This Document Contains Experiments 1 to 4 Name____________________________________________________Section________________Date___________ Experiment 1: Graphing Invitation to Inquiry The measurement of a quantity that can have different values at different times is called a variable. For example, the rate of your heartbeat, the number of times you breathe per minute, and your blood pressure are all variables because they can have different values at different times. In many situations, there are relationships that occur between variables. The rate of your breathing, for example, increases when you begin to exercise, so you could say that the breathing rate is in direct proportion to exercise up to a certain limit. Measurements of variables that increase or decrease relative to each other are in direct proportion will yield a straight line on a graph, and this relationship is said to be direct, or linear. There are more types of relationships between variables, and most can be identified as producing one of five basic shapes of graphs. These are identified, left to right, as no relationship, linear, inverse, square, and square root. y y y y y x x x x x After giving the possibilities some thought, look for relationships that might result in (1) a direct relationship, then (2) something other than a direct relationship. Make measurements, graph your data, then decide which of the five shapes the graph resembles. For example, compare your heartbeat rate before climbing any stair. Then after climbing 10 stairs, 20 stairs, and 30 stairs, what is the shape of a graph comparing the heartbeat rate and the number of stairs climbed? What does this mean about the relationship between the number of stairs climbed and your heartbeat? What other relationships can you find in the lab, outside, or between any two variables in everyday occurrences? Summarize your findings here: Background Refer to Figure 1.1 for terminology used when discussing a graph, and see Appendix I on page 391 for a detailed discussion about the terms. Unit for Figure 1.1 Procedure Position a meterstick vertically on a flat surface, such as a wall or the side of a lab bench. Be sure the metric scale of the meterstick is on the outside and secure the meterstick to the wall or lab bench with two strips of masking tape. Drop a ball as close as possible to the meterstick and measure (a) the height dropped, and (b) the resulting height bounced. Repeat this for three different heights dropped and record all data in Data Table 1.1 on page 6. In the data table, identify the independent (manipulated) variable and the dependent (responding) variable. Use the graph paper on page 9 to make a graph of the data in Data Table 1.1, being sure to follow all the rules of graphing (see Appendix I on page 391 for help). Title the graph, “Single Measurement Bounce Height.” After constructing the graph, but before continuing with this laboratory investigation, answer the following questions: (a) What decisions did you have to make about how you conducted the ball-dropping investigation? Must measure the height of the ball to the bottom of the ball, not to the top or center. Larger initial heights are better - more time to react. Must be sure not to give the ball any initial velocity. Better to make the height measurements away from the device so you can look almost horizontally at the height. Would you obtain the exact same result if you dropped the ball from the same height several times? Explain. No, not exactly. Not only do you have to worry about the things mentioned in (a), but the ball does not bounce the same every time. A different amount of energy will be lost each time. Did you make a dot-to-dot line connecting the data points on your graph? Why or why not? You should try to draw a smooth curve that best represents the data. A dot-to-dot line assumes that each value is 100 percent correct. This is not possible. A smooth curve tends to average out the errors. Could you use your graph to obtain a predictable result for dropping the ball from different heights? Explain why you could or could not. The graph should help you predict the bounced height relatively well. It will vary with different types of balls. In general, your predictions will be within 5-10 percent of the measured value, with the averaged data being better than the single measurement data. What is the significance of the origin on the graph of this data? Did you use the origin as adata point? Why or why not? The origin should be a data point in this case because the ball will not bounce if it is not dropped. Make at least three more measurements for each of the previous three height-dropped levels. Find the average height bounced for each level and record the data and the average values in Data Table 1.2 on page 6. Make a new graph of the average height bounced for each level that the ball was dropped. Draw a straight best fit line that includes the origin by considering