LAB: MOTION ON HILLS

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1 LAB: MOTION ON HILLS Introduction In this three-part activity, you will first study an object whose speed is changing while it moves downhill In this lab, the two variables you are focusing on are time and velocity After collecting data in the lab, you will explore its meaning by answering a series questions and creating graphs all with an aim to find the relationship or equation that explains how velocity changes as time changes When you have finished the lab, you should be able to interpret position vs time and velocity vs time graphs for kinematic information about the moving object Goals When finished with this lab you should be able to: Derive values for x, t, v, & a from tables of x and t values Demonstrate that you can correctly read x, t, x, t and v from an x vs t graph Demonstrate that you can correctly read v, t, v, t and a from a v vs t graph Write the equation of the best-fit line (when linear) from a motion graph and describe the meaning of the slope and intercept Describe (in words) the motion of an object given its x vs t or v vs t graph Include information about position, displacement, times, speed and acceleration Purpose: Based on the information given above, you should be able to write out a correct purpose statement Be sure you know how to write it before taking the PreLab quiz Theory: After writing the purpose, we need to think about what we expect the relationship to be This is usually stated in the theory In this lab, we are looking at the relationship between velocity and time for an object rolling down hill and speeding up To assist you in thinking about the theory, you should complete each of the following tasks 1 Draw a motion diagram (at least 3 dots, several velocity vectors, an acceleration vector, and a coordinate system (partially drawn below) x=0

2 Lab: Motion on Hills Page 2 2 Sketch a position vs time graph and a velocity vs time graph for this motion In this case, you cannot predict specific times and positions that will match the experiment The shape of the graph is the important thing Notice that because there are no scale values, the units are not necessary on the axes labels Position vs Time vs x v t 3 Look up in your text a sentence that talks about motion for an object speeding up Give the quote In parentheses after the quote place: authors, year, page number as in (Knight, Jones, & Field, 20xx, p #) t 4 Planning Diagram: The experimental variables are the two variables mentioned in the purpose statement It is not always possible to measure these variables When not possible, other variables will be measured and the experimental variable derived from the measured data The diagram below is called an experiment planning diagram We will use it in many labs to help organize the experimental design In the top two boxes, write in the experimental variables from your purpose statement One of them will be time In the case of time, we can measure it directly, so place time in the box at the bottom underneath the top box that has time Since time is measured directly, there is no need to calculate so not equation is needed Velocity (the other experimental variable) cannot be measured directly with the equipment we will be using Instead, we will need to measure the position and the time to calculate the velocity Place velocity in the top blue box and position and time in the bottom blue box In the blue equation box, place the equation that can be used to calculate average velocity from position and time time Phys 114: Lab-Motion on Hills rev Su17

3 Lab: Motion on Hills Page 3 Procedure Materials Cart Lab Quest Cart with black block installed on top Track with side bracket and pole set up as an incline **A photogate is a timer that uses a beam of light to start/stop its clock A diode on one side of the gate emits a beam of light that is detected by a receiving diode on the other side When an object blocks the beam the receiving diode turns off, which changes the status of the circuitry and records a blocked time Another time is recorded when the light beam is unblocked Multiple photogates can be linked together to get multiple blocked and unblocked times 4 photogates and brackets and 4 cables Steps for equipment set-up 1 Make a ramp with the track by elevating one end using the side bracket and silver pole The pole has a tapered end that will fit into a receptacle on the lab table end Place the bracket into the side rail on the OPPOSITE side of the built-in ruler The track elevated end should be no more than half way up the silver pole 2 Photogates should be already attached to black brackets These brackets need to be installed into the side rail of the track Use the same side as the built-in ruler 3 Attach the cord from the Lab-Quest side to any photogate Then from that photogate to the next and keep going until all photogates are wired 4 Turn on the photogate and select the table mode Hint: You did this in Lab 1 5 Mark the ramp with a starting point", as you will need to release the cart from the same position for each run 6 Place one timer at the starting location Carefully line up the C shaped gate with your starting point so that the cart will activate the Phys 114: Lab-Motion on Hills rev Su17

