If You Performed Investigations 1 and 2 Again With an Incline of 0 What Would

Report for Experiment three

Move in One and 2 Dimensions

Abstract

This experiment is designed to replicate Galileo experiment. The key equipment's used in this experiment were an air table to provide a frictionless environs and two pucks. While the carbon paper and spark timer were used in determining the distance traveled of the puck by providing dots at every time interval. By tilting the air table, move in both dimensions could be stimulated. The investigation support Galileo'south discovery that if an object being dropped vertically and some other was fired horizontally from the aforementioned height will reach the ground at the same time. Thus, the acceleration in the y management of both objects should exist equal. The acceleration of two different scenarios in this experiment autumn within range of error, which is approximately 0.56 g/due south². While the dispatch in the x direction should remain 0. By measuring the tiptop and length of table, the angle in which the table is tilted was adamant. This helped in computing the average gravitational dispatch which is equal to 9.04±0.277 thousand/s². The value did non fall within range of error with the theoretical value 9.80 m/due south². This could be due to systematic errors.

Introduction

In this experiment, nosotros will report ane and two-dimensional move of a puck on an inclined plane. This can be washed past using an air table with pucks along with a spark timer to determine the movement of the pucks in a nearly frictionless surface. The relationship between position, velocity, and acceleration is determined and analyzed in both dimensions.  The goal of this experiment is to create a better understanding of the component of motion equally it will replicate a famous experiment performed by Galileo that demonstrated that a ball fired horizontally at the aforementioned fourth dimension as a ball is dropped from same exact height will reach the ground at the same time. This experiment determined that the downward motion under the influence of gravity had to be independent of the horizontal motion.  In start two investigations, the given air table volition be tilted past a wooden block allowing the puck to slide across the table. The spark timer set at 100ms will be used to decide the movement of the puck by leaving marks of the pucks position on a carbon paper in a 100ms interval.

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The first investigation will mimic a ane-dimensional move of an object being dropped from a specific height by simply letting the puck slide down the air table. The change in the distance in the y component can exist calculated to determine the velocity of the puck at a given period. A ruler volition be used to measure the change in position between mark ane and 3, ii and iv, 3 and 5 until it reached the viii^th marker. Then, a plot was made to show the relationship of velocity and time in a 1-dimensional motion. The derivative of the equation of the line will exist equivalent to the acceleration of the puck.

The second investigation will mimic an object existence fired horizontally from the exact same pinnacle of the object from the kickoff investigation. This is done by simply pushing the puck from 1 side of the table to the other side, creating a parabolic marking on the carbon paper. The change in distance on the x and y management can be calculated which is and then used to find the velocity of the puck in both dimensions. A ruler will be used to measure the change in position betwixt mark 1 and 3, 2 and 4, 3 and 5 until information technology reached the eight^th marker. 2 plots are so made to show the relationship of velocity in the y and 10 direction and time in a two-dimensional motion.

Investigation 1

Setup and Procedure

The beginning investigation will carry out a one-dimensional motility of an object being dropped vertically from a specific pinnacle. The equipment used throughout the entire experiment are, pucks, spark timer, a carbon paper, a block, and a ruler. The air table will be tilted past placing a block on one of the legs to make an inclined airplane. A carbon newspaper is and then placed on the table which will be used to show the motion of the puck. An air hose is attached to the air valve. This will allow the puck to move in a nearly frictionless surface. The spark time is set up to the frequency of tenHz or 100ms. The puck is then placed on the upper left corner of the table. My partner and I worked collectively, where one is responsible for dropping the puck, and the other is to press and release the spark timer. The spark timer will be pushed at the aforementioned fourth dimension the puck is released and the spark timer is released correct earlier the puck hits the edge of the table. The motion of the puck is shown on the carbon paper, where the marks were to be numbered from the second to the end. This volition help determine the distance in the y management between mark one to 3, two to 4, three to five and and so on.