the general trend of the data points. Draw the straight line as close as possible to as many data points as you can. Try to have about the same number of data points on both sides of the straight line. Title this graph, “Averaged Bounce Height.” Compare how well both graphs, “Single Measurement Bounce Height” and “Averaged Bounce Height,” predict the heights that the ball will bounce for heights dropped that were not tried previously. Locate an untried height-dropped distance on the straight line, then use the corresponding value on the scale for height bounced as a prediction. Test predictions by noting several different heights, then measuring the actual heights bounced. Record your predictions and the actual experimental results in Data Table 1.3 on page 7. Use a new, different kind of ball and investigate the bounce of this different ball. Record your single-measurement data for this different ball in Data Table 1.4. Record the averaged data for the height of the bounce for the three levels of dropping in Data Table 1.5. Repeat procedure step 7 for the different kind of ball. Record your predictions and the actual results in Data Table 1.6 on page 8. Graph the results of the different kind of ball investigations onto the two previous graphs. Be sure to distinguish between sets of data points and lines by using different kinds of marks. Explain the meaning of the different marks in a key on the graph. What does the steepness (slope) of the lines tell you about the bounce of the different balls? The larger the slope, the bouncier the ball (less energy is lost during the collision with the floor). Results Describe the possible sources of error in this experiment. Difficulty in measuring the height (both height dropped and height bounced). Not dropping the ball from rest. Describe at least one way that data concerning two variables is modified to reduce errors in order to show general trends or patterns. Take several readings of the same data point and take an average. This helps eliminate random errors. Use a graph and draw a smooth curve to get the general relationship between the two variables. How is a graph modified to show the best approximation of theoretical, error-free relationships between two variables? Draw a smooth curve to "average out" the errors. If one data point does not seem to follow the general trend, it can be thrown out of consideration and the information should be re-determined. Compare the usefulness of a graph showing (a) exact, precise data points connected dot–to–dot and (b) an approximated straight line that has about the same number of data points on both sides of the line. The approximated straight line can be used to predict values that have not been measured. The dot-to-dot line does not give you any information about those points in between that aren't actual data points. Was the purpose of this lab accomplished? Why or why not? (Your answer to this question should be reasonable and make sense, showing thoughtful analysis and careful, thorough thinking.) (Consider whether the student has learned how to use the graph effectively. Have they determined what an impact the graph can have in the physical sciences and how to use their graph to make predictions?) Data Table 1.1 Single Measurement Data: 1st Ball Trial 1 2 3 Height Dropped _______________independent__variable Height Bounced ________________dependent _____variable Data Table 1.3 Predictions and Results: 1st Ball Trial 1 2 3 Single Measurement Data Dropped Predicted Measured Height Height Height Averaged Data Dropped Predicted Measured Height Height Height ________60 cm _______28 cm _ _______28 cm _ ________70 cm ________50 cm _______49 cm _ ________90 cm ________66 cm _______65 cm _ ________60 cm ________29 cm ________28 cm ________70 cm _51_______ cm 49_______ cm _ ________90 cm _______66 cm _ _______65 cm _ Data Table 1.5 Averaged Bounce Data: 2nd Ball Dropped Height Bounce Height Trial 1 Trial 2 Trial 3 Average __________60 cm __ ___________47 cm _ ___________47 cm _ ____________50 cm 62 cm 65 cm ____________ ____________ ____________78 cm ____________79 cm ____________48 cm 80 cm ____________ ___________100 cm _ 65 cm ____________ ____________80 cm 64 cm ____________ ____________79 cm Data Table 1.6 Predictions and Results: 2nd Ball Trial Single Measurement Data Averaged Data Dropped Predicted Measured Height Height Height Dropped Predicted Measured Height Height Height 1 2 3 _______50 cm _ _______41 cm _ _______40 cm _ _______70 cm _ ________56 cm _______57 cm _ ________90 cm _______72 cm _ _______73 cm _ ________50 cm ________40 cm ________40 cm ________70 cm ________56 cm _______57 cm _ ________90 cm ________72 cm _______73 cm_ Name____________________________________________________Section________________Date___________ Experiment 2: Ratios Invitation to Inquiry If you have popped a batch of popcorn, you know that a given batch of kernels might pop into big and fluffy popcorn. But