Linear Kinematics John Smith Kathy Hernandez (partner) Physics 1 Lab (Friday) Mr. Kiledjian 02/24/2006

Transcription

1 Linear Kinematics John Smith Kathy Hernandez (partner) Physics 1 Lab (Friday) Mr. Kiledjian 02/24/2006

2 Purpose: In this lab, we will investigate the relationship between displacements, velocity, and acceleration. We will prove that acceleration is the derivative of velocity, and displacement is the integral of velocity. We will also learn to graph using Excel and other skills like typing equations and tables in a Word document. Equipment: Calculator, Excel & Word Program, our Brains!!! Procedure: We are starting with a particle which is undergoing a 1 dimensional motion given by the following velocity function, v(t) = 200At 300Bt 2. The instructor gave our group the following values of A & B, A = 1, B = 1. Therefore, the velocity function reduces to v(t) = 200t 300t 2. Using calculus, we predict that the acceleration function, a theor (t) = v (t) = t. Also, we predict that the displacement function, X theor (t) = 100t 2 100t 3. In this lab, we want to know and prove that the derivative gives us the slope of a function and the integral gives us the area. Setting the theoretical acceleration function to zero gives us, t oatheor =.33 seconds which is also the same time that the velocity will reach its maximum. Therefore, we expect the acceleration function or the slope of the original velocity function to reach zero at.33 seconds, and the velocity function to reach its maximum. Also, setting v(t) = 0, we obtain t ovtheor =.66 seconds which is the time when we expect its velocity to reach zero and also the displacement X to reach its maximum value. Therefore, we expect that t MXTheor =.66 seconds. Lastly, setting the displacement equal to zero tells us when the particle will cross the origin. We obtain two values for t 0Xtheor = 0 & 1 second. The second of these values is the one of interest to us, since the first value is the starting time.

3 We took the v(t) equation that we had and calculated its value for every.05 seconds starting with t = 0 and ending with t = 1 second. Then we subtracted the velocities from each other skipping every other row, so we subtracted the 1 st row from the 3 rd, the 3 rd from the 5 th, so on. This gave us the change in the v between every other value. Then, we divided this by the time interval between every other velocity, which is.1 seconds. This gave us the slope of the secant line of the V vs. t graph. This is the 4 th column of the table, which we are calling, A avg. In the fifth column, we added every other velocity to each other and divided by 2. Graphically, this gave us the average height of the velocity function between every other interval. In the sixth column, we multiplied this average height by the interval,.1 seconds. This gave us the area under the graph of the velocity function for every other interval. Finally, we added all these areas up and obtained the displacement of the particle for every other time interval. Then, we took the values of v(t), a(t), x(t) and graphed them on Excel versus t and obtained the equations of these lines from Excel. We compared these to our theoretical results and obtained percent errors. DATA ANALYSIS: Time (sec) V (m/s) Δv (m/s) = v n + 2 v n A avg (m/s 2 ) = Δv/Δt V avg (m/s) = (v n v n )/2 Δx (m) = V avg Δt X (m) = Σ Δx 0 0 ΝΑ NA NA NA NA ΝΑ NA NA NA NA

4 Sample Calculation for Cells with Stars (4 th row) ΔV 4 = V 5 V 3 = = 11 m/s A avg4 = ΔV 4 /Δt = 11/.1 = 110 m/s 2 V avg4 = (V 3 + V 5 )/2 = ( )/2 = 45/2 = 22.5 m/s ΔX 5 = (V avg4 )(Δt) = (22.5)(.1) = 2.25 m/s X 5 = X 1 + X 3 + ΔX 5 = = 3.10 m Discussion: (Refer to the graphs on the next page!!) From Excel, we were able to obtain the following experimental functions, A exp (t) = -600t + 200, V exp (t) = 200t 300t 2, X exp (t) = 100t 2 100t 3 -.5t. Setting the experimental acceleration function equal to zero gives us, t oaexp =.33 sec. Therefore, % Error 1 = 100% x /.33 = 0%!!. To obtain the experimental value of when the velocity reaches its maximum, we take the derivative of the experimental velocity function and set it equal to zero. V exp (t) = t = > t MVExp =.33 sec. Therefore, % Error 2 = 100% x /.33 = 0%. To obtain the experimental value of when the displacement reaches its maximum, we took derivative of the experimental displacement function and set it equal to zero. X exp (t) = 200t 300t = > t MXExp =.6642 seconds. % Error 3 = 100% x /.66 =.63% error.

5 t (sec) V (m/s) t (sec) a (m/s^2) t (sec) X (m)

6 Finally, to calculate the experimental time when the displacement reaches zero, we set X exp (t) = 0 = 100t 2 100t 3 -.5t > t oxexp =.9950 seconds. Therefore, %Error 4 = 100% x /1 =.50% error. As can be seen above, most of the errors were very small, either 0% or smaller than 1%. This is great considering that we used large intervals of.05 seconds from 0 to 1 second. These errors are due to the fact that our intervals are not too small. To improve this, we could make very small intervals of like.001 seconds and then we would have a table with hundreds of rows. Another improvement would be to not round any numbers. Another improvement would be to use the trapezoidal method of approximating the integral. Also, another interesting result is that all of the R squared values came out to be 1 which means that the computer agrees with calculus that the best fit graph is the same equation as given by calculus. When I tried to put in another power of t, their coefficients came out to be very negligible. Therefore, the computer was telling us that calculus rules!!! Conclusion: This lab was really great, I loved it. I learned how to prove calculus starting from a velocity function and just plugging in numbers and creating a data table. Then, by subtracting and adding these numbers, we approximated the slopes and average areas of these numbers, and then plotted them. Along the way, this lab taught me how to type equations, tables, Greek alphabet, and how to construct graphs in Excel. The most interesting part of it was that when we asked Excel to graph these data and give us their equations, these equations came out to be VERY CLOSE to the equations obtained by calculus. Wow!! Overall % errors were very low, the accuracy of the graphs as demonstrated by the regression values was perfect. We have definitely shown calculus to be a force to be reckoned with. Go Newton!!!!