 Report for Experiment #3

Motion in One and Two Dimensions

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Eeswar

Lab Partner: Raymond
Lin

TA: Abraham Tishelman-Charny

01/30/2018

Introduction

Any
motion can be described through a breakdown of its x and y components. This
applies to both the velocity and acceleration of an object in motion. Some
motions are one-dimensional, meaning that they only have either an x or y
component. An example of this is a ball being dropped free-falling from the
roof of a building. There is no x component of acceleration present because
there is no horizontal motion, however there is clearly a y component present
as the ball is accelerating vertically. Not all objects in motion are as simple
and clearly defined as this example, however.
If a cannon shoots a ball up into the air, the ball will be moving both
horizontally and vertically until it reaches some maximum height, and then the
ball will continue to move in the x direction and decrease height in the y
direction. The object in this scenario is considered to be moving in two
dimensions. As one studies motion in two dimensions, the relationship between
the two components becomes apparent. In fact, Galileo conducted many
experiments and came to the conclusion that when firing a cannon ball from a
tower in a horizontal direction, the ball would hit the ground at the same time
as a ball dropped straight down from the same height. This lab tests the theory
concluded from Galileo’s experiment, in that downward motion under the
influence of gravity is independent from the horizontal motion.

Through
the use of one-dimensional modern equations, the objective of Investigation 1
was to analyze one dimensional motion. Using a frictionless air table tiled on
an incline plane, a puck is released from the top of the table and makes its
way to the bottom. Marks are made by the puck via electric sparks and a spark
timer, every 0.1 seconds. Using the distance between the two dots and the time
it takes, velocity is determined. This measurement is then applied to modern
kinematic equations to determine what the one-dimensional acceleration due to
gravity is. Investigation 2 is concerned with velocity and acceleration in both
x and y components. Using the same setup as in Investigation 1, the only
difference is that the puck is given a horizontal push as it makes its way to
the bottom of the air table. The same sequence of calculations are completed,
and through Investigation 3, the acceleration due to gravity (a known constant,
g = 9.8 m/s2) is derived from both a one dimensional, and
two-dimensional motion. These quantities are then compared to their error.
Ultimately, this lab will prove to familiarize students with the relationship
between objects in motion in one and two dimensions and enable them to see the
relationship between x and y components of objects in motion.

Investigation 1

To
set up the first investigation, the air table is leveled flat by turning the
air on and ensuring that the single puck being used doesn’t fall to one
specific edge of the table. Then a white piece of paper is placed on the table,
and the x and y axes are labeled. The table is propped onto a wooden block to
create the tilt which the investigation is based off of. The spark timer, which
is set to 100ms, is plugged in and when the puck is released from the top of
the table, the timer switch is manually held and then released just before the
puck hits the opposite end of the table. The spark timer leaves a mark every
100ms (?t), and the marks form a perpendicular line to the bottom of the table.
Eight dots are then labeled, and ?y, or the distance between dots 1 and 3, 2
and 4, 3 and 5, etc. till 8 is recorded into Table 3.1. This information was
used to find the average velocity through the equation v= distance/time, or  , where ?y occurs over two-time divisions so the
time interval between sparks, ?t, is
and relative errors for ?y was
determined and entered into the table.

Table 3.1 Measurements and Calculations for Investigation 1

range

?y(cm)

??y(cm)

??y/?y

t(ms)

?t

??t/?t

v(cm/ms)

?v
(cm/ms)

?v/v

1 to 3

1.9

0.05

0.02631

100

0.1414

0.002

0.0095

0.0002507

0.02639

2 to 4

2.8

0.05

0.01785

200

0.1414

0.002

0.014

0.0002515

0.01796

3 to 5

3.8

0.05

0.01315

300

0.1414

0.002

0.019

0.0002528

0.01330

4 to 6

4.7

0.05

0.01063

400

0.1414

0.002

0.0235

0.0002543

0.01082

5 to 7

5.7

0.05

0.00877

500

0.1414

0.002

0.0285

0.0002564

0.00899

6 to 8

6.9

0.05

0.00724

600

0.1414

0.002

0.0345

0.0002593

0.00751

After plotting Velocity vs. Time, the slope of the trend line was determined to be
5e-05,
which is equal to the acceleration, a.
Additionally, after using the IPL website, the graphical Error in Slope ( was determined to
be 6.09083e-7.
Errors in time and change in time are neglected due to low relative error
percentages, meaning that the values obtained are very close to the true value.
The relative error of time is found to be +/- 0.002. Using error propagation,
the relative error can be found for the velocity of each data point, which can
then be used to find the uncertainty in velocity as well. An example of this
calculation can be seen below for dots 1,3:

(1)

=

= 0.02638

This value is equal to , so in order to find , the value of the relative error is multiplied by the velocity. This
means multiplying (0.02638) (0.0095), to get 2.5066e-05 as the uncertainty of
velocity for those data points.

