uphys 1d kinematics

8/9/2019 UPhys 1D Kinematics

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University Physics Lab

Kinematics, Motion in One Dimension

Purpose of the Experiment

To be able to interpret graphs of position, velocity and acceleration vs. time.

To analyze the motion of a student walking across the room.

To measure a value of g , the acceleration of gravity at the Earths surface.

To understand the relationships between position, velocity and acceleration in

terms of value, slope and area under the curve.

Kinematics, Motion in One Dimension - 1


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Table of Contents

Background Information......................................................................................................

!inematics.......................................................................................................................Interpreting E"uations and #raphs..................................................................................$

%elationships between !inematic &uantities..................................................................'

(otion )ensor..................................................................................................................*

+relab...................................................................................................................................

The -ab............................................................................................................................../

+urpose of the E0periment............................................................................................./

1pparatus......................................................................................................................./

2iles................................................................................................................................+rocedure.......................................................................................................................3

+art I4 +osition vs.Time #raph (atching.................................................................3

+art II4 5elocity vs.Time #raph (atching................................................................$

+art III4 6ne 7imensional !inematics......................................................................'

+art I54 The Effects of )hifting the +osition #raph..................................................8

+art 54 %elationships between !inematic &uantities................................................3

1nalysis.........................................................................................................................3$

+art I4 +osition vs.Time #raph (atching.................................................................3$+art II4 5elocity vs.Time #raph (atching................................................................3'

+art III4 6ne 7imensional !inematics......................................................................39

+art I54 The Effects of )hifting the +osition #raph..................................................3*

+art 54 %elationships between !inematic &uantities................................................3



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ac!"roun# $nformation

!inematics

Kinematicsis the branch of physics that describes motion. It answers "uestions

concerning the motions of ob:ects such as4

;

;?ow fast is it going=>

;In which direction is it moving=>

;Is it speeding up, slowing down, or changing direction=>.

Bear in mind that while kinematics answers the "uestion ;?ow do ob:ects move=>, itdoes not answer the "uestion ;. 1nswering ; is left to the

branch of physics called dynamics, which is the sub:ect of a later lab.

!inematics answers these "uestions with three "uantities4 position, velocity, and

acceleration. +osition tells us where the ob:ect is. 5elocity tells us how fast it is going

and in what direction. 1cceleration tells us whether the ob:ect is speeding up, slowing

down or changing direction.

In this lab, for simplicitys sake, we will restrict ourselves to the idea of motion

along a straight line, also referred to as one@dimensional motion. It is important to first

understand one@dimensional motion before adding additional dimensions. 6nce you

understand one@dimensional motion, it is easy to figure out two@ and three@ dimensional

motions since they are simply two Aor three independent, perpendicular one@dimensional

motions. ?owever, for the purposes of this lab, the ob:ect will only be able to go

forwardCbackward or, alternatively, upCdown, and we dont have to worry about whether

or not it is turning. Two@ and three@dimensional motion will be dealt with in a later lab.



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Interpreting E"uations and #raphs

Donsider the kinematic e"uations of motion for constant acceleration. If we were

to plot these e"uations on a graph, they would have a particular geometric shape. By

comparing the kinematic e"uations to the e"uations for a parabola and straight line, you

can see this more clearly4

+osition vs. Time E"uation4 y=1

2a t

2+vO t+yO

E"uation of a +arabola4 y=A t2+Bt+C

?ere you can see that since A=a/2 , B=vO and C=yO , then the position vs.

time graph must be a parabola. 1lso notice that4

5elocity vs. Time E"uation4 v=a t+vO

E"uation of a )traight -ine4 y=m x+b

1gain, if I make the identifications that the slope m=a and the intercept b=vO ,

you can tell that the velocity vs. time graph must be a straight line. %emember this idea,

we will come back to it in many future labs.



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%elationships between !inematic &uantities

1dditionally, there is an important connection between the graph of position vs.

time and graph of velocity vs. time. This relationship is stated as follows4

The slope at any given point on a position vs. time graph is the value at the same point

on a velocity vs. time graph.

1 similar relationship e0ists between the velocity vs. time graph and the acceleration vs.

time graph4

The slope at any given point on a velocity vs. time graph is the value at the same point

on an acceleration vs. time graph.