The data provided are in meters, thus we demand to catechumen the fourth dimension from milliseconds to seconds to fit with the measurements.

Data and Analysis

Dot Interval	Time (s)	Δt (southward)	δΔt (due south)	Δy(m)	δΔy(m)	δΔy/Δy (m)	v (m/s)	δv/v	δv
one to 3	0.2	0.1	0.0002	0.044	0.0005	0.01136364	0.220	0.0113636	0.0025
ii to 4	0.iii	0.1	0.0002	0.055	0.0005	0.00909091	0.275	0.0090909	0.0025
3 to v	0.iv	0.1	0.0002	0.067	0.0005	0.00746269	0.335	0.0074627	0.0025
4 to vi	0.five	0.ane	0.0002	0.080	0.0005	0.00625000	0.400	0.0062500	0.0025
5 to seven	0.6	0.one	0.0002	0.089	0.0005	0.00561798	0.445	0.0056180	0.0025
6 to 8	0.vii	0.1	0.0002	0.100	0.0005	0.00500000	0.500	0.0050000	0.0025

Tabular array 1.ane – Measurement of altitude traveled and average velocity in each dot interval, because whatsoever necessary errors.

The above table shows the results of the one-dimensional motion of a puck. The distance mensurate between each time interval is shown by Δy, and the fault is ±0.0005 meters. The relative error is given by dividing the uncertainty of the distance over the distance, which is given past the equation: δΔy/Δy. The velocity was so calculated by taking the distance and dividing it by time.

$\mathrm{five}=\frac{\mathit{distance}}{\mathit{time}}$

But since the distance Δy is betwixt dots one and 3, ii and 4 etc, the time is therefore equal to 2 times the divisions equally shown in the equation below.

$v=\frac{\mathrm{\Delta y}}{2\mathrm{\Delta t}}$

The propagated error of the velocity was and then calculated by using the formula:

$\frac{\mathrm{\delta v}}{\mathrm{five}}=\sqrt{{\left(\frac{\mathrm{\delta \Delta }y}{\mathrm{\Delta }y}\right)}^{\mathrm{ii}}+{\left(\frac{\mathrm{\delta \Delta t}}{\mathrm{\Delta t}}\right)}^2}$

The propagated error of the velocity δv/v, and distance δΔy/Δy are equal every bit shown in Table i.1. The uncertainty of the velocity is and so given past multiplying the velocity past the propagated error, giving the value of ±0.0025 m/s

Figure i.ane – Plot showing the human relationship between velocity and time, including a trendline, equation of the line, and error bars.

The derivative of the equation of the line will result in the acceleration of the puck, which is 0.5643 m/south² . Then the IPL best fit line calculator was used to make up one's mind the slope and fault. The slope of the line was 0.5642x doubt of ±0.00598223m/s². Meaning that the slopes from equation of the line and IPL best fit line calculator are both inside the range of fault.

Investigation 2

Setup and Procedure

The 2d investigation analyzes a motion in 2 dimensional. This will stimulate an object existence fired horizontally, causing the object to feel velocity in both the 10 and y direction. The general setup of investigation 2 is the aforementioned to that of the first investigation. Still, in this investigation, the puck volition be pushed from ane side to the other and if washed correctly, the puck will motion parabolically from one side of the table to the other. The labeled x and y axis carbon paper will show marks in a parabolic pattern. The data was then used to determine the velocities of the puck in the x and y management as shown in Table 1.three.