**Side Note: the error
bars are present in the graph above; however, they are too small to be seen
with the scale shown on the y axis

(2)

(3)

(4)

(5)

In the end through plotting position vs. time, the slope, or
acceleration of the puck in motion in one dimension is determined to be 5e-5.
The angle of the air table has not yet been factored in to this number but will
be later in Investigation 3.

Investigation 2

The
goal of investigation 2 was to understand motion in two-dimensions, and to
collect more data to test the theory that downward motion under the influence
of gravity is independent of horizontal motion. In theory, means that once a
horizontal component is added to the falling puck on the air table, the
acceleration due to gravity should remain unaffected. Just like Galileo noted
in his canon ball experiment, the two balls hit the ground at the same time, as
they experienced the same vertical acceleration due to gravity. This should
mean that if the puck is given both horizontal and vertical components, if
studied over the same time interval as in the one-dimensional motion from
Investigation 1, the acceleration in the vertical direction (y) should be the
same. This idea is ultimately what was being tested for this investigation.

To
set up, the same piece of white paper was returned to the surface of the air
table, and the spark timer was turned on and set to 100ms. One lab partner
pressed the spark timer switch, as the other simultaneously launched the puck
with a horizontal push, and the spark timer was released just before the puck
hit the opposite end of the table. This time, the series of dots created by the
spark timer created a parabolic pattern, compared to the linear pattern in
Investigation 1. Starting around the third point, they were labeled one through
eight. The distance between marks 1 and 3 in the x direction and y direction
were recorded into tables 3.2 and 3.3, as well as from 2 to 4, 3 to 5, etc.
(just like in Investigation 1, the distances were recorded over two-time
divisions).  The determined errors were
also recorded and added to the tables below. Four sets of graphs were created
from the data collected; x vs. t, y vs. t, vx vs. t, and vy vs.
t. Velocity components were determined by using the same formula as in
Investigation1,  or .

Table 3.2: Data in the Horizontal
Direction (x)

Dot range

?x(cm)

??x(cm)

??x/?x

vx (cm/ms)

?vx (cm/ms)

1 to 3

7.3

0.05

0.006849

0.0365

0.0002604

2 to 4

7.1

0.05

0.007042

0.0355

0.0002598

3 to 5

6.8

0.05

0.007352

0.034

0.0002590

4 to 6

6.5

0.05

0.007692

0.0325

0.0002583

5 to 7

6.4

0.05

0.007812

0.032

0.0002580

6 to 8

6.4

0.05

0.007812

0.032

0.0002580

Table 3.3: Data in the Vertical
Direction (y)

Dot range

?y(cm)

??y(cm)

??y/?y

vy (cm/ms)

?vy (cm/ms)

1
to 3

8

0.05

0.00625

0.04

0.0002624

2
to 4

8.8

0.05

0.005681

0.044

0.0002650

3
to 5

9.6

0.05

0.005208

0.048

0.0002677

4
to 6

10.1

0.05

0.004950

0.0505

0.0002696

5
to 7

10.8

0.05

0.004629

0.054

0.0002723

6
to 8

11.7

0.05

0.004273

0.0585

0.0002760

Table 3.4: Velocity and time

Dot range

t(ms)

?t

??t/?t

v (cm/ms)

?v (cm/ms)

?v/v

1 to 3

100

0.1414

0.002

0.0365

0.0002604

0.007135

2 to 4

200

0.1414

0.002

0.0355

0.0002598

0.007321

3 to 5

300

0.1414

0.002

0.034

0.0002590

0.00762

4 to 6

400

0.1414

0.002

0.0325

0.0002583

0.007948

5 to 7

500

0.1414

0.002

0.032

0.0002580

0.008064

6 to 8

600

0.1414

0.002

0.032

0.0002580

0.008064

**Error Bars are Present in ALL GRAPHS;
however, they are too small to be seen along the increments of the y axes.

From
an inspection of x vs. t and y vs. t, the two are different because the slope
of the x vs. t graph is basically a straight line meaning that the puck did not
move nearly as much over the time period, as the puck did in the y direction,
as shown by the increasing graph, showing that it was moving at a constant
velocity. As time increased so did the distance covered.  Looking at the plot of vy vs t,
the acceleration due to gravity (the slope of the trend line), ay, is 4e-05 m/s2. The
error of this slope is 0.005976, meaning that the
calculated acceleration due to gravity doesn’t agree within errors. Looking at
vx vs. t, the error in slope is 0.005976. The velocity is constant
because the slope of vx is within the error number of zero, as the
true value of horizontal acceleration is supposed to be zero.

