on the centre of the dot being measured
and read the position off of the ruler.
iii. Record the positions in Table 1.
For now, fill in only the position column and leave the uncertainty on the measurement
blank. You will determine the uncertainty in Question 2.
Figure 2. Position of the puck
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Lab 4: Uniform Motion
Time
(s)
Position
+/-
(mm)
Displacement
+/-
(mm)
Average Velocity
+/- (mm/s)
0
±
± ±
0.100
±
± ±
0.200
±
± ± 0.300
±
± ± 0.400
±
± ± 0.500
±
± ± 0.600
±
Table 1: Position, displacement and average velocity as a function of time
Question 2:
Today we will assume that there is no uncertainty due to the time (i.e. the sparks occurred at
exactly 0.1 s intervals).
a. Is there an instrumental uncertainty (in mm ) for the position measurements? Hint:
what instrument was used to measure the position in mm? Describe the source of this
uncertainty:
What is the value of the instrumental uncertainty? _______ mm
In addition to an instrument uncertainty, there may be several other uncertainties in the position
measurement due to other sources. Today we will consider one other source of uncertainty: an
observational uncertainty due to a curvature in the path of the puck.
Take a close look at the path of the puck dots in your spark table data. Is the path perfectly
straight? You can check this with a piece of paper or some other straight edge. Line up the
paper so that the edge is centered and cuts through both the dot at and the dot at . See Figure 3
for an example. Do all the dots in between these two also lie exactly on the line formed by the
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Lab 4: Uniform Motion
paper edge? If there is even a slight deviation of the middle points above or below the paper
edge, this is a sign that the puck path was slightly curved.
Figure 3. Use a piece of paper or other straight edge to test whether the path of the puck was
perfectly straight. In this example, the fact that the middle dots do not lie on the paper edge
indicates that the path of the puck was slightly curved.
b. After performing this test, is there any curvature at all in the path of the puck?
If there is even a slight curvature to the path of the puck, this indicates that there will be
difference between the true distance that the puck travelled and the position of the later dots as
measured by a ruler, as illustrated in Figure 4.
Figure 4. Comparing the true path of the puck (red) to the distance travelled as measured by a
straight ruler (black).
To measure the true path of the puck, we would need to find the length of the arc of the puck
path (the red line). This is possible, but it takes a bit of work. Today, we will use an
approximation to find the difference between the position measured by the ruler and the true path
of the puck. We will record the position measured by the ruler as the measured position and the
difference between the paths will be used as the observational uncertainty.
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Lab 4: Uniform Motion
Take a look at Figure 4 again. Notice that both the ruler path and the true puck path travel the
same distance horizontally. That is, they begin and end at the same horizontal point. The
difference with the two paths is that the puck path (red) experiences a vertical displacement near
the centre of the path. In addition to travelling horizontally, the puck following the red path is
displaced upwards vertically and then travels back down to its original vertical position. The
total upwards vertical displacement occurs at the maximum of the puck path arc. It is illustrated
in Figure 5, where it is called . We can approximate the additional path that the puck took as the
additional distance travelled up to the maximum height at the top of the curved path and then
back down; that is, twice the vertical spacing between the two lines at the path maximum, or .
Figure 5. Measuring the maximum vertical difference between the puck path and the measured
position according to the ruler.
Pause for a moment and think about this. Make sure you understand why the difference in the
two path lengths is approximated by .
c. Why is the difference between the black and the red path lengths is approximated by ?
d. Measure using your spark table data.
_____________ mm
e. We will approximate the observational uncertainty on the ruler measurements as . What
is the value of the observational uncertainty?
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Lab 4: Uniform Motion
_______mm
f. This is only an approximation of the deviation between the ruler measurement and the
path the puck took. For the measurement of some positions, this approximation is a large
overestimation of the actual difference between paths. For other positions, this is a
reasonable estimation of the path difference. For which position measurements (to ) is
the approximation the most reasonable and the least reasonable?
