Report for Experiment #3

Motion in One and Two Dimensions

Eeswar

Adluru

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

multiplied by 2. Additionally, errors

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

another, despite the additional horizontal component added to the puck in

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.