The second statement above is easily proven if you recall that the slope is simply the rise

over the run for a given graph4

slope on v vs . t=rise

run=

v

t=

(a t2+vO)(a t1+vO )t2t

1

=a(t

2t

1)

t2t

1

=a=valueonavs.t

Thus, using the first statement above, if one knows an ob:ects position as a function of

time, it is a simple matter to compute the velocity by finding the slope. The process can

be repeated with the velocity graph to find the acceleration.

Is it also possible to reverse these relationships= 2or e0ample, if we know the

acceleration, can we find the velocity, and then repeat the process to find the position as a

function of time= The answer is yes, and to do so we need to consider the area under the

curve. These reverse relationship between acceleration and velocity is summarized as4

The area under the curve for a given time interval on an acceleration vs. time graph is

the change in value (delta-v) over the same time interval on a velocity vs. time graph.

-ikewise, a nearly identical relationship e0ists between velocity and position4



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The area under the curve for a given time interval on a velocity vs. time graph is the

change in value (displacement) over the same time interval on a position vs. time

graph.

In this lab, we will use these concepts to analyze the motion of a ball.


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(otion )ensor

6ne of the most effective methods of describing motion is to plot graphs of

position, velocity, and acceleration vs. time. 2rom such a graphical representation, it is

possible to determine in what direction an ob:ect is going, how fast it is moving, how far

it traveled, and whether it is speeding up or slowing down.

In this e0periment, you will use a (otion 7etector to determine this information

by plotting a real time graph ofyourmotion as you move across the classroom, as well as

a ball in freefall. The (otion )ensor measures the time it takes for a high fre"uency

sound pulse to travel from the sensor to an ob:ect and back.

2igure 4 +1)+ort (otion )ensor.

sing this round@trip time and the speed of sound, you can determine the distance to the

ob:ectF that is, its position. The Dapstone data ac"uisition software will perform this

calculation for you. It can then use the change in position to calculate the ob:ects

velocity and acceleration. 1ll of this information can be displayed either as a table or a

graph.

(otion 7etectors are useful tools for measuring the distances to ob:ects and

e0ploring the laws of kinematics. They are easy to use if you keep two limitations in

mind4

. (inC(a0 7istance4 The (otion 7etector will have difficulty measuring distances to

ob:ects that are less than about 15cm or greater than 8m away from the sensor.

3. 6bstructions4 The sound pulse is emitted in a cone from the speaker on the front of the

detector. (ake certain that the cone only intersects the ob:ect you are trying to

monitor. !eep other ob:ects, such as the detectors cord, the edge of the lab table, or

stools out of the sound cone.



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Prelab

+relab 4 2or this "uestion, go to the website4

http4CCphet.colorado.eduCenCsimulationCmoving@man

Gou will need Hava on your computer. Gou can download this program for free at4

http4CCwww.:ava.comCenCinde0.:sp

%ote& (ac 6) users will need to use the 7ownload button on the +hET website.

6nce the simulation is downloaded to the computer, open up the 7ownloads

directory. Then, right@click Aor control@click on the simulation file and select

6pen with from the pop up menu, and open the simulation with Hava.

2or each situation listed below, set up the appropriate parameters in The (oving

(an program under the Dharts tab, and then press +lay to watch the mans motion.

6n a separate graph for each situation listed below, sketch the position vs. time and

velocity vs.time curves plotted by the computer. Be sure to label your a0es clearly.

a. The man at rest Astanding still at an initial position of x=5m . 2or clarity,

+ause and Dlear your graphs between each run.

b. The man moving Awalking in the positive direction with a constant speed.c. The man moving Awalking in the positive direction with a greater constant

speed than in part Ab.

d. The man moving Awalking in the negative direction with a constant speed.

e. 1n ob:ect accelerating Aaccelerating in the positive direction, starting from

rest Ause a small acceleration of 0.5m/s2 or 1m/ s2 and zoom in and out

on the graphs as needed using the magnifier J and @ symbols ne0t to the

graphs.



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+relab 34 Donsider a ball that is tossed vertically upward into the air. Think about the

changes in motion a ball will undergo as it travels straight up and down. 6n a graph,

sketch your prediction for the position vs. time graph in the vertical direction. 7escribe

in your own words what this graph means.


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The Lab

+urpose of the E0periment 4

To be able to interpret graphs of position, velocity and acceleration vs. time. To

analyze the motion of a student walking across the room. To measure a value of g , the

acceleration of gravity at the Earths surface. To understand the relationships between

position, velocity and acceleration in terms of value, slope and area under the curve.

