CUNY Kingsborough Community College Linear Uniform Acceleration Motion Lab Report

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CUNY Kingsborough Community College

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motion of objects that have constant acceleration is common in nature, Motion with constant acceleration is known uniform accelerated motion. The acceleration for this type of motion is defined as Av (5.3) a V₂-V t₂-t, vo 1 Vi are the final and initial instantaneous velocities at the instants t, and t, respectively. The acceleration is positive if the velocity increases with time and is negative if the velocity decreases with time. To simplify our notation, let us take the initial time t, to be zero: t = 0. This is effectively starting a stopwatch at tq. We can then let t = t, be the elapsed Fig. 5.2. Experim acceleratic where and V2 time. The initial and final velocities will now be represented by v, and v. For these assumptions equation (3) becomes (5.4) Procedure V-Vo a= t We can solve this equation for v and obtain (5.5) V = Vo + at. It can also be shown that the displacement x for uniformly accelerated motion at a later time t is given by at? (5.6) x = Vot+ The aim here i AL and the times The second air This goal can be a cart to pass betwee 1. Set up the dyn 2. Measure the le 3. Connect the Connect the D 5. Place Photog 6. Set up the se Open Da 2 We may combine equations (5.5) and (5.6) to eliminate t and obtain a relationship for a, x, V., and v. We first solve equation (5.5) for t and then substitute the resulting expression into equation (5.6). After simplification we finally get v2 – vă = 2ax (5.7) 4. or X= z? - vo (5.8) 2a The objectives of this laboratory activity can now be achieved by determining the velocity and the acceleration according to equations (5.2) and (5.4) and then subsequently verifying the kinematic equations (5.6) and (5.8). The actual experimental configuration is shown in Fig. 5.1. This experiment requires two photogate timers to measure the average velocities of a moving cart in two positions on an inclined dynamics track. A flag of known length is placed on the top of the cart, and the photogate timers are mounted so that the flag obscures the light beam of the photogate as it passes through. When the flag interrupts the light beam to the photocell of the photogate, the timer is turned on. When the flag completely passes through the photogate, the timer turns off. Therefore, the photogate timer records the time At for a flag of given length AL to pass through the photogate. Measuring the time taken for the flag of known length to pass through the photogate and dividing the flag length by this time gives the average velocity of the cart as Find the result th again, a In the E the initi 1, when from th “Unblo Now in need to by the sequer Now i moves AL V= At (5.9) Photo “Bloc To IE “Cho icon your an Because the cart is accelerated during the measurement, the calculated velocity is the average of the true velocity. Since we are using the short length flag AL, the average velocity approaches the true Photogate 1 Flag Photogate 2 instantaneous velocity of the cart. The shorter the flag, the better the Computer approximation. Thus, average velocity can be used as an instantaneous velocity By using two photogate timers in tandem and this technique of measuring velocity, we determine the initial velocity vo and the velocity v at any Fig. 5.1. Setup for measurements of the velocity and acceleration of the moving cart and verification of the kinematic equations. later time t from equation (5.9), and, as a result, determine the acceleration of the 7. Place the button a and bet Data Ta 8. Displac initials 34 bject Linear Uniform Accelerated | Motion Lab 5 Learning Objectives In this laboratory activity you explore the relationship between position, velocity, and acceleration for a motion with constant acceleration. There are two objectives. The first is to measure the velocity of an object. The second is to measure the acceleration of a moving object and verify the kinematic equations for uniformly accelerated motion. Required Equipment O Ruler Dynamics cart track or air track Dynamics cart or glider Science Workshop Interface box Two Photogate accessories Set of 5 metallic rods of different length Learning Outcomes You will be able to: Understand the concept of acceleration. Explore the relationship between position, velocity, and acceleration. Analyze the graphs of position and velocity versus time for an object's motion. Determine the instantaneous velocity and acceleration of an object. Explain how the slope of the velocity graph relates to the acceleration. Use a computer and time sensors (photogates) for a precise measurement of time. Use Excel for analysis and find the equation of the trend line. Verify the kinematic equations for uniformly accelerated motion. Theoretical Background As an object moves, the value of its coordinate changes with time. Consider an object with the initial position's coordinate being x, at some time t, and the final position's coordinate being x, at some later time tz. The displacement between these two positions is defined by Ar = x2 – X;. The physical quantity that shows the rate at which the position changes is known as the velocity. The average velocity of the object is defined as the ratio of the displacement Ar to the time interval At: Ax (5.1) V av = At where At = tą - t . Note that the displacement and the average velocity may be either positive or negative depending on whether x, is greater or less than x . A positive value indicates motion to the right and a negative value indicates motion to the left. An instantaneous velocity v of the object indicates how fast the object moves and the direction of the motion at each instant of time. The magnitude of the instantaneous velocity is called the instantaneous speed. If you measure the average velocity of the object over smaller and smaller distances, the value of the average velocity approaches the value of the object's instantaneous velocity, and we can write (5.2) Ar V= At In other words, as At becomes smaller, Ar becomes smaller as well, but their ratio approaches a constant value. When the velocity of the object is changing with time, the object is said to be accelerating. The simplest accelerated motion is a linear motion with constant acceleration, when the velocity changes at the same rate throughout the motion. The 33 is known as Av=V - Vo (5.3) Slope = a= V – Vo Vo 1 t t celeration is tation, let us the elapsed Fig. 5.2. Experimental determination of the acceleration from the slope of the graph. cart from equation (5.4). On the other hand, analysis of equation (5.5) shows the linear dependence between the velocity v