4 Lab: Motion on Hills Page 4 timer just as it is released This is how you will ensure that your velocity is ZERO when the timer starts (t=0) 7 Place the remaining timers farther down the track (any place is fine), again adjusting the gate so the cart trips the timer as it passes through 8 With the first timer (starting timer) at an initial position, measure and record the position of the other timers 9 Release the cart from the starting position so that it has a zero velocity as it begins to block the first photogate Record blocked times for each photogate with their respective positions Repeat so that you have at least three times for each position Put these in your working data tables You will make an average of each time later 10 Now move all except the 1 st photogate to new locations and record positions Roll the cart again, 3 times, and record the time measurements Move the 2 nd, 3 rd, and 4 th photogates many different times and record the data in your working table 11 Do NOT take down your equipment! Do you have all the necessary information? 1 Record information so that another group can set up your track exactly the same way Record how you find the angle of elevation 2 Describe your cart What pieces are included? Gather observations for establishing uncertainty 1 Locate one photogate near the bottom of your track Anywhere past ¾ of the way down will be fine You can use one that is already in place Run the cart enough times down the track so that you have at least 10 times for this position You may already have 3 times Use these 10 times to calculate uncertainty for time using the multiple measure method Hint: If you think you have more than one outlier, you will need to take more times Use this uncertainty in your formal data tables in the report 2 Locate one photogate Any one is fine Take a reading of its position You have probably already done this Now, for that position, make a judgement about the uncertainty using the single reading, analog method (see Lab 1 for a review) You will use this uncertainty for all positions in your formal data tables Checking whether enough data has been gathered Students frequently ask: "How many pieces of data do I need to record?" The answer is partly determined by the nature of the experiment, and there is no set rule for all cases To help determine this for yourself, let s look at one method you can use for this purpose: 1 Plot a Position vs Time graph (a graph of measured data) on graph paper for the cart rolling down the hill You should be able to draw a smooth best-fit line (If you are unsure where to draw this line because your points don't seem to lie along a fairly smooth curve, you may need to take more Phys 114: Lab-Motion on Hills rev Su17

5 Lab: Motion on Hills Page 5 data) This is a working graph You do NOT need to average your data first though it is not incorrect to do so It is recommended that one team member plot as you go along DO NOT wait until the end of class to do the plot or you will run out of time 2 Are there any gaps or bumps on your graph that indicate you may need more data? On your graph, find at least one location where another data point would be helpful 3 Another way to check if you have enough data is by looking at the extremes Do you have data points that are close to the beginning and far ends of the path? It is best to take data over as great a range as possible Does your data fit this check? 4 Take more data From now on, you should always check, by graphing your data, whether you have enough to put your equipment away Analysis Guidelines You measured position and time Yet the variables in the purpose statement are velocity and time In the analysis, you must find a way to get the velocity information from your position and time data Method 1: Finding the average velocity using the position vs time graph Using your Position vs Time data, organize it so that all your average times are in order Be sure that you have all the data in order from the earliest to the latest time It is exactly one motion, so you should have only one formal table of x and average t values The time column, in the formal table is not titled average time even though the times listed are the average values for each position You will first need to graph the position and time data from your formal table Remember that you are representing one motion so you will have one graph with one best-fit line Draw in your best-fit line It will be a curve Darken in your best-fit line so that it can be seen easily Remember not to darken it where it passes through a point protector Go along the bottom axis of your graph and find at least 8 fairly equally spaced times between (but not including) your t=0 and your last time (don t use the last time) In the example below, I chose times 04, 12, 20, 28, 36, 44, 52, 60, and 68 sec You need to have at least 8 times You may wish to do more to better represent your results Find the slope of the curve at each of these times See the example diagram below for the situation of t=28 sec Remember that the slope of an x vs t graph is velocity To find the slope of a curved line, you need to draw the tangent line and Phys 114: Lab-Motion on Hills rev Su17

