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. 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.
Lab: Motion on Hills Page 2 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 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?
Lab: Motion on Hills Page 3 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 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.
Lab: Motion on Hills Page 4 Method 1: Calculating the average velocity between adjacent positions using the data table. Using your Position vs. Time data, organize it so that all your average times are in order. Notice that the time column is not titled average time. Yet, you averaged several runs to get the best time to use in the formal table. Calculate the time intervals. Choose consecutive times for your time intervals: t 1 and t2, then t2 and t3, etc. For example, to calculate the time interval between the 1 st and 2 nd times use the equation: t = t 2 t 1. Now continue for the time interval between the 2 nd and 3 rd time, 3 rd and 4 th, etc. In the same way, you should be able to calculate the displacements. Now you will need to calculate the average velocity within each time interval: The equation for average velocity is: v = x. To calculate the average velocity in t the time interval between the 1 st and 2 nd times. Use the corresponding time interval and displacement. Then repeat for the interval between the 2 nd and 3 rd times, 3 rd and 4 th, and so on until all your formal data has been used. Calculate the time that is best associated with the average velocity you calculated in each time interval. For example, in the interval between the 1 st and 2 nd 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. For example, notice in the example table below that the 2 nd and 3 rd times are 0.090 sec. and 0.125 sec. The time directly in the middle between them can be calculated by using the equation for middle or average values: t middle = t 3 + t 2 2 0.090 + 0.125 t middle = 2 t middle = 0.1075 After rounding it is 0.108 sec. Tabulate all your average times, positions, and calculated data. Remember that formal data tables often contain both measured data and derived (calculated) data. Plot the velocity and the Middle time values on a velocity vs. time graph. Use point protectors but do not draw a best-fit line yet. Do not panic! These values will not be nicely aligned. The reason is that you are getting a larger amount of uncertainty in your velocity values because of the uncertainty in the positions and times. The is the reason we now turn to Method 2. Measured Data Time (s) Position (cm) ±0.021 ±0.1 0.000 20.0 0.090 24.5 0.125 28.4 Time Interval (s) Displacement (cm) Derived Data Mid Time in interval (s) 0.090 4.5 0.045 50 0.035 3.9 0.108 111 Velocity (cm/s)
Lab: Motion on Hills Page 5 You will need to add many more rows to your formal data table, as many as the different positions that you have. Notice that all columns have proper headers and that rounding is the same in each column. Also note that the measured data is rounded to correspond to the uncertainty. You will have different uncertainty and different values. DO NOT assume that your values will match the example. Method 2: Finding the average velocity between adjacent positions using the position vs time graph. You will first need to graph the position and time data from your formal table. That is, graph the data from the left two columns in the formal data table you made in method 1. Draw in your best-fit line. It will be a curve. Go along the bottom axis of your graph and find 5-7 fairly equally spaced times between (but not including) your t=0 and your last time. In the example below, I chose times 0.8, 2.0, 2.8, 4.4, and 6.0 sec. You need to have at least 5 times. You may wish to do 6 or 7 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=2.8 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 find its slope. Note: The diagram below does not match the data from the example above. Yours, however should match your table. Slope = (9 ( 1))m (5 0.8)sec = 2.4 m s You would then plot 2.4 m/s as the velocity at 2.8 sec on a velocity vs time graph. 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, Method 2 velocity and time values to draw in your best-fit line. It should be linear. If it is not, see your instructor for assistance. Notice that they
Lab: Motion on Hills Page 6 are much better in representing the actual velocities. It is because they came from the best-fit line and not from the individual data points that were above and below the best-fit line due to uncertainty. 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. 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. c) You are given a Velocity vs. Time graph 3. 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.