Predicting the Path of a Projectile Motion

1. Ana Leyva, Andrew Imm, Jonathon
The purpose of this experiment was to come up with an equation that would help us predict the point of impact of a projectile motion.
 For this experiment we are trying to calculate the distance that a ball will travel before it hits a wooden plank sitting against a table at a certain angle. In order to calculate this we first have to find the velocity that the ball travels at after it is launched. Then we will use that velocity and the angle to calculate at what distance the ball will hit the board.

We first started off this lab by setting up an apparatus that would allow us to launch the ball. Then we launched the ball to see where it would land on the ground. We then put a piece of carbon paper on top of a white sheet of paper that would allow us to see where the ball is hitting the ground. We then got the average distance from the end of the table to the spot of the ground where the ball hit. We used this distance  and the height of the table to calculate the velocity of the ball when it was launched.

After calculating the initial velocity of the ball we now had to figure out at what distance the ball would hit a plank that was leaning against the table. In order to figure this out we first had to realize that any distance on the plank of wood could be represented as x = dcos(theta) and y = dsin(theta). We then used the equation V_initial *t = dcos(theta) .5gt^2 = dsin(theta) V_initial = 1.53 m/s^2 and theta = 50.0. We then used those equations to solve for d. This d gives us the distance at which the ball is expected to hit. 

 We got the uncertainty in d and v_initial by taking the ln of the v_initial and d equations and then taking the derivative of those two equation. We the plugged in numbers to get the uncertainty in V_initial and d.

My calculation for the distance that the ball would land away from the plank of wood was pretty close but it was less than the distance that I calculated even taking into consideration the propagated uncertainty. The distance I calculated was .8848 m +- 0.00269 with the actual distance being .86m +- 0.1 m. The reason why my distances aren't exact could be because of a calculation error or because my measurements weren't as exact as they could have been. Other than that slight movements of the table and/or apparatus could have also contributed to the differences of the distance 

Modeling the fall of an object with air resistance

Ana Leyva, Andrew Imm, Victoria
3/13/17
By doing this experiment we were trying to find a relationship between the air resistance force and speed.
With this experiment we were trying to find the terminal velocity of 1,2,3,4,and 5 coffee filters as they fall from  a second story balcony. We calculated the terminal velocity by first getting a linear fit for our 5 position vs time graph for all five coffee filters. Once we did the linear fit we got the slope for each five of those graph and got a value for our terminal velocity. Then we plotted a velocity vs Fnet graph to get the values of k and n. We did this because our formula F_resistance=kv^n lets us know that if we plot a velocity vs F_resistance graph we will get our values for k and n. We will then use these values in order to use excel and to find the terminal velocity of our coffee filters.

Linear fit for two coffee filters

We started this experiment by heading to the design and technology building and capturing on video different numbers of coffee filters falling from the balcony. From these videos we were able to get a position vs time graph for 1, 2, 3, 4, and 5 coffee filters falling. From these graph we were able to figure out the terminal velocity for the coffee filter by doing a linear fit on the end portion of our graph. Given the equation F_resistance = kv^n  we realized that if we plotted a velocity vs. Fnet graph and did a (y=ax^b) auto fit we would be able to find the values for k and n.

After this was done we used the values of k, and n to plug them into excel in order to find another way to predict the terminal velocity of our coffee filters. We set up our excel sheet by setting delta t, m, g, k, and n as set values that could later be adjusted. We then used these values to calculate t, v, F_net, a and delta v.  t was calculated by adding t_initial + delta t, v = v_initial + delta v, F_net = m*g-k*v^n, and finally delta v = a*delta t plugging all of this in correctly let us see the terminal velocity for the different number of coffee filters.

The terminal velocity that we calculated from excel and those that we got from the graph were not off by too much, which means that we have a pretty good model. They were all mostly correct to the first decimal place.

