March 15, 2017 Lab 5: Trajectories

Lab 5: Trajectories

Chris Ceron, Amy Chung, John Choi

Purpose

To apply our understanding of projectile motion to predict the impact point of a ball on an inclined board

Materials

• Aluminum "v-channel"
• Marble
• Board
• Ring stand
• Clamp
• Paper
• Carbon Paper

Theory/Introduction

We set our apparatus by connecting two v channel rails to create a ramp. The first rail was set at an incline to provide the marble with velocity. The second rail allowed the ball to launch horizontally.

Part 1: Determine the initial velocity of the ball

Procedure:

In order to record where the marble would hit the floor, we launched the ball and noted where the ball lands. Then we taped a piece of paper on that location, and added carbon paper on top to create marks each time the marble landed. 

We then launched the ball five times from the same place and ensured that the ball would land virtually in the same place each time.

After repeating the process five times, we recorded the height of our apparatus by taping a piece of string with a weight onto the edge of the table (also where the marble launched out of the v-channel) and measuring along the string. The weight on the string allowed for the string to fall straight down and perpendicular to the floor. To record the distance the marble traveled, we set our meter stick to start from the end of the hanging string, and measured to the center of the five landing points. We assigned an uncertainty value by measuring the distance between the farthest landing point the center point. Using kinematics, we were able to calculate the initial velocity of the marble.

Calculations

Because the marble was launched horizontally, we know that the angle in which it was launched was at 0 degrees.

This means that the velocities in the x and y direction are:

To calculate for the initial velocity, we first had to calculate time. We did this by using a kinematics equation in respect to the y component of the trajectory.



Since our marble was launched horizontally, we know that the velocity in the y direction is 0. The acceleration of the marble is due to gravity. The height of the ball is delta y.



Once we had our value for t, we used a kinematics equation in respect to the x component to solve for the initial velocity.



We know that the acceleration of the ball in the x component is 0



Measured Data





Initial Velocity: 1.619313504 m/s.

Part 2: Determine where the marble will strike an inclined board

Procedure:

We added a board such that one end touched the end of the table, and the other end on the floor (like the figure above). To ensure the board did not move, we added weights to the end of the board and used duct tape on the weights.We then launched the ball, noted where the ball landed on the board, and placed a new piece of paper with carbon paper on top. Once our apparatus was set, we launched the ball five times and measured the distance d.

Calculations

To mathematically find the distance along the wooden board, we expressed distance in terms of its x and y components





We then used the same kinematics equations as part one, starting with the x component. Using this kinematics equation, we solved for t.



We plugged our value of y into our kinematics equation for y
'




We then solved for d



Measured Data

Conclusion

Through the mathematical approach, we calculated that distance d was 0.938339123 m, and our experimental distance was within that range. There were a few sources of error within our calculations. For part one, our landing points were more scattered than we anticipated. We ran the experiment several times to receive more accurate data. Our final landing points were relatively closer than our initial landing points, and our calculated uncertainty was 1.7% for the final points. We encountered the same issue in part two of our experiment, and had a calculated uncertainty of 2.0%.

Our main reason for our uncertainty was due to the impact point on the two v-channels. Instead of smoothly transitioning from one rail onto the next, we noticed the marble would hit the second v-channel potentially causing it to lose some velocity and land very scattered in relation to the other test cases. However, we found that even with these uncertainties our experimental values were still reasonable and within our mathematical values. 

The intersection of both v channels

March 13, 2017 Modeling the Fall of an Object Falling with Air Resistance

Lab 4: Modeling the Fall of an Object Falling with Air Resistance

Chris Ceron, Amy Chung, John Choi

Purpose

To determine the relationship between air resistance force and speed.

Theory/Introduction

We expect that the air resistance force on a particular object is dependent on the object's speed, its shape, and the material it is moving through. This can be modeled through the following power law:

k and n are unknown variables

The k term takes the shape and area of the object into account.

To find the unknown variables, we had to create several scenarios where we calculate the terminal velocities within each scenario. The objects used in the scenario had to have the same size and shape, but also needed different masses. Once these objects reached their terminal velocity, the downward pull of gravity exactly balanced the upward force of air resistance.