Data and Analysis

Dot Interval	Fourth dimension (s)	Δt (s)	δΔt (s)	Δy(chiliad)	δΔy(m)	δΔy/Δy	Δx(g)	δΔx(yard)	δΔx/Δx
1 to 3	0.ii	0.1	0.0002	-0.023	0.0005	-0.021739	0.095	0.0005	5.26E-03
ii to four	0.3	0.ane	0.0002	-0.038	0.0005	-0.013158	0.099	0.0005	5.05E-03
three to 5	0.iv	0.1	0.0002	-0.049	0.0005	-0.010204	0.092	0.0005	5.43E-03
four to vi	0.5	0.1	0.0002	-0.060	0.0005	-0.008333	0.091	0.0005	5.49E-03
5 to seven	0.six	0.one	0.0002	-0.070	0.0005	-0.007143	0.088	0.0005	five.68E-03
vi to eight	0.7	0.i	0.0002	-0.081	0.0005	-0.006173	0.079	0.0005	half-dozen.33E-03

Tabular array ane.2- Measurement of distance traveled in both dimensions in each dot interval, considering any necessary errors.

The in a higher place table shows the results of the two-dimensional motion of a puck. The distance in the y direction is Δy and the altitude in the 10 direction is Δx, which were measured direct from the carbon paper. Where their uncertainty in both dimensions is equal to ±0.0005 g. the relative error can be derived past using the formulas $\frac{\mathit{\delta \Delta y}}{\mathit{\Delta y}}$

and $\frac{\mathit{\delta \Delta x}}{\mathit{\Delta x}}$

. Their values are shown in Table 1.2

Effigy 1.2 – Plot of the human relationship of Δx and t, including a trendline, equation of the line, and error bars at each point.

The to a higher place plot conspicuously shows that distance in the x direction increases for each 0.i seconds time interval. In a frictionless surround, the change in the distance in the x management should be constant. Still, the change in the x direction in both the graph and the tabular array shows a slight variation. This could be due to friction of the puck and carbon paper or but only air friction. This error could have not been avoided, as it is caused by the surroundings.

Figure 1.three – Plot of the relationship of Δy and t, including a trendline, equation of the line, and error bars at each point.

The above plot clearly shows that distance in the y direction decreases for each 0.1 seconds time interval. From Figure one.3, the trend of the marks is parabolic dissimilar the distance in the x management (linear). The puck has a negative acceleration due to the acceleration of the puck in the downwards direction. While, the distance of ten vs time is linear which ways that it has a abiding and positive velocity and therefore it should feel zero acceleration in the horizontal direction.

Dot Interval	Time (s)	Δt (s)	Vy (thousand/south)	δVy/Vy	δVy	Vx (m/s)	δVx/Vx	δVx
i to iii	0.2	0.1	-0.115	0.005267	-0.00061	0.475	0.021740	0.010327
two to four	0.3	0.1	-0.19	0.005054	-0.00096	0.495	0.013159	0.006514
iii to 5	0.4	0.ane	-0.245	0.005438	-0.00133	0.460	0.010206	0.004695
four to six	0.five	0.1	-0.3	0.005498	-0.00165	0.455	0.008336	0.003793
v to vii	0.6	0.1	-0.35	0.005685	-0.00199	0.440	0.007146	0.003144
6 to viii	0.seven	0.1	-0.405	0.006332	-0.00256	0.395	0.006176	0.002440

Tabular array 1.3- Adding of vertical and horizontal velocities and their respective relative errors

The higher up table shows the results of the vertical and horizontal velocity. By using the measured distance in the x and y management of the puck at every time interval, the vertical and horizonal could be determined by using the formula:

$5=\frac{\mathrm{\Delta y}}{2\mathrm{\Delta t}}$

The relative error for both the velocity in the y and x direction can be calculated by using the formula:

$\frac{\mathrm{\delta v}}{\mathrm{5}}=\sqrt{{\left(\frac{\mathrm{\delta \Delta }y}{\mathrm{\Delta }y}\right)}^2+{\left(\frac{\mathrm{\delta \Delta t}}{\mathrm{\Delta t}}\right)}^2}$

The propagated error is given every bit δVy/Vy and δVx/Vx, where the uncertainty can be calculated past multiply the velocity (in the y direction and x direction) by the specific propagated error to each velocity.  It tin also be noted that if the environment was completely frictionless, the velocity in the x management should be equal in every time interval. Only as shown in the Table one.3, the velocity in the x direction slightly differ which could exist due to air resistance of friction of the carbon paper on the puck.