Investigation 3

The
main purpose of the third investigation was to calculate the accelerations due
to gravity using the results obtained from Investigations 1 and 2. A formula
for acceleration due to gravity, g,
was derived using the angle at which the air table was tilted, and eventually
this was used to calculate g1 and g2, as well as their uncertainties.
G1 and g2 were then averaged to find gavg, and
additionally the uncertainty in that quantity was found. The relationship
between the angle of the air table and acceleration due to gravity can be
expressed by the equation , and can be found without actually measuring the angle of the air table
through the equation , where h2 is the height of the tallest vertical point of the
table, and h1 is the height of the lowest vertical point of the
table, and L is the horizontal length of the air table. The net height of the
table was found to be 3.38 cm, and the length of the table
was found to be 66.6 cm. This meant that  was equal to 0.0508. This numerical
value was then plugged into the equation above in order to solve for the
acceleration due to gravity, by plugging in the value for  and the value of ay. Values from both
investigations were used accordingly in order to solve for two accelerations
due to gravity, g1 and g2. G1 was found to be 9.39 m/s2 and g2 was found to be 9.97 m/s2.
Gavg was found by adding those two quantities together and dividing
by two, to get 9.68 m/s2 as the average acceleration due to
gravity. This however is not the end of the investigation.

Next,
uncertainty in g1 and g2 were to be found. First,
uncertainty in  was found to be 1.48
x 10-4, which was calculated using the propagated error
formula shown in Investigation 1. Using the same propagated error formula, and
the equation , the uncertainty associated with each g1 and g2
were found.  was found to be 0.120192
by multiplying g1 by its relative error.  was found to be 0.12127
and was also derived by multiplying the value for g2 by its relative
error. Lastly, using the formula , the average uncertainty of acceleration due to gravity was found to be
0.0854.

Our
value of gavg, 9.68 m/s2, is lower than
the accepted value which is 9.81 m/s2, and unfortunately the
uncertainty of the average acceleration due to gravity does not agree with our
value, however if our error was doubled, then the value calculated from
Investigations 1 and 2 would work. This discrepancy could be due to a number or
reasons, but to name a few; underestimation of initial measurements, or friction.

Conclusion

The
overarching goal of this lab was to prove the relationship between objects in
motion in one, versus two-dimensions. The x component of the puck moving in two
dimensions had little to no effect on the motion in the vertical direction. By
using an air table, a virtually “frictionless” surface was created in order to
help simulate a scenario where air resistance and friction don’t exist, and the
only forces acting on the puck are either pure gravity, or pure gravity plus
horizontal push.

In
the end, the value for average acceleration due to gravity was in the ballpark
of the actual acceleration due to gravity, which is 9.81 m/s2. The
calculated value does not agree within the error; however, this does not change
the fact that the values from both Investigations were relatively close to one
Investigation 2. This Lab was therefore successful in proving Galileo’s theory,
and showing that what happens in the x direction does not affect an objects
acceleration due to gravity.

Some
possible errors which may have impacted the results of this lab, are that the
air table could’ve not been 100% leveled. In our first trial in experiment 1,
the dots moved slightly horizontally, even though they were only supposed to
have moved only vertically, indicating that the balance was off. Additionally,
if we were to do this lab again, we would shift the points picked to closer to
the release of the puck, to use as data points in Investigation 2 in order to
have our graph of y vs. t appear to be more parabolic in shape, as it is
supposed to be.

Questions

1) The total time to reach the bottom
of the air table for Investigations 1 and 2 are found to be around 1.8seconds.
These times will all be the same regardless of the mass of the object, as they
are all accelerating in the same vertical direction due to the same
gravitational force, so it will take them the same amount of time to reach the bottom,
whether there is a horizontal component involved or not.

2) . The accelerations are the same.

3) The meaning of the intercept of my
line with the v-axis, is that this is the initial velocity of the puck at time
0 s. We expect this value to be close to the origin because it occurs at time
0.

4) With an incline of , the trajectories would look very different. First, there would be no
vertical drop, so the puck would not move, meaning that the position vs. time
graphs would be horizontal lines.

5) If there was insufficient air flow through
the table, this would mean friction is more likely to occur and oppose the
accelerations. This would cause deceleration and cause the slope of my graphs
to become less slanted, and more horizontal. 