Non-Constant Acceleration Activity

1.Ana Leyva, Andrew Imm, Victoria Bravo
2. The purpose of this lab was to find a numerical approach that allowed us to find the time and distance it takes for an object to stop. The exact question was,"Find how far the elephant goes before coming to rest."
3. In order to find the distance that it takes for the object in this problem to stop we were given some information about the problem. The information that was given to us was: V_initial= 25 m/s, Force= 8000 N, Mass = 6500 kg, and that B_burning rate = 20 kg/s. With this given information we were able to find the distance and time it took for the elephant to stop. We were able to do this two ways analytically, and numerically. The analytical way was by finding an equation for the acceleration and then integrating it too find the v(t) equation and the x(t) equation which allowed us to find the two thing we were looking for. The second way we were able to do it was by plugging in the information that was given to us into excel and using those numbers to find the acceleration, average acceleration, change in velocity, velocity, average velocity, change in displacement, and distance all in a certain time. Once we had this done we were able to check in our excel sheet for the time at which the velocity was closest to zero and find the distance that the elephant had traveled at that time.
4. First, we started off by creating some set variables whose values weren't going to be changing. These were going to be those initial values that were given to us and change in time. After that, we set a row that showed us what variable we were going to be calculating in each column. Our first column was going to calculate time, followed by acceleration, average acceleration, change in velocity, velocity, average velocity, change in displacement, and distance. We then calculated all of these by using a = F/(m-bt), a_avg = (a_f - a_i)/2, delta v = a_avg* delta t, v = v_i - delta v,         v_avg = (v_f - v_i)/2, delta x = v_avg* delta t, and x = x_i + delta x. This allowed us to find the time and distance it took for the elephant to come to rest by looking at the the point where the velocity was closest to zero.

5.                   Using delta t = 1 we get that the elephant stops at 19 s and travels 248.4 m

          Using delta t = 0.1 we get that the elephant stops at 19.6 s and travels 248.7 m

               Using delta t = 0.05 we get that the elephant stops at 19.65 s and travels 248.7 m 

                 With the analytical approach we had gotten t = 19.69 s and x = 248.7 m 

In conclusion our excel sheet did a really good job in helping us find the time and distance it took for the elephant to come to rest. Our closest result gave us a time of 19.65 and a distance of 248.7. This result agreed with the answer we got with an analytical approach. The way one would be able to tell if our time interval was small enough is by looking at how much the distance changes by with the different time interval. In our case we knew that using a delta t of 0.1 was good enough because there were more than ten numbers times where the velocity was close to zero that gave us 248.7 m when the distance was rounded. If the elephant were to change and have an initial mass of 55000 kg, the burning rate was to change to 40 kg/s, and the force was to go up to 13000 N, the elephant would travel a distance of 164.0 m before coming to rest. 

   

Calculating Density and the Error in Density Measurement

1. Ana Leyva and Christian Valencia
3/6/17
2. For the first part of this lab we were trying to calculate the density of two metal cylinders. In my case I had to find the density of an iron and aluminum. After we found the density of both of those cylinders we calculated the propagated error in both of the density measurements.
3. The first thing that we did for this lab was to measure the diameter, height ,radius,and mass of the two cylinders. Once we had these measurements we calculated the density by dividing the mass from  the volume. Then, we still had to calculate the propagated uncertainty of these density measurement.

                                                                     Caliber

 4. This lab had a very simple procedure. First, we calculated the diameter and height of the cylinder using a caliber. We used a caliber because it allowed us to find the diameter and height to two decimal places.

 Next, we calculated the mass of the cylinder using a scale that gave us the mass to first decimal place. After that, we used the density formula to calculate the density of the two metal cylinders. Once we calculated the density we were able to calculate the uncertainty in the density measurements.

 We did this by getting the density formula and multiplying everything by the natural log. We then took the derivative of everything and got dp/p. We then squared and took the square root of the other side. When we got the answer of this we multiplied it by the density to get the uncertainty of measurements. This gives us how off our density might be off by.

8. In conclusion my results weren't too far off from the actual density. For my iron cylinder I got a density of 7.27 +- 0.12 g/cm^3 with the actual density being 7.87 g/cm^3. For the aluminum cylinder I got a calculated density of 2.64 +- 0.049 g/cm^3 with the actual density being 2.70 g/cm^3. The difference in these measurements might be due to us not having pure iron and aluminum cylinder. These cylinders also have small amounts of other measurements mixed into them. This might be one of reasons why our calculated densities are different than the actual densities. Another reason for these differences might be an error in measurement of calculations.    

Determining the Value of g

1. Ana Leyva
3/1/17
2. With this experiment we were trying to calculate the acceleration of a free falling object.
3. By taking the marks that the spark generator created on a piece of spark sensitive tape we plotted the distance between the marks into an excel chart. using the values on the excel chart we then created a velocity vs time graph and a distance vs time graph which both helped us find the acceleration of the free falling object. The slope of our velocity vs time graph will give us the acceleration and the ax^2 value of the polynomial fit of the distance vs time graph will also give us a value for the acceleration when you multiply it by two.