and the elapsed time t. Using this fact we can experimentally find acceleration of the cart by measuring its velocity at different positions and the corresponding elapsed time t. A plot of the velocity v versus the elapsed time t gives a straight line that intersects the v axis at point vo, as shown in Fig. 5.2. According to equation (5.5), the slope of this straight line is equal to the magnitude of the acceleration. Therefore, by finding the slope from the graph we can determine the acceleration. In this way, we experimentally determine the acceleration. omes (5.4) Procedure (5.5) AL and the times At, and At it takes the flag of known length to pass through the photogates. The aim here is to determine the initial vo and final v velocities of the moving cart by measuring the length of the flag The second aim is to determine the acceleration of the moving cart and verify the kinematic equations (5.6) and (5.8). (5.6) ve equation (5.7) (5.8) ccording to לל, to measure s placed on as it passes en the flag or a flag of hrough the cart to pass between two photogates, and the distance x between the photogates. 1. Set up the dynamics track and the photogates as shown in Fig. 5.1. This goal can be achieved by using the measured velocities, and w of the moving cart, the time interval t it takes for the 2. Measure the length of the flag AL and record the result in the data Data Table 5.1. 3. Connect the Digital Photogate I accessory plug into Digital Channel 1 on the Science Workshop Interface box. 4. Connect the Digital Photogate 2 accessory plug into Digital Channel 2 on the Science Workshop Interface box. 6. Set up the sensors in the software. 5. Place Photogate 2 at a distance of x=0.50 m below Photogate 1. Record this distance in Data Table 5.1. Open DataStudio Window. Click “Create Experiment”. Find the “Photogate” in the Sensors list in the Experiment Setup window. Double-click the “Photogate”, and as a result the “Photogate” icon appears below Digital Channel 1 of the Interface box 750. Double-click the "Photogate” again, and as a result the “Photogate” icon appears below Digital Channel 2 of the interface. In the Experimental Setup Window click “Timers”, and as a result the “Timer Setup” window appears. To measure the initial velocity V, of the cart you need to measure the time At, the cart's flag requires to pass through Photogate 1, when the Photogate 1 is first blocked by the flag and when it is unblocked as the back end of the flag moves away from the Photogate 1. In the timing sequences choices click Ch. 1 and select "Blocked”. Click once again and select "Unblocked". Now in the “Timer Setup” window click +New and Timer 2 appears. To measure the final velocity v of the cart you need to measure the time At the cart's flag requires to pass through Photogate 2, when Photogate 2 is first blocked by the flag and when it is unblocked as the back end of the flag moves away from Photogate 2. In the timing sequences choices click Ch. 2 and select "Blocked”. Click once again and select “Unblocked”. Now in the "Timer Setup” window click +New again and Timer 3 appears. To measure the time t, when the cart moves between Photogate 1 and Photogate 2 (Photogate 1 is first blocked by the flag and then the flag blocks Photogate 2), in the timing sequences choices click Ch. 1 and select “Blocked”, then click Ch. 2 and select “Blocked”. Finally click “Done” and all timing will be set. To record your measurements of time, double-click the Table icon in the Display window, and as a result the "Choose a Data Source” window appears. Choose "Timer l” and click OK. (You can also click-and-drag the Table icon from the Display window to the Data window for Timer 1). Repeat this procedure for Timer 2 and Timer 3. For your measurements you only need “Elapsed time”. Click on the “Clock” icon in each table menu, and as a result you will record just “Elapsed time”. Resize the tables to fit your screen. (5.9) Since we ag AL, the s the true e cart. The better the average tantaneous timers in measuring e initial v at any and, as a 7. Place the cart at the starting position. The cart must be released from the same point for each run. Click the "Start" button and release the cart from the starting point. The times Ato, At, and t the flag takes to pass through the photogates and between the photogates will be immediately displayed. Click the "Stop” button. Record the values At,, At, and t in 8. Displace Photogate 2 by 0.1 m down and record the distance x in Data Table 5.1. Again release the cart from the same initial starting position and repeat step 7. Data Table 5.1. on of the 35 . sie 21 9. Increase the displacement of the second Photogate by an additional 10 cm to a maximum separation of 0.9 m between the Photogates. Computations and Data Analysis 1. Use equation (5.9) and compute the velocities v, and v based on the length of the flag AL and measured Ato, At, and t. 2. Use equation (5.4) and compute the acceleration of the cart for each run. We assume here that vo and v are the instantaneous velocities of the moving cart. Find the average acceleration. Record your results in Data Table 5.1. 3. Plot a graph of the velocity v, versus the time t. If you are using Excel, display the equation of the trend line on the chart to find the slope. From the slope of the graph find the acceleration and compare this with the average acceleration by computing the percent difference. 4. Using the experimental values of t, V., and a, determine x from equation (5.6). Compare this result with measurement for x by computing the percent difference. Make conclusions. 5. Plot a graph of the distance x versus the time t. 6. Using the experimental values of vo , V, and a, determine x from equation (5.8). your Questions 1. 2. 3. Does a speedometer in a car measure speed or velocity? Does an odometer in a car measure distance or displacement? If the velocity of an object is zero, does its acceleration have to be zero? A cart has a constant acceleration of a=2 m/s”. At a certain instant its speed is 5 m/s. What is its speed 2 s later and 2 s earlier? What is its average speed during the time interval of 4 s? 2 Section Date Student's name Data Table 5.1. Measurement of the acceleration of the moving cart and verification of the kinematic equations Length of the flag AL = m Distance Distance Acceleration % difference for Time v2 – v? Time at2 X = 9 V-Vo x= Vot + Distance between photogates X, > tama х At, Ato, 2a 2 Velocity AL VE At m/s Velocity AL VO Διο m/s a = Time between photogates t, S - t m S т m/s? m Average acceleration, m/s² Slope of the graph, m/s² Acceleration from the slope of the graph, m/s² Percent difference for acceleration 37
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Explanation & Answer