In our lab, we used coffee filters:

To collect our data, we used the video capture feature within LoggerPro, and created six different cases by dropping multiple coffee filters stacked together in the Design Technology Building. Our cases went from dropping 1 coffee filter to 6 coffee filters. 

Black drape was used to clearly see the falling coffee filter

Through the videos we captured, we were able to create graphs for each scenario that allowed us to plot position vs. time graph that would allow for us to calculate the terminal velocity. 

Procedure

The professor was in charge of dropping the filters and adjusting each case by adding a coffee filter. The black drape had two pieces of blue tape that were each one meter apart from each other. This served as a scale for our position vs. time graph. In order to calculate the gravitational force, we needed to know the mass for each coffee filter. If we were to take the mass of an individual coffee filter, our uncertainty would be large. To reduce the uncertainty, the professor took the mass of 50 coffee filters, which allows for us to divide both the mass and the uncertainty of an individual coffee filter.

Once we recorded each case, we used LoggerPro to plot the position of the coffee filter after every three frames. This created a position vs. time graph. In each graph, we could calculate the terminal velocity by taking a linear fit in the more linear part of the graph. Each graph would become more linear within the end of the graph because the coffee filters would reach terminal velocity and move at a constant distance per time as a result of air resistance force balancing the gravitational force.

Position vs. Time Graphs for Each Scenario:

Graph for one coffee filter: Terminal velocity is -1.136 m/s

Graph for two coffee filters. Terminal velocity is -1.715 m/s

Graph for three coffee filters. Terminal velocity is -2.035 m/s

Graph for four coffee filters. Terminal velocity is -2.439 m/s

Graph for five coffee filters. Terminal velocity is -2.578 m/s

Graph for six coffee filters. Terminal velocity is -2.786 m/s

Once we calculated the terminal velocity for each case, we created a Force of Air Resistance vs. Terminal velocity graph that allowed for us to find our unknown variables k and n. 

Results:

k = A = 0.005303 ± 0.0008248

n = B = 2.195 ± 0.1675

Numerical Calculations

Our next approach was to model the fall of the object with air resistance and predict the terminal velocity in each case through Microsoft Excel. To set this up, we used the numerical approach given in our lab handout

Once we set up our excel sheet, we used a time interval of 1/125th of a second and found where our velocity remaining constant. 

Terminal velocity for one coffee filter

Terminal velocity for two coffee filters

Terminal velocity for three coffee filters

Terminal velocity for four coffee filters

Terminal Velocity for five coffee filters

Terminal velocity for six coffee filters

Conclusion

Comparison of the two terminal velocities we calculated through LoggerPro and Excel are relatively close, but are different due to some uncertainties in our measurements and calculations. When plotting the position of the coffee filter as it was falling, we are subject to human error as our plots may not be as accurate as we would like. In the video recording, the coffee filters would occasionally fall outside of the black drape, making it harder to accurately plot a position within that frame. When plotting our terminal velocities to calculate for k and n, our correlation was .9942. Although this is a good correlation, it is still not the highest and most accurate.

Our ratio of uncertainty in our n value also caused a difference between the two terminal velocities, as the ratio of dn/n was approximately 7.6%. In order to calculate a more accurate terminal velocity through LoggerPro, we would need to plot more accurate points in order to find a more accurate k and n value.

March 08, 2017 Lab 3 Non-constant Acceleration

Lab 3 Non-constant Acceleration

Chris Ceron, Amy Chung, John Choi

Purpose

To calculate the final position of an elephant traveling with a non-constant acceleration using a numerical approach (Microsoft Excel) as shown below.

Theory/Introduction

An elephant on frictionless roller skates reaches the bottom of a hill and arrives on level ground with an initial velocity of 25 m/s. At that point, a rocket mounted on the elephant's back generates a constant 8000N thrust in the Elephant's opposite direction of motion. The mass of the rocket changes with time due to the burning of fuel. 