1pparatus


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2iles

+art I4

University Physics Lab/1D Kinematics/Graph Matching.cap

+art II4


+art III4

University Physics Lab/1D Kinematics /Bouncing Ball .cap

+art I54


+art 54




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+rocedure

+art I4 +osition vs. Time #raph (atching

. Donnect the (otion )ensor to +1)+ort of the '/ Interface, and set the rangeswitch on top of the (otion )ensor to -ong %ange Ai.e. the +eople symbol. Then,

place the (otion )ensor so that it points toward an open floor space at least 3m

long, as shown in 2igure 3. +osition the computer monitor so you can see the screen

while you move in back and forth in front of the (otion )ensor.

2igure 3 4 #raph matching.

3. 1im the (otion )ensor at your midsection when you are standing in front of the

sensor. Gou will hold the provided whiteboard in front of your body to improve your

results, since reflections from different parts of your body as you move can mimic

rapid motion back and forth.

. )tart Dapstone and power on the '/ Interface. -oad the file4


1 graph of position vs.time will appear. The ob:ect of this e0ercise is to walk either

towards or away from the detector so that your motion duplicates the graph on the

screen.



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$. 6n each of the three +osition tabs in Dapstone, there is a graph you will try to match.

2or these position graphs, you can see the starting distance on the graph, and you

should try to start at that position. Try to match the graph by moving forward or

backward. The )core display will show how closely you match the graph. The

closer to 100 , the better. It is fairly easy to get a score above 95 on the position

plots.

'. 1fter starting data collection by clicking %ecord, there will be a three@second

countdown before data recording begins to allow you to get in position.


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+art II4 5elocity vs. Time #raph (atching

. 6pen the 5elocity tab in Dapstone. 1 graph of velocity vs. time will be displayed.

1s in +art I, the ob:ect of this e0ercise is to walk either towards or away from the

detector so that your motion duplicates the graph on the screen.

3. 6n each of the four 5elocity tabs in Dapstone, there is a graph you will try to match.

2or these velocity graphs, start about 0.5m from the (otion )ensor. Try to match

the graph by moving forward or backward. The )core display will show how

closely you match the graph. The closer to 100 , the better. (ost groups will find

the velocity graphs are more difficult to match than the position graphs. 1s a result,

scores above / are considered good for the velocity graphs.

. 1fter starting data collection by clicking %ecord, there will be a three@second

countdown before data recording begins to allow you to get in position.


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+art III4 6ne 7imensional !inematics

. Donnect the +1)+ort (otion )ensor to +1)+ort of the '/ Interface. )et the

range switch on top of the (otion )ensor to -ong %ange Ai.e. the +eople symbol.

3. +ower on the '/ Interface and start Dapstone. 6pen the e0periment file4


Three graphs with a common horizontal time a0is are displayed. The top graph is a

position vs.time graph, and the lower two are inactive. The data collection is set for

5 seconds at 100 samples per second.

. )et up the following apparatus. +lace the Motion Sensor on the ring stand and then

ad:ust the positioning dial on the side of the sensor so that the sensor is pointing down

and very slightly away from the lab table. The sound-emitting portion of the sensor

should be 1m off the floor. Use the meter stick to check this.

2igure 4 )etup for +art III.



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$. (ake absolutely certain that the (otion )ensor is not picking up the cord, the table,

or any other ob:ect besides the floor that is 1m below the sensor. Dlick %ecord in

Dapstone while the (otion )ensor is aimed at the floor as per the previous step. 6n

the position vs.time graph, there should be a single fluctuating dot at on the vertical

a0is at 100cm . If not, ad:ust the positioning of your (otion )ensor and try

again. se the 7elete -ast %un button on the Dontrols palette to remove any

unwanted data runs.

'. Dapstone is set to start taking data once the ball falls underneath a certain threshold

height. )o, starting with the ball about 20cm above the floor, click %ecord and

let the ball fall towards the ground underneath the sensor. %eposition the (otion

)ensor if needed and be sure to move your hands out of the way after you release theball. Gou need to record a minimum of four clean bounces on the distance vs. time

graph. This step may re"uire some practice, and smaller bounces around 10 or

20cm are better than larger ones. But keep doing it until you get it right. se the

7elete -ast %un button on the Dontrols palette to remove any unwanted data runs.