Hello there, Please find the attachements below. Attached are all parts required of the report.The data sheet you submitted has some calculation errors so I did the calculations again based on the measurements you provided in the sheet. Also Graphing and computations are shown in the excel file labeled data analysis.I hope I was able to assist you prefectly. Please come back with any question you need.Good Luck !!

Lab Report
Lab No.5 – Linear Uniform Accelerated Motion

Part 1: Explanation of the Formulas Used in the Experiment
1. Average and Instantaneous Velocities
𝑣𝑎𝑣 =

∆𝑥
∆𝑡

… (5.1)

Average velocity (𝑣𝑎𝑣 ) is defined as the ratio between displacement (∆𝑥) and time interval (∆𝑡). Such that (∆𝑥 = 𝑥2 −
𝑥1 ) is the difference between the final and initial positions, and (∆𝑡 = 𝑡2 − 𝑡1 ) is the difference between times at final
and initial positions.
Instantaneous velocity is the average velocity over an extremely small time interval. Hence, if (∆𝑡) approaches zero
then the average velocity (𝑣𝑎𝑣 ) will approach the Instantaneous velocity (𝑣).
𝑣=

∆𝑥
∆𝑡

… (5.2)

In the experiment, this equation is used in this form:
𝑣=

∆𝐿
∆𝑡

… (5.9)

Where (∆𝐿) is the length of the flag used in the experiment.

2. Average Acceleration
Average acceleration in a uniform accelerated motion is given by the rati...


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