another batch might not be big and fluffy and some of the kernels might not pop. Popcorn pops because each kernel contains moisture that vaporizes into steam, expanding rapidly and causing the kernel to explode, or pop. Here are some questions you might want to consider investigating to find out more about popcorn: Does the ratio of water to kernel mass influence the final fluffy size of popped corn? (Hint: Measure mass of kernel before and after popping). Is there an optimum ratio of water to kernel mass for making bigger popped kernels? Is the size of the popped kernels influenced by how rapidly or how slowly you heat the kernels? Can you influence the size of popped kernels by drying or adding moisture to the unpopped kernels? Is a different ratio of moisture to kernel mass better for use in a microwave than in a conventional corn popper? Perhaps you can think of more questions about popcorn. Summarize your findings here: Figure 2.1 Background The purpose of this introductory laboratory exercise is to investigate how measurement data are simplified in order to generalize and identify trends in the data. Data concerning two quantities will be compared as a ratio, which is generally defined as a relationship between numbers or quantities. A ratio is usually simplified by dividing one number by another. Procedure Part A: Circles and Proportionality Constants Obtain three different sizes of cups, containers, or beakers with circular bases. Trace around the bottoms to make three large but different-sized circles on a blank sheet of paper. Figure 2.3 Mark the diameter on each circle by drawing a straight line across the center. Measure each diameter in mm and record the measurements in Data Table 2.1 on page 23. Repeat this procedure for each circle for a total of three trials. Measure the circumference of each object by carefully positioning a length of string around the object’s base, then grasping the place where the string ends meet. Measure the length in mm and record the measurements for each circle in Data Table 2.1. Repeat the procedure for each circle for a total of three trials. Find the ratio of the circumference of each circle to its diameter. Record the ratio for each trial in Data Table 2.1. The ratio of the circumference of a circle to its diameter is known as pi (symbol π), which has a value of 3.14… (the periods mean many decimal places). Average all the values of π in Data Table 2.1 and calculate the experimental error. Part B: Area and Volume Ratios Obtain one cube from the supply of same-sized cubes in the laboratory. Note that a cube has six sides, or six units of surface area. The side of a cube is also called a face, so each cube has six identical faces with the same area. The overall surface area of a cube can be found by measuring the length and width of one face (which should have the same value) and then multiplying (length)(width)(number of faces). Use a metric ruler to measure the cube, then calculate the overall surface area and record your finding for this small cube in Data Table 2.2 on page 23. The volume of a cube can be found by multiplying the (length)(width)(height). Measure and calculate the volume of the cube and record your finding for this small cube in Data Table 2.2. Calculate the ratio of surface area to volume and record it in Data Table 2.2. Build a medium-sized cube from eight of the small cubes stacked into one solid cube. Find and record (a) the overall surface area, (b) the volume, and (c) the overall surface area to volume ratio, and record them in Data Table 2.2. Build a large cube from 27 of the small cubes stacked into one solid cube. Again, find and record the overall surface area, volume, and overall surface area to volume ratio and record your findings in Data Table 2.2. Describe a pattern, or generalization, concerning the volume of a cube and its surface area to volume ratio. For example, as the volume of a cube increases, what happens to the surface area to volume ratio? How do these two quantities change together for larger and larger cubes? As the volume of a cube increases, the surface area to volume ratio approaches zero. Part C: Mass and Volume Ratios Obtain at least three straight-sided, rectangular containers. Measure the length, width, and height inside the container (you do not want the container material included in the volume). Record these measurements in Data Table 2.3 on page 24, in rows 1, 2, and 3. Calculate and record the volume of each container in row 4 of the data table. Width Figure 2.3 Measure and record the mass of each container in row 5 of the data table. Measure and record the mass of each container when “level full” of tap water. Record each mass in row 6 of the data table. Calculate and record the mass of the water in each container (mass of container plus water minus mass of empty container, or row 6 minus row 5 for each container). Record the mass of the water in row 7 of the data table. Figure 