Sketch of problem

Analytical Approach (Given):

The analytical approach begins with modelling an equation for acceleration as a function of time. Then we take the integral of acceleration to find a velocity function, and integrate once more to find a position function. We then use the function to solve for the time at which the velocity of the elephant is zero, and plug the value into our position function to find the distance traveled in that time.

Numerical Approach:

Using Microsoft Excel, we were able to create functions that calculate when the elephant's velocity is zero, and the position at that time. Through the analytical approach, we know that our time is 19.69075 seconds and the distance the elephant traveled was 248.7 meters. 

Data Breakdown:

Rows:

1 - Initial mass of both the elephant and the rocket

2 - Initial velocity of the elephant

3 - Fuel burn rate

4 - Thrust of the rocket, measured in Newtons

5 - The change of time within our time column

Columns:

A - Time

B - Acceleration

C - Average acceleration

D - The change of velocity

E - Velocity

F - Average Velocity

G - The change in position

H - Position

Data

Data using time interval of 1 second

Data using time interval of .1 second

Data using time interval of .05 second

Conclusion

In this data, we search for when the velocity of the elephant (column E) goes from positive to negative. At some point in between those rows, the velocity of the elephant reaches zero. As we reduce our time intervals, the point of the velocity going to zero gets closer, and our approach in finding the distance becomes more accurate. With a small enough interval, we could find the exact distance the elephant travels. 

1. Through our numerical approach, we were able to accurately find the position of the elephant. The position value found through the analytical approach was 248.7 meters. Through our data, we found the position value to be 248.6966±0.0005m.


2. As our time intervals got smaller, we could see the position of the elephant to change only in the thousandths place. To continue to get more accurate results, you could continue to decrease the time intervals. 


3. Determine how far the elephant would go if its initial mass were 5500kg, the fuel burn rate is 40 kg/s, and the thrust force is 13000 N.

We adjusted our parameters and found that the distance the elephant now travels is 164.03356±0.0009m.

March 6, 2017 Lab 6: Propagated Uncertainty in Measurements

Lab 6: Propagated Uncertainty in Measurements

Chris Ceron, Amy Chung, John Choi

Purpose

The purpose of this lab is to calculate the density of metal cylinders and the propagated error in our measurements. 

Vernier Caliper with micrometer measurements

The two cylinders used, Copper and Zinc respectively 

Items Used:

• Metal (Copper and Zinc) cylinders
• Vernier Caliper with micrometer measurements
• Electronic Scale

Theory/Introduction

This lab applies the use of propagated uncertainties to accurately calculate the range of the densities for our metal cylinders. Propagated errors are found in every measurement taken. To find the total propagated uncertainty, we first found the individual uncertainty with each measurement of height (cm), diameter (cm), and mass (g).

Individual Uncertainties

Our Vernier Caliper used micrometer measurements, which divides each millimeter with 10 division markers. This allowed us to measure the height and diameter of each cylinder to the nearest hundredth, Our electronic scale was accurate to the nearest hundredth's place. The result of the individual certainties were:

Height(cm): 0.01 cm
Diameter(cm): 0.01 cm
Mass(g): 0.01 g

Method Used to Calculate Total Propagated Error

To calculate density, we used the following formula:

Because we measured the diameter of the cylinder and not the radius, we expressed r in respect to the diameter and simplified to end up with:

To find the total propagated error, we took the natural log on both sides, and used a natural log identity to separate each variable:

Taking the derivative of both sides resulted in:

Using the square root form, the final result of the formula resulted in:

Data

	Mass (g)	Height (cm)	Diameter (cm)
Cylinder 1	28.93	3.28	1.29
Cylinder 2	81.48	4.89	1.59

Calculations

Calculation of density and propagated error for both cylinders

Conclusion

We were able to calculate the propagated error of the density using the uncertainty of each measurement as shown above. To determine our greatest source of error, we went back and calculated which measurement gave us the greatest uncertainty. The uncertainty of each of our measurements were by a unit of .01; however, the uncertainty within the measurement of the diameter gets doubled by a coefficient of two, producing a greater number within the delta rho calculation which causes greater uncertainty in our calculation.

Our value of density for zinc:


Our value of density for copper: