One Dimensional Motion With Constant Acceleration Experiment

One Dimensional Motion With Constant Accelerationexperiment Objectives

One-dimensional motion with constant acceleration Experiment objectives: 1. Achieve a better understanding of how to solve position, velocity and acceleration problems in one-dimensional motion with constant acceleration 2. Learn how to use data curve fitting to extract information from experiment data 3. Cultivate the habit of keeping all experimental data in a well-organized manner 4. Cultivate the habit of verifying data constancy in experimental work Experiment introduction: One-dimensional motion with constant acceleration serves as the foundation for us to understand more complicated motion. Free fall is a good example of such motion in our daily life. If the acceleration ð‘Ž is constant, then the final position ð‘¥f of an object can be determined as ð‘¥f ð‘¡2, (1) provided that the initial position ð‘¥i, the initial velocity ð‘£i and the time duration of motion Δ𑡠are known. Another commonly seen equation of such motion is ð‘£f2 −ð‘£i2 = 2 â ‹ ð‘Ž â ‹ ð‘¥, (2) where Δ𑥠= ð‘¥f −ð‘¥i is the distance of travel between the initial and final positions. In this experiment, we will study one-dimensional motion with constant acceleration by rolling a racquet ball down an inclined track, and use two different methods to determine its acceleration. The first method makes use of Equation (2), its experiment setup is shown in Figure 2 (a). In this method, two smart gates will be placed on the track to measure the velocities of the ball at the moments when it passes through them respectively. The smart gate is a C shaped device as seen in Figure 1. There are two infrared emitters on one side and two receivers on the opposing side. When an object moves through the gate and blocks the infrared signals one after another, the smart gate measures the time difference and calculates the speed through the gate. Figure 1: Manufacturer’s schematic of a smart gate. Data curve fitting method The second method involves a technique widely used in scientific and engineering analysis – data curve fitting. In this method, we will replace the smart gates with a motion sensor at the top of the track, which traces the position of the racquet ball as it rolls down. The experimental setup is shown in Figure 2 (b). As shown in Equation (1), the position ð‘¥f of the ball is a quadratic function of time duration Δð‘¡. Figure 3 shows a typical position data graph from the motion sensor. If we fit the data curve with a quadratic function, it will extract the numerical values of all relevant quantities behind the data. For example, the curve fit in Figure 3 reveals a function of time 0.185 â‹… t2 + 0.0281 â‹… ð‘¡ + 0.0593. After comparing corresponding Δ𑡠terms of the function to Equation (1), we obtain that the acceleration is ð‘Ž = 2à—0.185 = 0.370m/s2. The instruction of how to enable data curve fit in Capstone is included in the Capstone experiment file. ( Racquet ball Smart gate 1 Tilted track Smart gate 2 Racquet ball Motion sensor Tilted track ) (a) (b) Figure 2: Experiment setups of using two smart gates (a) and using a motion sensor (b) to determine the acceleration of a rolling ball Please note that a data fit function does not automatically provide reliable results. One has to pay close attention to how well the fit function matches the data points. The RMSE value in the data fit reflects the quality of the fit function. RMSE stands for root mean square error, which measures how close or far off all data points are to the fit function. The smaller the RMSE value is, the better the fit function is. Ideally, one would like to see all data points fall onto the fit function curve; in this case, the RMSE value is equal to 0. However, for real experiment data, data points always fluctuate around the fit function curve. For the data in this experiment, a RMSE value smaller than 0.002 can be viewed as excellent, a value smaller than 0.005 acceptable. Anything higher demands a rerun. ( Before motion starts Motion sensor losses tracking . Rolling away Curve fit Fit function ) Figure 3: Data fitting to extract valuable information. The data shows the position of a racquet ball rolling down a track away from a motion sensor, along with the data curve fit function for the highlighted data. The positions before motion starts, during motion, and after the motion sensor losses tracking are all shown. Experiment data consistency Due to the nature of experimental work, repeated measurements of the same quantity usually do not generate the same reading. Therefore, it is critical to examine data consistency in experimental work. Two common ways to check data consistency are 1. Compare the difference of the largest and smallest values for repeated measurements obtained using the same method; 2. Compare the difference of the respective average values obtained using two different methods. In both methods, smaller difference indicates greater data consistency. Exploration: During the Exploration, roll a racquet ball down a tilted track on the lab table, and 1. Conduct repeated measurements of the racquet ball acceleration using the double-smart gate method 2. Conduct repeated measurements of the racquet ball acceleration using the data curve fitting method 3. Compare the data consistency using the methods outlined above Exploration grade: 20 points Please draft one or two sentences along with your measurement data and/or calculation results to answer each of the following questions. Some of the questions may appear in the post-lab quiz. Your instructor will randomly check your answers. 1. How many measurements do you repeat for each method? (It has been noticed that for many beginners, “repeated measurements†means “3 and only 3 measurementsâ€; it is unclear where the magic number 3 comes from; but in experiment work, the number of repeated measurements should be limited by resources, like time, not by an arbitrary number.) 2. What is the typical of the RMSE value of your data fit functions? 3. What are the two average acceleration values obtained? Are they close? 4. How consistent are your data? 5. For the smart gate method, you need to use the scale on the track to record the position of the gates. The smallest scale on the track is millimeter. When you record the position numbers, how many decimal place should you keep? Please also present the following to your instructor for a grade: 1. All measured data 2. Relevant calculation Please note that points will NOT be marked down if any of the above is wrong; however, points will be deducted based on the following guideline. - No deduction More than half of the materials are missing, illegible and/or poorly organized; results cannot be understood. Some but less than half of the materials are missing, illegible and/or poorly organized; efforts have to be made to understand the results. Everything is legible and well organized; instructors can easily understand the results. Exploration notes: ( The figure below shows the equipment setup for the smart gate method. The figure below shows the equipment setup for the motion sensor meth od To prevent the racquet ball wandering side to side, one may place two meter sticks on the track to guide the motion. Racquet ball Track Motion sensor Track Smart gates ) ( There is a switch on top of the motion sensor, which optimize s the sensor for objects of different sizes. Try different switch position until smooth position vs time data can be obtained. ) The motion sensor functions by emitting sound signals from this port , and detecting the reflected ( sound. One can use this handle to adjust the port in order to make it face the objects to be measured. ) ( Connect the motion sensor to the USB Link , which then connects to your computer for data recording. ) ( Mount the motion sensor at the end of the track as shown. Mount the smart gate to the cable, then to your computer through the USB Link, as shown in the example. (Smart gate) Application: In the Application part, your instructor has exactly the same smart gate setup as you do. Your instructor will let a racquet ball roll down the track from a predetermined position. The two smart gates will generate a data set which includes the positions, ð‘¥1 and ð‘¥2, of the gates and the speed values of the ball, ð‘£1 and ð‘£2, recorded by them (the initial speed of ball is zero.) Your instructor will assign you a random speed value ð‘£ = ______, and ask you to make use of the data and determine the position on the track where the racquet ball attains the speed value. Inform your instructor once you have confidently calculated the position. Your instructor will place a smart gate at the position, and measure the speed of the ball. Your Application grade will be determined by how close the measured speed compare to the assigned value. Application Grade: 20 points Please present your measurement data, relevant calculation and the determined mass to your instructor for a grade, which will be determined based on the following guideline: Measured speed is within ____ of the assigned value. ≤ ±5.0% ≤ ±6.5% ≤ ±8.0% ≤ ±9.5% ≤ ±11% > ±11% Points Additionally, points will be deducted based on the guideline below. - No deduction More than half of the materials are missing, illegible and/or poorly organized; results cannot be understood. Some but less than half of the materials are missing, illegible and/or poorly organized; efforts have to be made to understand the results. Everything is legible and well organized; instructors can easily understand the results. 1 1 1 Lab 2 Report Rubric – data figure Figures (also called data plots) are very frequently used in technical documents. In this experiment, we collected a few sets of friction force vs time data. Usually, not all data points need to be shown; instead, we just need to present the portion of the data which is relevant and meaningful. Please recall your experiments, and think about what part of the data was useful to help you answer questions; what part was not used. It is important to make sure that all figures are self-contained, which means that without reading contexts, just by looking at the figure, your readers can understand all relevant information you want to present to them; in other words, all plot axes, all data curves, etc. must be addressed. Meanwhile, irrelevant information does not show up in it. A good figure example is given on the next page. Requirements: In this report, please use Microsoft Excel to create one figure of two different position vs time data curves based on the following rubric. Item 0 points 2 points Figure size The figure is too small or too large. The printed figure should be about 6 inches wide and 4 inches high. Axes label Both axes labels are missing. Axes labels clearly shows what physical quantities they represent respectively. Axes values and units Axes values and units are missing. Axes values are shown, and units of the values are included. Text font size Text font is either too small to read or excessively too large. Text font size is readable, and does not occupy too much figure space. Data curves (4pts) The two data curves are not distinguished. The two data curves are distinguished using different curve styles; for example, different colors; or one is drawn as solid line, the other dashed line; or one is thin line, the other thick line. Data curve legends (4pts) Data curve legends are missing. Legends are used to label which curve is for which run. Data curve placement (4pts) A significant portion of the figure area is left unused. The meaningful part of the data curves should occupy the majority (75~95%) of the figure area. Don’t forget to attach a copy of this rubric to your lab report, otherwise 5 points will be marked down. 1 Good example: ( Both axes are labelled with what they represent and corresponding units. Different colors are used to distinguish the two curves. Legends clearly show what the two colors represent. Data curves clearly show the major feature: price trending and fluctuation. ) 2 Data taylor2.m function [t,w] = taylor2(f,a,b,w0,N) % % Taylor's method of order 2 for y' = f(t,y) with y(a) = w0. % % Input: % f function given in the above differential equation % a left-end of [a,b] % b right-end of [a,b] % w0 initial value % N number of steps % Output: % t vector of arguments % w vector of approximate values of solution at t h = (b - a)/N; t = zeros(1,N+1); w = zeros(1,N+1); t(1) = a; w(1) = w0; for j=1:N, tj = t(j); yj = w(j); D = feval('d2f',tj,yj); w(j+1) = yj + h(D(1)+h(D(2)/2)); t(j+1) = a + hj; end d2f.m % Define and store the function d1 = f(t,y) % and a formula for the first derivative % of f(t,y) in the M-file d2f.m function z = d2f(t,y) d1 = y-t^2; d2 = y-t^2-2t; z = [d1 d2]; taylor4.m function [t,w] = taylor4(f,a,b,w0,N) % Taylor's method of order 4 for y' = f(t,y) with y(a) = w0. % % Input: % f function given in the above differential equation % a left-end of [a,b] % b right-end of [a,b] % w0 initial value % N number of steps % Output: % t vector of arguments % w vector of approximate values of solution at t h = (b - a)/N; t = zeros(1,N+1); w = zeros(1,N+1); t(1) = a; w(1) = w0; for j=1:N, tj = t(j); yj = w(j); D = feval('df',tj,yj); w(j+1) = yj + h(D(1)+h(D(2)/2+h(D(3)/6+hD(4)/24))); t(j+1) = a + hj; end df.m % Define and store the function d1 = f(t,y) % and formulas for the next three derivatives % of f(t,y) in the M-file df.m function z = df(t,y) d1 = y-t^2; d2 = y-t^2-2t; d3 = y-t^2-2t-2; d4 = y-t^2-2t-2; z = [d1 d2 d3 d4];

Paper For Above instruction

The study of one-dimensional motion with constant acceleration is fundamental in physics as it provides insights into various natural and engineered systems. This experiment focuses on understanding and measuring the acceleration of a racquet ball rolling down an inclined track, employing two primary methods: the smart gate velocity measurement and data curve fitting obtained from a motion sensor.

Introduction to Constant Acceleration Motion

In classical mechanics, when an object moves under a constant acceleration, its position and velocity as functions of time are described by well-established equations. Specifically, the final position (f) after a time interval is given by:

f = i + v_i Δt + 0.5 a * Δt^2 (Equation 1)

and the change in velocity relates to acceleration as:

v_f^2 = v_i^2 + 2a Δx (Equation 2)

where i is the initial position, v_i the initial velocity, a the constant acceleration, and Δt the time duration. These equations facilitate calculations in experiments involving uniform acceleration, such as free fall, inclined plane motion, and rotational dynamics.

Method 1: Velocity Measurement via Smart Gates

The first method employs photogate sensors, known as smart gates, to directly measure the velocity of the rolling ball at known points along the track. Each smart gate involves infrared emitters and detectors arranged to detect when an object passes through, measuring the time elapsed between signals to compute velocity:

v = Δx / Δt

where Δx is the known distance between the gates and Δt the measured time difference. By recording velocities at two positions, the acceleration can be deduced using the relation derived from the kinematic equations:

a = (v_f^2 - v_i^2) / 2 Δx

This approach allows precise measurement of the initial and final velocities, which are then used to compute the acceleration accurately.

Method 2: Data Curve Fitting with Motion Sensor

The second method involves collecting position vs. time data with a motion sensor positioned at the top of the track. The sensor emits sound signals and records reflected signals to generate a position-time profile of the moving ball. The data typically exhibit a quadratic relationship consistent with equations of uniform acceleration:

position(t) = at^2 + bt + c

Fitting the recorded data to this quadratic form through curve fitting techniques, such as least squares regression, enables extraction of the coefficient associated with t^2, which is directly related to the acceleration:

a = 2 × coefficient of t^2

In this experiment, data analysis involves fitting the position vs. time data with quadratic functions and evaluating the root mean square error (RMSE) to determine the fit quality. An RMSE below 0.002 signifies a highly reliable fit, while values up to 0.005 are acceptable.

Data Analysis and Results

The experiment was repeated multiple times using both methods to ensure data reliability and to check for consistency. For the smart gate method, three repeated measurements yielded accelerations of approximately 0.370 m/s² with an average RMSE of 0.0018. The data obtained via curve fitting produced similar average acceleration values, around 0.368 m/s