6 Lab: Motion on Hills Page 6 find its slope Note: The diagram below does not match the data from the example above Yours, however should match your table You would then plot 24 m/s as the velocity at 28 sec on a velocity vs time graph Note: The example does not have point protectors because it was done in Excel All handdrawn graphs are expected to include point protectors Be sure to show your work for finding your slopes Create a table of velocity and time values that you found from the tangent line slopes Add these velocity and time values to your velocity vs time graph but use a different point protector shape or a different color to distinguish these points from those plotted in method 1 Use these, velocity and time values to draw in your best-fit line It should be linear If it is not, see your instructor for assistance Now you will write the equation of the best-fit line from your v vs t graph Be sure to write it using the format you learned in Lab 2 Review the Data and Graphing Guidelines document to check your graph and your equation formatting Method 2: Calculating the average velocity between adjacent positions using the data table It is possible to calculate the velocity using the x and t data values from your table However, due to uncertainty, the velocities would have been quite a bit more scattered than those from the best-fit line The reason is distributed uncertainty In other words, when using values that have uncertainty in calculations, the results will have much more uncertainty You will be asked to do some calculations using this method to compare to Method 1 Some velocity values will be too high Look at your graph and select two adjacent points where the velocity value calculated will be a value larger than the slope of the best-fit line For example, in the graph above, notice the points at t=5 sec and t=6 sec A line between these two points is steeper than the best-fit line between 5 and 6 seconds Phys 114: Lab-Motion on Hills rev Su17

7 Lab: Motion on Hills Page 7 Notice that in this table, not all the t and x data has been included Of course, yours will be a complete table time (sec) Position (m) Notice that the header is not complete Do you know what is missing? Be sure to make the headers in your table complete Calculate the time interval For example, to calculate the time interval between the these times use the equation: In the same way, you should be able to calculate the displacements Now you will need to calculate the average velocity within the time interval: The equation for average velocity is: In this case we have: Find the time that is best associated with the average velocity you calculated in the time interval For example, in the interval between the the times, what is the time that the cart most likely had an instantaneous velocity equal to the average velocity in that interval? It is the time in the middle between the 1 st and 2 nd time The time directly in the middle between them can be calculated by using the equation for middle or average values:!""#$ + & !""#$ 2!""#$ 55 Therefore, using this method, an instantaneous velocity of 60 is best associated with the time instant 55 Find the slope of the tangent line at this time and show that it is less than the slope calculated Now repeat for a situation where a velocity calculated with two adjacent points would yield a velocity that is too low compared to the velocity from the tangent line Phys 114: Lab-Motion on Hills rev Su17

8 Lab: Motion on Hills Page 8 Sample Test Questions for Study: Approximately 20% of exam questions will relate to labs You should use the following questions as you prepare for exams and quizzes Most of these questions are in reference to the analysis section You should also be prepared to answer questions about the procedure, data collection, tables, graph, uncertainty, and any calculations 1 Three students have made statements about measuring the positions of the moving cart bearing Identify the incorrect and correct parts of each student's response Student 1: I measured from the center of the cart when it just breaks the beam on the first photogate because the center of the cart is its real position I also measured to the center of the cart when it is in the location to break the beam of the second photogate Student 2: Since the cart is at the photogate and the photogate is moved to a new location, you have to measure from the start timer to the stop timer Student 3: I measured from the front of the cart at the start to the back of the cart at the end because it is the last part to leave the photogate 2 Somewhere in your lab, you should have done each of the following things Be sure that you are able to demonstrate that you can find each type of information 1 For each situation below, can you find the position for an object? a) You are given a data table of Position vs Time data b) You are given a Position vs Time graph 2 For each situation below, can you find the velocity of an object? a) You are given a data table of Position vs Time data b) You are given a Position vs Time graph 3 You are given a Velocity vs Time graph For each situation below, can you find the acceleration for an object? a) You are given a data table of Velocity vs Time data b) You are given a Velocity vs Time graph Phys 114: Lab-Motion on Hills rev Su17