 4. The procedures for this experiment were pretty simple we had to measure the distance between the marks that were created on the paper. The dots on the paper corresponded to the the position of the falling mass every 1/60th of a second. After that we had to enter the distance into an excel spreadsheet. Once that information was all enter we calculated the change of x, the mid interval time, and the mid interval speed on that same excel sheet. We did this differently for each column. For the time column we started at zero and increased each one after that by 1/60 of a second. For the distance column we started at zero and input all of our measurement after calculating them with a ruler. For the change in x column we subtracted the distance from the row we were on from the row before it. The mid time interval was calculated by adding the time from the row we were on by 1/120. Finally, the mid-interval speed was calculated by dividing the change in x by 1/60. After all the values were calculated we created a velocity vs time graph and a position vs time graph. By creating a linear fit and a polynomial fit of order 2 we were able to get the values of acceleration from the two graphs. For the velocity time graph the acceleration was given by 960.65 and for the distance vs time graph the acceleration was given by 484.485. However, you have to multiply the acceleration for the distance vs time graph by two because the acceleration is given by .5ax^2 so the acceleration for that graph is 968.97.

7. The reason why we graphed a position vs time graph and a velocity vs time graph was because those two graphs were going to help us find the acceleration of the free falling object.
8. The acceleration that we got from our velocity vs time graph was 960.65 cm/s^2 and for our position vs time graph we got 968.97 cm/s^2. Both of these values are less than the expected value which was 981.0 cm/s^2. The absolute difference was -20.35 cm/s^2 and -12.03 cm/s^2. The reason why our answer wasn't what we expected it to be could be due to air resistance which wasn't taken into account when doing this experiment.
Questions:
1.

2. In order to get the acceleration due to gravity  from my velocity vs time graph I had to get a linear fit from my data point and get the slope of that line. In this case the slope for that line was 960.65 cm/s^2.
3. The acceleration due to gravity can be found using a polynomial fit of the second degree for the position vs time graph. The formula for the polynomial fit is y = y_initial + v_initial*t +.5at^2 from this equation we set 484.485t^2 equal to .5at^2 and we get a= 968.97

27-Feb-2017:Finding a relationship between mass and period

1. Lab 1 Finding a relationship for mass and period

Ana Leyva, Christian Valencia

Completed on 2/27/17

2. With this experiment we are trying to find a correlation between the period and the mass of an object. We are trying to do this by first finding out how the period of an object is affected when weight is added to the pendulum we are using to measure the period. Once this is found we will come up with an equation that will allow us to approximate the mass of an object when the period is known and will allow us to approximate the period when the mass is known.

3. In this experiment we are trying to come up with an equation that will do the best job at approximating the mass of an object when the period is known and vice versa. In order to do this we first had to find the period when mass was added to the pendulum that was being used to calculate the period. Once we had that information we put it into a ln(M+Mtray) vs lnT graph. By putting it in that graph and messing around with what we thought was the mass of the tray we were able to get a linear fit on that graph with a correlation of 0.998.We then got three different values of the Mtray(minimum, intermediate, and maximum) that gave us a correlation of 0.998 and also gave us the slope and y intercept of the graph.

4. At the beginning of this experiment we set up a pendulum and photogate that would allow us to calculate the period of the pendulum. We used Logger Pro, the pendulum, and the photogate to help us calculate the period. We then started to record the period of the pendulum when there was no mass added. Then we added 100g and once again calculated the period. We kept on adding 100g to our pendulum and calculating our period until we got to 800g. When we had the period for all the masses we created a ln(M+Mtray) vs lnT graph. We then adjusted the values of the Mtray to try to get a near perfect correlation of 0.998. With each value of Mtray that we got that gave us a near perfect correlation we also got the slope and y intercept of the line when the Mtray was a certain mass. The values that we got from our line allowed us to come up with a mathematical model for the behavior of the pendulum.

5.

With these graphs we were able to get the slope and the y intercept for two Mtray values. These two graphs and another one that is not pictured allowed us to come up the the equation in the pictures below. The slope value N and the y-intercept A were plugged into the equation:

 lnT = n* ln(M+Mtray) + lnA Then the equation were maipulated to make finding T and M easier. 

Conclusions:

Based on the last part of the experiment that asked us to find the mass of two unknowns my equations did not do that great of a job in predicting the mass of an object. I believe that the reason why my equations weren't very accurate has to do with maybe the way that the mass was put into the pendulum. When we were trying to calculate the period of my wallet it was very hard to place my wallet in a way that would distribute the mass of my wallet perfectly on the pendulum. I believe this is one of the reasons why my calculations were off. I believe that most of the errors that contributed to the uncertainty of my calculations had to do with human errors.