2.4 Use a graduated cylinder to measure the volume of water in each of the three containers. Be sure to get all the water into the graduated cylinder. Record the water volume of each container in milliliters (mL) in row 8 of the data table. Calculate the ratio of cubic centimeters (cm3) to mL for each container by dividing the volume in cubic centimeters (row 4 data) by the volume in milliliters (row 8 data). Record your findings in the data table. Calculate the ratio of mass per unit volume for each container by dividing the mass in grams (row 7 data) by the volume in milliliters (row 8 data). Record your results in the data table. Make a graph of the mass in grams (row 7 data) and the volume in milliliters (row 8 data) to picture the mass per unit volume ratio found in step 5. Put the volume on the x-axis (horizontal axis) and the mass on the y-axis (the vertical axis). The mass and volume data from each container will be a data point, so there will be a total of three data points. Draw a straight line on your graph that is as close as possible to the three data points and the origin (0, 0) as a fourth point. If you wonder why (0, 0) is also a data point, ask yourself about the mass of a zero volume of water! Calculate the slope of your graph. (SeeAppendix II on page 387 for information on calculating a slope.) ∆ y (800 - 400) g Slope = ∆ x = (800- 400) mL = 1.0 g/mL Calculate your experimental error. Use 1.0 g/mL (grams per milliliter) as the accepted value. You can expect less than 10 percent error, probably less than 5 percent. Density is defined as mass per unit volume, or mass/volume. The slope of a straight line is also a ratio, defined as the ratio of the change in the y-value per the change in the x-value. Discuss why the volume data was placed on the x-axis and mass on the y-axis and not vice versa. Because if you don't have a volume of water, you do not have a mass. Volume is the independent variable and mass is the dependent variable. Was the purpose of this lab accomplished? Why or why not? (Your answer to this question shouldshow thoughtful analysis and careful, thorough thinking.) (Student answers will vary.) Results What is a ratio? Give several examples of ratios in everyday use. A relationship between numbers or quantities. Examples: 100 cents per dollar, 60 seconds per minute, 365 days per year. How is the value of π obtained? Why does π not have units? By taking the ratio of the circumference of a circle to the diameter. Both circumference and diameter are measured in the same units and when you divide the circumference by the diameter the units cancel out. Describe what happens to the surface area to volume ratio for larger and larger cubes. Predict if this pattern would also be observed for other geometric shapes such as a sphere. Explain the reasoning behind your prediction. Surface area to volume ratio approaches zero for larger and larger cubes. This pattern would also be true for other shapes because surface area is proportional to length squared and volume is proportional to length cubed so surface area/volume is propotional to 1/length which goes toward zero as the object gets larger. Why does crushed ice melt faster than the same amount of ice in a single block? There is more surface area for the smaller pieces of ice than the single block, the air is in contact with more of the ice, so it melts faster. Which contains more potato skins: 10 pounds of small potatoes or 10 pounds of large potatoes? Explain the reasoning behind your answer in terms of this laboratory investigation. The 10 lbs of small potatoes have more potato skins. There is more total surface area for the same smaller potatoes than the larger potatoes. Using your own words, explain the meaning of the slope of a straight-line graph. What doesit tell you about the two graphed quantities? The slope of a straight-line graph tells you how one quantity changes when the other variable changes. In this case, the slope equals 1.0 g/mL. This tells me that the mass of water in grams equals the volume of the same water in milliliters. Explain why a slope of mass/volume of a particular substance also identifies the density of that substance. Density is mass/volume. The slope equals the change in mass divided by the change in volume. This is the same as density. Problems An aluminum block that is 1 m × 2 m × 3 m has a mass of 1.62 × 104 kilograms (kg). The following problems concern this aluminum block: Figure 2.5 l. What is the volume of the block in cubic meters (m3)? Volume = (length)(width)(height) = (3 m)(2 m)(1 m) = 6 m3. What are the dimensions of the block in centimeters (cm)? 300 cm by 200 cm by 100 cm Make a sketch of the aluminum block and show the area of each face in square centimeters (cm2). What is the volume of the block expressed in cubic centimeters (cm3)? (300 cm)(200 cm)(100 cm) = 6,000,000 cm3. What is the mass of the block expressed in grams (g)? 