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In addition, the velocity in the y direction should be increasing subsequently each time interval considering the gravitational acceleration should speed upwards the velocity past nine.viii m/s every second. All the same, since the puck is moving in a downwardly direction, the velocity in the y direction should get more negative after each time interval. This is clearly shown in the Table 1.3.

Figure ane.four – Plot of the human relationship of velocity in the ten direction and t, including a trendline, equation of the line, and mistake bars at each betoken.

The graph demonstrated the relationship between the velocity in the x management and fourth dimension. Equally stated higher up, the graph is almost showing a pattern that is parallel to the 10 axis. Nonetheless, due to friction, the result showed otherwise. The dispatch of the graph is -0.1629 m/s². The IPL best fit line calculator was then used to determine the uncertainty of the acceleration which is ± 0.0114945 1000/s².

In a non-frictionless surface, the acceleration should be equal to 0. This can exist taken to be systematic error of friction in testing.

Figure 1.5 – Plot of the relationship of velocity in the y management and t, including a trendline, equation of the line, and error confined at each point.

The graph demonstrated the relationship between the velocity in the y direction and time. The plot is showing a downwards linear blueprint. The velocity in the y direction should be increasing subsequently each time interval because the gravitational dispatch should speed up the velocity by 9.viii m/s every second. Since the puck is moving in downward direction, the velocity in the y management should become more than negative in this case.

The y intercept of the line should equal the initial velocity of the puck in the y direction.

The acceleration in the y direction is -0.5671 m/s². The IPL best line calculator was then used to determine the dubiety of the acceleration which is ± 0.00327275 m/s².  This can be due to a systematic fault and/or random error in measuring the altitude of the dots on the carbon newspaper.

Equally a cheque, we compared the acceleration of Vy and the acceleration measured in investigation ane. The acceleration of both investigations should exist almost equal or fall within the range of mistake, since they are experiencing the aforementioned gravitational dispatch value of 9.eight m/s². The acceleration from the y management in investigation 2 is equal to -0.5671 m/due south² ±0.00327275, while the accusation for investigation 1 is 0.5643 m/due south² ±0.00598223. The negative sign shows the direction of the puck; thus it can exist dismissed in this case. The acceleration of both investigations is about entirely equal.

Investigation 3

Setup and Procedure

No necessary setup was required for this investigation, as only calculation was performed. The goal of this investigation to summate the acceleration due to gravity past using our results from investigation 1 and ii. This tin exist done past measuring the length and height of the incline.

Data and Analysis

The goal of this investigation is to calculate the acceleration of complimentary fall both investigations and their incertitude. And so both gravitational accelerations are used to obtain their average and their respected errors.

The formula to derive yard using the angle of the tabular array is:

$\left[\mathrm{i}\right]\mathit{ g}=\frac{\mathit{ay}}{\mathit{sin\theta }}$

Sinθ can be obtain past diving the height past the length of the table (also known every bit the hypotonus). The height and length obtained in this investigation is 0.04 and 0.645 m respectively with an uncertainty of ± 0.0005 m. Thus, sin will equal:

$\mathrm{Sin\theta }=\mathrm{}\frac{0.04}{0.645}=$

0.062016

With this information, k of both investigations could be calculation using the formula [1]. Using the y acceleration of Investigation 1 at 0.56m^two/s, $g1=\frac{0.56}{0.062016}=\mathrm{nine}.04m^2/\mathrm{southward}$

.

Using the y acceleration of Investigation 2 at 0.56m²/due south, $g\mathrm{ii}=\frac{0.56}{0.062016}=9.04m^2/\mathrm{south}$

.

If done correctly, the gravitational acceleration of both should be equal, which was obtained in this case. However, the gravitational acceleration is supposed to exist 9.lxxx ${\mathrm{chiliad}}^2/s$

, which was non obtained. This could be due to a systematic mistake that as a group could not figure out. It could also be since one of the legs of the air tabular array given to us was broken, making the tiptop of the tabular array uncertain.