6.)Kot all of the recorded data on your graph is useful. Gou need to neatly display only

four of your cleanest, largest bounces on the top graph. To do this, you can either4

+an and )cale4PanAi.e. use the hand cursor on the graph area to move the

viewable area around andscaleAi.e. use the double arrow cursor on the

horizontal and vertical a0es to change the scale of the graph to show the four

bounces.

?ighlight and 1utoscale4 Dlick the Highlight range of points in active data

button on the menu bar at the top of the graph. A resizable Highlighter tool will

appear on the graph. Highlight the four bounces you wish to use, and then click

the Scale axes to show all data button. When you are done, right click on the

Highlighter tool and select Delete Highlighter from the pop up menu.



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*. Display the velocity vs.time graph. To do this, click on the left vertical axis label of

the middle graph, and select v (cm/s) from the pop up menu that appears. Again,

pan and scale the graph to show just the useful section of the data.

. Display the acceleration vs.time graph. To do this, click on the left vertical axis label

of the bottom graph, and select a (cm/s2) from the pop up menu that appears. Again,

pan and scale the graph to show just the useful section of the data.

8. The motion of an ob:ect in free fall is modeled by y=1

2g t

2+vO t+yO , where y

is the vertical position, yO is the initial vertical position, vO is the initial

velocity, t is time, and g is the acceleration due to gravity A 9.8m/s2 . This is

a "uadratic e"uation whose graph is a parabola. Gour graph of position vs. time

should be parabolic. To fit a "uadratic e"uation to your data4

)elect the position graph and then click the ?ighlight range of points in active

data button on the menu bar at the top of the graph. 1 resizable ?ighlighter tool

will appear on the graph. ?ighlight a portion of the position vs. time graph that is

parabolic, selecting :ust the free@fall portion of the graph Ai.e. the smooth parabola

between the spikes where the ball bounces off the floor.

Dlick the )elect curve fits to be displayed button on the menu bar at the top of

the graph. In the pull down menu that appears, select &uadratic fit from the list

of models. 1 summary of the fit will appear in the position vs. time graph

window. (ove the summary bo0 out of the way if needed. 7o not move the

?ighlighter tool until +art I5.

/. The e"uation displayed on the screen is y=A t2+Bt+C .


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. In the region where the ball was in free fall, the graph of velocity vs.time should be

linear. To fit a line to this data, highlight the free@fall region of the motion and then

choose -inear from the list of models. the corresponding

point on the velocity graph as well as the corresponding point on the acceleration

graph A;1cceleration at (a0imum ?eight>. Kote that all three points occur at the

same time on the horizontal a0is.

$. %etain copies of these three graphs displaying the annotations, statistics, best@fit line

and curve fit information.

'. 7o not move the three ?ighlighter tools, leave the graphs set up as they are and go

directly to +art I5.



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1verage these points

+art I54 The Effects of )hifting the +osition #raph

. To further understand the relationship between position, velocity and acceleration, you

are going to shift the position of the ball by a constant amount in the vertical direction

and see how this shift affects the velocity and acceleration graphs. To do this, we willad:ust the position graph to make the floor correspond to the origin A y=0 of our

coordinate system.a. Dlick on the position graph window to select it. se the 1dd Doordinates

ToolNN1dd DoordinatesC7elta Tool button to find the height of the lowest

part of the bounce for three bounces, as shown below. 1verage these three

values.


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c. Dlick on the top window. 7elete the ?ighlighter tool here. +an and scale the

graph to show your shifted position vs. time data with the bounces occurring

at y=0 , as shown below4

2igure ' 4 )ample of data after zeroing.

3. +lace a new ?ighlighter tool on the position graph over the e0act same points you

used in +art III.


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+art 54 %elationships between !inematic &uantities

. Dopy the following data table into your lab notebook.

Slope on Positionvs

. Time cm/salue on elocit! vs. Time cm/sPercent "ifference

Slope on elocit! vs. Time cm/s2

alue on Acceleration vs. Time cm/s2

Percent "ifference

3. 6n your position vs. time graph showing your four clean bounces, click on the

7etermine slope of line between endpoints of interval surrounding data point button

at the top of the graph area. sing the )lope tool that appears, pick a point on the

left half of your position vs. time graph at which to measure the slope. sing the

snap to arrow to guide your placement, put the tool at that point.

. Ke0t, select the velocity vs. time graph and click on the 1dd Doordinates ToolNN1dd

DoordinatesC7elta Tool button. +lace the Doordinates tool that appears at e0actly

the corresponding point on the velocity graph to the point you chose on the position

graph in the previous step Ai.e. the point on the velocity graph should be at the e0act

same time coordinate and therefore directly underneath the point you chose on the

position graph. sing the snap to arrow to guide your placement, put the tool at

that point.