1.62 × 104 kg × 1000 g/1 kg = 1.62 × 107 g What is the ratio of mass (g) to volume (cm3) for aluminum? mass/volume = 1.62 × 107 g/6 × 106 cm3 = 2.7 g/cm3 Under what topic would you look in the index of a reference book to check your answer to question 6? Explain. Check the value of mass density for aluminum. Data Table 2.2 Area and Volume Ratios Small Cube Medium Cube Large Cube Surface Area (cm2) Volume (cm3) Ratio of Area/Volume ________________24.4 ________________8 ________________3.0 (cm2)/(cm3) ________________96 ________________64 ________________1.5 (cm2)/(cm3) ________________386 ________________512 ________________0.75 (cm2)/(cm3) Name____________________________________________________Section________________Date___________ Experiment 3: Motion Invitation to Inquiry Have you ever seen an entire stage covered with dominoes lined up, one after another and winding around into interesting patterns? The entertainer tips over one domino, which falls into another, which falls into the one next to it . . . and on until in a short time all the dominoes have fallen over. How far apart should the dominoes be spaced for maximum speed? Is it possible to vary this speed by changing the spacing? One domino causes a falling row to continue falling by hitting its neighbor, so the limit to how far apart the dominoes are spaced must be the length of a domino. The other limit would be zero space between two adjacent dominoes, so the limits to the spacing between two adjacent dominoes must be somewhere between zero and one domino length. Thus, it would be convenient to record spaces between dominoes as a ratio of domino lengths, that is, the space between dominoes in cm divided by the length of one domino in cm. You will need to determine how you plan to space the dominoes, as well as how many dominoes are needed to measure the speed. By making a graph and doing some calculations, can you predict how many dominoes would be needed—and at what spacing—to make a row that takes exactly 2 minutes to fall? Ratio of spacing length to domino length (spacing/domino length). (example 4.0 cm) (Spacing in domino ratios: from 2.0 cm/4.0 cm = 0.5) Figure 3.1 Background In this investigation you will analyze and describe motion with a constant velocity and motion with a nonconstant velocity. First, motion with a constant velocity will be investigated by using a battery-operated toy bulldozer, or any toy car or truck that moves at a fairly constant speed. Data will be collected, analyzed, and a concept will be formalized to describe what is happening to the toy as it moves. Figure 3.2 compares the distance vs. time slopes for motion with a constant velocity, with a nonconstant velocity, and with no velocity at all. Note that the slope for some object not moving will be a straight horizontal line. If a vehicle is moving at a uniform (constant) velocity, the line will have a positive slope. This slope will describe the magnitude of the velocity, sometimes referred to as the speed. The line for a vehicle moving at a nonconstant speed, on the other hand, will be nonconstant as shown in Figure 3.3. A nonconstant speed is also known as accelerated motion, and the ratio of how fast the motion is changing per unit of time is called acceleration. Taking measurable data from a multitude of sensory impressions, finding order in the data, then inventing a concept to describe the order are the activities of science. This investigation applies this process to motion. Figure 3.2 Procedure Part A: Constant Velocity on the Level Use masking tape to secure a length of paper, such as long sheets of computer paper, rolled butcher paper, or adding machine tape across the floor. The paper should be long enough so the motorized toy vehicle used will not cross the entire length in less than 8 to 10 seconds. Thus, the exact length of paper selected will depend on the vehicle and battery conditions. (Note: Erratic increases or decreases of speed probably mean that a new battery is needed.) The paper will be used to record successive positions of the toy at specific time intervals. Paper Motorized toy Figure 3.3 One person with a stopwatch will call out equal time intervals that are manageable but result in atleast five or six data points for the total trip. Another person will mark the position of the toy vehicle on the paper when each time interval is called. To avoid interfering with the motion of the toy, mark the position from behind each time. This also means that the starting position should be marked from behind. Other means of measuring velocity that might be used in your laboratory, such as the use of photogates and computer software, will be explained by your instructor. Measure the intervals between the time marks, recording your data in Data Table 3.1 on page 34. Make a graph that describes the motion of the toy vehicle by placing the distance (the dependent variable) on the vertical axis, and time (the independent variable) on the horizontal