Their uncertainties can be found using the rules of fault propagation in Appendix A by commencement finding mistake for the $\mathit{sin}\left(\theta \right)\left[2\right],$

and and so for the $g$

.

$\left[2\right]\mathit{}\frac{\mathit{\delta sin\theta }}{\mathit{sin\theta }}=\sqrt{{\left(\frac{\mathit{\delta h}}{h}\right)}^{\mathrm{ii}}+{\left(\frac{\mathit{\delta l}}{l}\right)}^2}=0.012524$

The errors of which are derived using the propagation formula: $\frac{\mathrm{\delta g}}{\mathrm{g}}=\sqrt{{\left(\frac{\mathrm{\delta a}}{\mathrm{a}}\right)}^2+{\left(\frac{\mathrm{\delta sin\theta }}{\mathrm{sin\theta }}\right)}^{\mathrm{ii}}}$

Uncertainty of ${\mathrm{grand}}_1$

is:

$\frac{\delta {\mathrm{one\; thousand}}_1}{g_1}=\sqrt{{\left(\frac{\delta a_1}{a_{\mathrm{i}}}\right)}^{\mathrm{ii}}+{\left(\frac{\mathit{\delta sin\theta }}{\mathit{sin\theta }}\right)}^2}=0.016408$

Uncertainty of $g_2$

is:

$\frac{\delta g_2}{g_{\mathrm{ii}}}=\sqrt{{\left(\frac{\delta a_2}{a_2}\right)}^{\mathrm{two}}+{\left(\frac{\mathit{\delta sin\theta }}{\mathit{sin\theta }}\right)}^{\mathrm{ii}}}=0.059054$

The resulting error for chiliad_i is ±0.016408mⁱⁱ/southward, and the mistake for chiliad_two is ±0.059054m²/due south. Finally, we need to summate the averages of the accelerations and their uncertainties. Nosotros can do this by using a normal average calculation and then that ${\mathrm{grand}}_{\mathit{av}}=\frac{{m_{\mathrm{one}}+g}_2}{2}$

, or ${\mathrm{yard}}_{\mathit{av}}=\frac{\mathrm{nine}.04+9.04}{2}=9.04{\mathrm{chiliad}}^2/\mathrm{southward}$

. Since both accelerations are equal, their average dispatch will non change.

Incertitude of ${\mathrm{1000}}_{\mathit{av}}$

is:

$\delta g_{\mathit{av}}=\frac{\sqrt{{\left(\delta g_1\right)}^2+{\left(\delta g_2\right)}^2}}{2}=0.277\frac{m}{s^{\mathrm{two}}}.$

The last acceleration is equal to 9.04±0.277 m/southward². This value does not concur with the theoretical value of gravity 9.81 m/s². This is mainly caused by a systematic error when measuring the length and height of the table. Also noting that the air table had a deformed leg, which could upshot In the change in height. However, the calculated doubtfulness is equal to ±0.277 which ways the experiment was done well but not perfect. 1 of the reasons for the error is frictional force of the puck with the carbon paper, or even just air friction. And the other would exist a systematic fault as stated to a higher place.

Conclusion

Fifty-fifty though the experiment was performed well, in that location will however be uncertainties and errors. This experiment replicated Galileo experiment past using a puck and an air table. A carbon paper and a spark timer were used to mark the movement of the puck every 100ms. A replica of the i-dimensional motion was performed in the first investigation. Values of velocities were calculated by measuring the change in distance of the puck at every time interval. This provided the data to calculate the acceleration of the puck by taking the derivative of the equation of the line. The calculated acceleration for the start investigation is 0.5643 1000/s² with an uncertainty of ±0.00598223m/south²

In the 2nd investigation, a replica of Galileo second dimensional move was performed. The velocities in both the x and y direction were calculated past measuring the change in distance at each time interval (in the ten and y management). The human relationship between velocity and fourth dimension for each management were graphed separately. This provided the data to summate the acceleration of the puck by taking the derivative of the equation of the line. The calculated acceleration for the velocity in the ten direction -0.1629 m/s² ±0.0114945  The calculated acceleration for the velocity in the y direction -0.5671 m/southward² ± 0.00327275 The dispatch values of investigation 1 and investigation 2 in the y direction were compared to evidence if the experiment was washed correctly. Both accelerations were within range of fault.