$. %ecord the slope from the position graph and the value from the velocity graph in

your data table.

'. %epeat )teps 3 and . This time, however, pick a point on the right half of the graph,

use a )lope tool on the velocity graph and a Doordinates tool on the acceleration

graph.

9. %ecord the slope from the velocity graph and the value from the acceleration graph in

your data table.



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*. %etain a copy of all three graphs with their common a0is, slope and coordinate

information displayed.

. %ight click on each of the )lope and Doordinate tools and delete them.

8. Dopy the following data table into your lab notebook.

Area #n$er Curve on elocit! vs. Time cmChange in alue on Position vs. Time cmPercent "ifference

Area #n$er Curve on Acceleration vs. Time cm/sChange in alue on elocit! vs. Time cm/sPercent "ifference

/. 6n your velocity vs. time graph, click the ?ighlight range of points in active data

button. ?ighlight a section of data near the left side of the graph. Be sure to e0pand

the ?ighlighter so as not to miss any data points in the range you picked. The data

you select will be highlighted in yellow like a highlighter pen. Then, click the

7isplay area under active data button to display the area under this section of the

velocity curve.

. Ke0t, on the position vs.time graph, click the 1dd Doordinates ToolNN1ddDoordinatesC7elta Tool button. %ight click on the Doordinates tool that appears,

and turn it into a 7elta tool by selecting )how 7elta Tool from the pop up menu.

+lace one side of the 7elta tool at the point on your position graph that corresponds

e0actly to the left side of the range of your velocity graph you selected in the previous

step. +lace the other side of the 7elta tool at e0actly the other side of the range.

1gain, use the snap to arrow to guide your placement. A?int4 Be sure the range of

points that are bracketed by the 7elta tool here in the position graph correspond

e0actly to the yellow highlighted range of points in the velocity graph, otherwise your

results will have a systematic error.



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3. %ecord the area under the curve from the velocity graph and the change in value

from the position graph in your data table.

. %epeat )teps 3 and . This time, however, pick a range on the right half of the

graph, use a ?ighlighter tool on the acceleration graph and a 7elta tool on thevelocity graph. Dareful4 7o not let the range you select to highlight beginor endin

one of the large upward spikesO Because the bounces happen very "uickly, there is

not enough data to work with and the results will be off. The range can includeentire

spikes, however.

$. %ecord the area under the curve from the acceleration graph and the change in value

from the velocity graph in your data table.

'. %etain a copy of all three graphs with their common a0is, area under the curve and

delta information displayed.



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1nalysis

+art I4 +osition vs. Time #raph (atching

. 7escribe in your own words how your lab group had to walk to produce each graph.

Be detailed, but brief, in your answer. )pecifically, connect the shape of each graph

to the way you had to walk to reproduce the graph. Be very descriptive and strive to

write your e0planations so that a physics student at another school could understand

your meaning. 7o not presume that the grader will know what you mean.

3.


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+art II4 5elocity vs. Time #raph (atching

. 7escribe how you walked for each of the graphs that you matched.

3.


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+art III4 6ne 7imensional !inematics

. -ooking at the annotations on your graph, what was the velocity of the ball at the top

of its motion=


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error=|experimental valuetheoreticalvalue|

theoreticalvalue 100

+art I54 The Effects of )hifting the +osition #raph

. Dompare each of your values for the "uadratic curve fit coefficients A , B and

C of your position graph before and after the shift of the position graph.


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+art 54 %elationships between !inematic &uantities

. Domplete your data tables by calculating the percent differences between the

appropriate values.

difference= |value 1value2|(value1+value 2)/2

100

3. ?ow do low percent differences in your e0perimental values taken from an actual

bouncing ball support the relationships between the kinematic variables listed below=

E0plain thoroughly.

The slope at any given point on a position vs. time graph is the value at the same

point on a velocity vs. time graph.

The slope at any given point on a velocity vs. time graph is the value at the same

point on an acceleration vs. time graph.

The area under the curve for a given time interval on an acceleration vs. time

graph is the change in value Adeltav over the same time interval on a velocity vs.

time graph.

The area under the curve for a given time interval on a velocity vs. time graph is

the change in value Adisplacement over the same time interval on a position vs.

time graph.

uphys 1d kinematics

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