axis. Draw the best straight line as close as possible to the data points. Calculate the slope and record it someplace on the graph. Part B: Constant Velocity on an Incline This investigation is similar to part A, but this time the toy vehicle will move up an inclined ramp that is at least 1 m long. Elevate the ramp with blocks or books so that 1 meter from the bottom of the ramp is 10 cm high. As in part A, one person with a stopwatch will call out equal time intervals that are manageable, but result in at least five or six data points for the total trip. Another person will mark the position of the toy vehicle on the paper when each time interval is called. To avoid interfering with the motion of the toy, mark the position from behind each time. Also mark the starting position from behind. Measure the intervals between the time marks, recording your data in Data Table 3.2 on page 35. Elevate the ramp to 20 cm high and repeat procedure step 2. Elevate the ramp to 30 cm high and again repeat procedure step 2. Make a graph of all three sets of data in Data Table 3.2. Calculate the slope of each line and write each somewhere on the graph. Part C: Motion with Nonconstant Velocity You will now set up a track for collecting data about rolling balls. This track can be anything that serves as a smooth, straight guide for a rolling ball. It could be a board with a V-shaped groove, U-shaped aluminum shelf brackets, or two lengths of pipe taped together, for example. The track should be between 1 and 2 m long and supported somewhere between 10 and 50 cm above the table at the elevated end (see Figure 3.4). In this investigation, a longer track will mean better results. You should consider 1 m as a minimum length. Your instructor will describe a different procedure if your lab has photogates, computer software, or different equipment. Figure 3.4 You will select a minimum of six positions on the ramp from which to release a steel ball or marble. One position should be the uppermost end and the others should be equally spaced. Hold a ruler across the track with the ball behind it, then release the ball by lifting the ruler straight up the same way each time. Start a stopwatch when the ball is released, then stop it when the ball reaches the bottom of the ramp. A block at the bottom of the ramp will stop the ball and the sound of the ball hitting the block will signal when to stop the stopwatch. Measure the distance and time for three data runs, then average the data for each of the six positions. Record the data in Data Table 3.3 on page 36. Make a graph of the data with time on the x-axis. Results Explain for each part of this investigation how you know if there is or is not a relationship betweenthe variables according to your graphs. All the graphs indicate a relationship between the variables. If there were no relationship the data points would be scattered all over the place with no organization. For motion with a constant velocity, how do the changes in distance compare for equal time intervals? Is this what you would expect? Explain. Equal changes in distance for equal time intervals. This is to be expected. Velocity is defined to be change in distance/change in time. If this ratio does not change in time, then the velocity is constant. This is the case here. What is the rate of travel of the toy over (a) a flat surface, (b) a surface elevated 10 cm high, (c) a surface elevated 20 cm high, and (d) a surface elevated 30 cm high? rate of travel = velocity (or speed) speed = slope of distance vs. time graph The speed of the toy over a flat surface and various elevated surfaces will depend on the type of motorized car, conditions of batteries, and other variables. In general, the speed should decrease as the angle of incline increases. For motion with a nonconstant velocity, how does the total distance change as the total time increases; that is, do they both increase at the same rate? Explain the meaning of this observation. Total distance increases at a faster rate than the total time. Considering nonconstant velocity, how do the changes in distance compare for equal time intervals? Changes in distance are not constant for each time interval. The changes in distance increase as the total time increases. Was the purpose of this lab accomplished? Why or why not? (Your answer to this question should show thoughtful analysis and careful, thorough thinking.) (The student should be able to describe the distance between constant velocity and nonconstant velocity in terms of the graphs drawn.) Going Further In part of this investigation, you learned that v = dt . Using this equation, explain how you can find the time for a trip when given the average speed and the total distance traveled; t = d v (Must make sure the distance units match.) the total distance traveled when given the time for a trip and the average speed; d = t v (Must make sure the time units match.) the average speed for a