For the 3rd investigation, the gravitational acceleration for each investigation and its average (in the y direction) was calculated by measuring the top and length of the table. By doing these measurement, the angle of the incline was adamant which is necessary in solving for the gravitational acceleration. The calculated average dispatch was $g_{\mathit{av}}=9.04\frac{m}{s^{\mathrm{ii}}}\pm 0.277\mathrm{}$

. The actual value did non fall within mistake of the theoretical value of $9.81\frac{g}{s^{\mathrm{ii}}}.$

  This is mainly due to systematic mistake when measuring the distance of the puck at each time interval. This could likewise be due to a slight friction between the puck and the carbon paper. Some other issue is that the air table given had a unstable leg, which could bear on the acme of table and affect our results. To get a more than accurate gravitational acceleration, the experiment needs to be done several times to get consequent measurement and to be given a stable air tabular array.

Questions

1. Discover the total time to reach the bottom of the air tabular array for Investigation 1 and 2. Are there times all the same, as shown in the picture of the cannon? Why or why not?
   1. Using the kinematics equations, x = v_0xt + $\frac{\mathrm{i}}{2}$

      at², and y = five_0yt + $\frac{\mathrm{one}}{2}$

      at². We can summate the total time to reach the bottom for both investigations. The merely blazon of acceleration acting on the puck in the y direction of both investigations is the gravitational acceleration.

   2. The total time to reach the lesser for investigation 1 is:
   3. The total time to reach the bottom for investigation 2 is:
   4. The slight difference in fourth dimension is probable due to random mistake.
2. If the acceleration of a puck falling down on an incline of bending $\theta$

   (where $\theta$

   < ${45}^0$

   ) is ${\alpha }_{\theta }$

   , what is ${\alpha }_{\mathrm{two}\theta }$

   ?

   1. Using formula for acceleration of a puck falling down an incline of angle for $a_{\theta },$ $a_{\theta }=g*\mathit{sin}\left(\theta \right)$

      , and for $a_{2\theta }=g*\sin \left(\mathrm{ii}\theta \right)=g*2\mathit{sin\theta cos\theta }={2a}_{\theta }*\cos \theta .$

3. You determined the acceleration from the gradient of you lot plot v vs. t. What is the meaning of the intercept of your line with the five axis? Do yous expect this value to be close to the origin?
   1. The initial velocity will be equal to the intercept of y axis. In the get-go investigation I would expect the initial velocity to equal cypher. While in the second investigation, I would expect the initial velocity to equal the value of the push button of the puck.
4. If yous performed Investigation 1 and 2 again with an incline of $\theta =0$

   , what would the trajectories look similar in each case?

   1. If the angle of incline equals zero, then there would exist no motion or trajectories in investigation one every bit the puck volition non motility. While in investigation 2, the puck will experience a constant velocity, equally in that location is no gravitational acceleration.
5. If there were insufficient air menses through the air table, how would the slope of your graphs exist affected?
   1. Bereft air flow will crusade the puck to feel a frictional strength. This would mean that the gradient of the y velocity versus time graphs would exist less steep and the x velocity versus time graphs would exist more than steep. Thus, slowing down the puck.

Honors Questions

1. How would your results of Investigation 1 and 2 change if the masses of the pucks were doubled?
   1. If the mass of the puck were double would not affect the result of acceleration.

References

• O.Batishchev and A.Hyde, Introductory Physics Laboratory, p.263, Hayden-McNeil, 2017.

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