Measurement in which is the largest overestimation of the path difference:
Reason:
Measurement in which is the most reasonable estimation the path difference:
Reason:
g. Is this observational uncertainty a systematic or random uncertainty? Recall that a
systematic uncertainty causes measurements to be consistently too high or too low
compared to the true path of the puck. In the case of a random uncertainty, some position
measurements will be too high while others will be too low compared to the true path
length of the puck. For example, think about your measurements , , and using the ruler
path. Do you think they all three are either too high or too low compared to the true path
of the puck? Or did and both have equal chances of being either too high or too low
compared to the true path?
Circle: Systematic Random
Reason:
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Lab 4: Uniform Motion
h. When there is more than one type of uncertainty in a measurement, we need to account
for both in the absolute uncertainty. There are several different analysis techniques for
this, the best technique depends on the specific experiment. For today’s experiment, add
the two uncertainties using the propagation formula for adding quantities:
Show your calculations:
Total absolute uncertainty on the position measurement: _______
i. Fill in the uncertainty of the position measurements in Table 1 using in e).
Question 3:
Calculate the displacement and velocity using your position measurements. Record these
quantities in the second and third column of Table 1.
Be sure to propagate the uncertainties in position to an uncertainty in displacement and average
velocity. Assume there is no uncertainty in time. Recall:
if: then
if: R = X / c then when c is a constant value
a. To calculate the displacement, you must subtract two positions from each other. Which
equation should you use to propagate the uncertainty on the displacement? Show a
sample calculation below:
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Lab 4: Uniform Motion
b. To calculate the velocity, you will divide the displacement by the time interval. The time
does not have an uncertainty, so it is treated as a constant. Which equation should you
use to propagate the uncertainty on the velocity? Show a sample calculation below:
Group Checkstop – You must complete up to this point at a minimum before meeting with your
group members and instructor during the synchronous lab hour.
Question 4:
a. Plot the graph by hand of position-versus-time using mm spaced graph paper. Be sure to
label your axis, provide a title, and include error bars. The length of your error bars
must match the actual size of the absolute uncertainty. Use “nice” numbers when
choosing your axis scale, so that the smallest tick marks represent a multiple of 1, 2, 5, or
10.
b. Draw a best fit line
c. Based on the graph you plotted, what type of motion did the puck undergo? Explain what
information you used to conclude this.
d. We want to determine both the average velocity and its uncertainty using this
position-versus-time using this graph. In order to do that, draw two additional fit lines:
one to represent the largest reasonable slope and one to represent the smallest reasonable
slope. Refer to the section on Estimating Uncertainties of Slopes Using a Graphical
Method in the Lab Standards Manual for instructions.
Question 5:
a. Calculate the smallest reasonable slope. Indicate the two points on the line that you use,
marking them as P1 and P2. Do not use the origin or any data points. Show your work
including the units:
Smallest reasonable slope: = ____________
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Lab 4: Uniform Motion
b. Calculate the largest reasonable slope. Indicate the two points on the line that you use,
marking them as P1 and P2. Do not use the origin or any data points. Show your work
including the units:
Largest reasonable slope: = ____________
c. To calculate the average slope, take the average of the smallest and largest slope:
=
d. To calculate the uncertainty on the slope, calculate half of the difference between the
smallest and largest slope:
e. According to your graph, what is the average velocity of the puck with uncertainty? Use
the appropriate number of decimal places based on the precision of the uncertainty.
f. Is the average velocity, as calculated using the fit lines on the graph, consistent with each
of the average velocities of the smaller intervals as calculated in Table 1? Compare the
graphed result with each velocity interval in Table 1:
g. Attach the graph at the end of this document in the Appendix.
Lab Complete. Upload both this document and your conclusion statement as a PDF
to their respective submission links.
Appendix
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Lab 4: Uniform Motion
Include the following in this section:
● Spark table data with your markings
● Position vs Time graph with fit lines and all other required labels.
Conclusion :
When commenting on consistency, compare the velocities that you calculated using two
different methods (graphing and measuring short intervals) as summarize in Question 5f
of the Worksheet. Are they consistent with each other (i.e., do their ranges overlap)?
Submit this document as a PDF to the “Lab 4 Conclusion” assignment link in the
“Assignment” folder on Blackboard.
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