trip, no matter what units are used to describe the total distance and the total time of the trip. v = dt . (Then the average speed will have the units of distance/time.) Data Table 3.1 Distance and Time Data for Battery-Powered Toy over a Flat Surface Total Time (s) Total Distance (cm) __________0 __________10 __________2 __________3 __________30 __________4 __________39 __________5 __________51 Data Table 3.2 Distance and Time Data for Battery-Powered Toy over Elevated Surfaces Total Distance (cm) Time (s) 10 cm Elevation20 cm Elevation30 cm Elevation __________0 cm __________0 cm __________0 cm 0 __________ __________15 cm __________13 cm __________12 cm 2 __________ __________31 cm __________29 cm __________25 cm 4 __________ __________48 cm __________42 cm __________36 cm 6 __________ __________65 cm __________55 cm __________47 cm 8 __________ Data Table 3.3 Time and Distance Data for Rolling Ball on Ramp Distance from Bottom (cm) Time Trial 1 (s) Time Trial 2 (s) Time Trial 3 (s) Time Average (s) __________ 200 __________ 8.93 __________ 8.89 __________ 8.96 __________ 8.93 __________ 160 __________ 7.96 __________ 8.02 __________ 7.98 __________ 7.99 __________ 120 __________ 6.90 __________ 6.96 __________ 6.93 __________ 6.93 __________ 80 __________ 5.66 __________ 5.62 __________ 5.70 __________ 5.66 __________ 40 __________ 4.04 __________ 3.97 __________ 4.02 __________ 4.01 __________ 2.78 __________ 2.84 __________ 2.81 __________ 20 __________ 2.80 Name____________________________________________________Section________________Date___________ Experiment 4: Free Fall Invitation to Inquiry Find out how well you can predict the motion of falling objects. First, select some objects such as a rubber ball, a sheet of notebook paper, and a large metal paper clip. Predict, then study the detail of each object falling independently. For example, what happens to each as they fall? Then compare the motion of the objects side by side. Is it possible to cause them to fall together, at the same time? Use measurements to construct a graph or graphs that show what is going on between the variables involved in falling objects. Then use the graph to show how to place three or four objects on a long cord. Attach them so when the cord is hung from a high place, then dropped, the objects make a constant plop, plop, plop sound when they hit the ground. Figure 4.1 Background In this experiment you will calculate the acceleration of an object as it falls toward the earth’s surface. An object in free fall moves toward the surface with a uniform accelerated motion due to gravity, g. The value of g varies with location on the surface of the earth, increasing with latitude to a maximum at the poles. The value of g also varies with elevation, decreasing with elevation at a certain latitude. The average, or standard, value of g, however, is usually accepted as 9.8 m/s2 or 980 cm/s2. When you measure the total distance that an object moves during some period of time, you can calculate an average velocity. Average velocity is defined as v = ∆∆dt where ∆d is the total distance (final distance minus initial, or df – di) and ∆t is the total time (final time minus initial, or tf – ti). In this experiment you will be measuring the velocity of an object that falls from an initial distance and time of zero, so ∆d = df – 0 and ∆t = tf – 0. For the case of a falling object, v = ∆∆dt = dtff i−− tdi since di= 0 and ti=0 ∴ v = dtff . Thus, you can calculate the average velocity of an object in free fall from the total distance traveled and the time of fall. When an object moves with a constant acceleration, you can also find the average velocity by adding the initial and final velocity and dividing by 2, v = vf i2+v . By substituting the other expression for average velocity, we have v = vf i2+v and v = dtff ∴ vf i2+v = dtff . Since the initial velocity of a dropped object is zero, then vi is zero and we can solve for the final velocity of vf, and vf df 2df = ∴ v = . 2 tf f tf In this experiment you will measure the distance a mass has fallen during recurring time intervals according to a timing device. This data will enable you to calculate the instantaneous velocity at known time intervals. Plotting the velocity versus the time, then finding the slope will provide an experimental value of g. Procedure Spark timer Paper tape (Note: in some setups the paper moves; in others the sparker moves.) mass Figure 4.2 You will experimentally determine the acceleration due to gravity and compare it to the standard value of 980 cm/s2. The procedure may vary with the apparatus used. For example, you might use an apparatus that consists of a device to measure the free-fall of an object with a spark timer that will mark a paper tape at equal time intervals. As a mass accelerates downward it will leave a trail of spark marks at equal time intervals.You will draw a perpendicular line through each mark, then identify the first mark as your reference line. The first mark is identified as the place where df = 0. Other means of measuring velocity that might be used in your laboratory, such as the use of photogates and computer software, will be explained by your instructor. For spark mark (or ink dot) trails measure the total distance (df) by using the beginning mark as a reference line. On page 48, record in Data Table 4.1 the distance in cm of each mark from the reference line. For each spark, record the time (t) that elapsed between the marks as determined by the spark timer. Your instructor will provide exact information for your timer. Most timers are set to operate on 60 Hz, making a spark every 1/60 second. Thus, the second spark would have occurred 1/60 second after the first, the third spark mark would have occurred 1/60 plus 1/60, or 2/60 (0.033 s) after the first mark. Fill in Data Table 4.1 with the total distance and time data for each mark, and calculate and record the velocity at each spark. Repeat the experiment two more times with two more paper tapes, completing Data Table 4.2 (page 49) and Data Table 4.3 (page 50). Results Look over the data in Data Table 4.1, 4.2, and 4.3, think about what the data means, then select the Data Table that seems to have the “best run” data. State which table was chosen and explain the basis for your choice. Students should choose the run in which the velocity increases at a constant rate from one time to the next. In the example data, this would be run #2. Using the data table from the best run, make a graph with velocity (v) on the y-axis and elapsed time (t) on the x-axis. (Note: Because the first spark was probably not made at the actual time of release, the line on your graph will probably not have a y intercept of 0.) Find the slope and record it here, along with any notes you may wish to record. slope = ∆∆tv = 805/60scm/s −−112/60scm/s = 1020cm/s2 The answer should be close to 980 cm/s2. Use the calculated slope and the accepted value of 980 cm/s2 to calculate the experimental error. ×100% = 4% difference Was the purpose of this lab accomplished? Why or why not? (Your answer to this question should show thoughtful analysis and careful, thorough thinking.) Was the student (within 10%) correct in determining the value of g? Did the student explain why the slope of the velocity vs. time graph should equal the acceleration, which is g for free fall? Going Further What is your reaction time? One way to measure your reaction time is to have another person hold a meterstick vertically from the top while you position your thumb and index finger at the 50 cm mark. The other person will drop the meterstick (unannounced) and you will catch it with your thumb and finger. Accelerated by gravity (g), the stick will fall a distance (d) during your reaction time (t). Knowing d and g, all you need is a relationship between g, d, and t to find the time. You know a relationship between , , and from = d v t v d / t. Solving for givesd d = vt. Any object in free-fall, including a meterstick, will have uniformly accelerated motion, so the average velocity is v = vf + vi 2 Substituting for the average velocity in the previous equation gives d = vf +2 vi ( )t The initial velocity of a falling object is always zero just as it is dropped, so the initial velocity can be eliminated, giving d = v2f ( )t Now you want acceleration in place of velocity. From a = vf − vi t and solving for the final velocity gives vf = at The initial velocity is again dropped since it equals zero. Substituting the final velocity in the previous equation gives d = at( )t 2 or d = at2 Finally, solving for t gives t = Measuring how far the meterstick falls (in m) can now be used as the distance (d) with g equaling 9.8 m/s2 to calculate your reaction time (t). Data Table 4.1 Free Fall Run Number One Spark Number Distance (cm) Time of Fall (s) Computed Instantaneous Velocity* (cm/s) 1 0.2 0.0167 12.0 2 0.5 0.0333 18.0 3 1.2 0.0500 42.0 4 2.2 0.0667 60.0 5 3.4 0.0833 72.0 6 4.9 0.1000 90.0 7 6.7 0.1167 108.0 8 8.7 0.1333 120.0 9 10.9 0.1500 145.0 10 13.7 0.1667 164.0 *From vf = 2tdff Data Table 4.2 Free Fall Run Number Two Spark Number Distance (cm) Time of Fall (s) Computed Instantaneous Velocity* (cm/s) 1 0.1 0.0167 12.0 2 0.5 0.0333 30.0 3 1.3 0.0500 52.0 4 2..2 0.0667 66.0 5 3.4 0.0833 82.0 6 5.0 0.1000 100.0 7 6.6 0.1167 113.0 8 8.7 0.1333 131.0 9 11.0 0.1500 147.0 10 13.6 0.1667 163.0 *From vf = 2tdff Data Table 4.3 Free Fall Run Number Three Spark Number Distance (cm) Time of Fall (s) Computed Instantaneous Velocity* (cm/s) 1 0.2 0.0167 12.0 2 0.5 0.0333 18.0 3 1.2 0.0500 42.0 4 2.2 0.0667 60.0 5 3.4 0.0833 72.0 6 4.9 0.1000 90.0 7 6.7 0.1167 108.0 8 8.7 0.1333 120.0 9 10.9 0.1500 145.0 10 13.7 0.1667 164.0 *From vf = 2tdff Solution Manual Experiment for Integrated Science Bill W. Tillery, Eldon D. Enger , Frederick C. Ross 9780073512259
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