PHYSICS 15a, Fall 2006 SPEED OF SOUND LAB Due: Tuesday, November 14 GENERAL INFO The goal of this lab is to determine the speed of sound in air, by making measurements and taking into consideration the errors involved in these measurements. You will work in pairs. You can sign up your group s two names either in lecture or in office hours. Each pair works together on their lab report, and the due date is Tuesday, November 14 (but please don t wait until the last minute; try to pretend it s due a week or two earlier). There are various methods for measuring the speed of sound. For example, you can use high-tech timing equipment, or you can utilize resonant frequencies in tubes. But we re going to do it the old-fashioned way here: You and your lab partner will stand a large distance away from each other. One of you will make a noise, and the other will measure the time it takes to hear the sound. That s about it. Nothing fancy going on. There are, however, a few things you ll have to worry about... THINGS TO CONSIDER Temperature: The speed of sound in air depends on the temperature. A reasonably accurate expression for the speed as a function of temperature is v = (331 + 0.6 T C ) m/s, where T C is the Celsius temperature. For example, a cool October day of 40 F 4 C yields v 334 m/s, whereas a nice warm day of 70 F 21 C yields v 344 m/s. The difference in these values is only about 3%, but be sure to note the temperature when you do your experiment. Wind speed: Sound needs air to propagate in (unlike light, which requires no medium). If the air as a whole is moving with respect to the ground, then this speed must be added or subtracted from the v = (331+0.6 T C ) m/s speed given above. You are strongly advised to pick a day without much wind, but not so much because the wind affects the speed, but because it makes it much harder to hear the sound your lab partner makes. Therefore, assuming that you ve picked a day with only about a 5-10 mph (that is, 2-3 m/s) wind, this will be at most a 1% effect. But make your best estimate of this effect in any case. Since it is much easier to hear things downwind (due to an interesting effect of refraction), the timer should stand downwind, unless this puts the timer close to the sound of cars, in which case you ll have to experiment to find the spot that works best. Reaction time: We all have some nonzero reaction time. After you hear a sound, a signal needs to travel from your ear to your brain, and then from your brain to your hand to tell your finger to stop your stopwatch. This all takes time. Experimentally, it is found that a normal reaction time is about 0.19 s for a light stimulus and about 0.16 s for a sound stimulus. A reaction time of less than 0.10 s is considered impossible. In elite track meets, if a sprinter registers a reaction time of less than 0.10 s (from when the gun goes off to when pressure from the feet is detected on the starting blocks), then he/she is disqualified, because it must have been the case that he/she was guessing when the gun would fire. In making a time measurement, you will need to start and stop your stopwatch. As 1
far as the starting goes, the reaction time isn t so much of an issue. If your lab partner makes a large and very visible motion when making the sound, then you can anticipate when the sound occurs, so you are in fact not reacting to anything. You can therefore assume that your starting time is the true starting time. (This depends, of course, on the speed of light being essentially infinite for the present purposes.) There will inevitably be a slight error in the starting time, but it will be on the order of a few hundredths of a second, and it will probably average out to zero. But you and your partner should practice a bit on this. (Can you think of an experiment you can do to figure out what the general size of this error actually is?) How do we deal with the inevitable reaction time during the stopping process? There are two ways you can determine your reaction time. Your lab partner holds a meter stick vertically, and you place your thumb and forefinger on either side of it at, say, the 50 cm mark. Your partner then drops the stick, and you catch it. If the stick falls a distance d during this process, then your reaction time can be calculated from the usual relation, d = gt 2 /2. Let x be the distance you stand from the sound source, and let t(x) be the reading on your stopwatch (which includes your reaction time). If you take readings for various values of x, and if you plot the results on a t vs. x graph, then you can draw a best-fit line (more on this below) and see where it intersects the t-axis (the vertical axis). This intercept gives the hypothetical value of t that you would obtain in the case of x = 0, that is, in the case where you are standing right next to the sound source. In other words, the t-intercept is your reaction time. The desired speed is simply the reciprocal of the slope of the t vs. x line. (Equivalently, it would be the actual slope of the x vs. t graph if you had put t on the horizontal axis.) Since the speed of sound is what we re concerned with in this lab, technically we need only concern ourselves with the slope, and there is actually no need to even mention the reaction time, which is simply a byproduct of the graphical analysis. But it s an interesting byproduct, so you should definitely see what it turns out to be. Note that this second method of determining the reaction time is based on the assumption that the reaction time is independent of x. This is a reasonable assumption, provided that x isn t too small. If x is too small, then the sound might get to you too quickly for you to be able to physically move your finger fast enough to press stop after you ve finished pressing start. Since the speed of sound is around 340 m/s, an x value of 100 m corresponds to a transit time of about a third of a second. So you shouldn t go too far below 100 m. But feel free to see how small you can make x and not have things go haywire. Note that in order to keep things consistent, the same person should do all the timing. EQUIPMENT Two boards (or whatever you want to smash together to make a big sound) Stopwatch String 2
Meter stick YOUR TASKS As you do the experiment, you will inevitably find that certain ways of doing things work better than others. It is therefore advisable to start the lab soon, so that if you discover a much better way of doing things, you ll still have time to implement it. Once you ve figured out what works best, the actual gathering of data won t take too long. But no matter what method you end up using, we want you to do at least the following: 1. Read the Lab Companion handout posted on the web. 2. Measure your reaction time via the first method above. 3. Note the temperature and wind speed. 4. Take about 10 time measurements (or more if you want to) each for at least 8 different distances. These distances should be in the 100 m to 200 m range, but if you re able to go above this range, by all means do so. If you find that 100 m is too short a distance to obtain reliable data, then start with a larger distance. Be sure to practice a few times before you actually start taking data. Also, in this modern day of cell phones, you might want to take advantage of these in order to communicate over the large separation. It beats shouting. 5. Make a table of your measurements, and plot the t vs. x data (with error bars) on graph paper, and find the best-fit line (the t-intercept should hopefully be roughly equal to the reaction time you obtained with the meter stick). Also, find the upper and lower bounds on the slope and t-intercept. (More on error bars, best-fit lines, etc. below.) 6. Discuss any other possible errors (random or systematic) that might affect your results. 7. Your lab report should contain a description of everything you do. Assume that you are explaining the lab, from start to finish, to a fellow student who doesn t know anything about it. You can divide the various parts of the report between the two of you, but both partners should understand everything in the report (just hand in one copy for both of you). The usual structure of a lab report is (but you don t have to follow this structure exactly, as long as you include everything in one way or another): Introduction and Goal Procedure and Theory Data Analysis and Results Discussion and Conclusion In addition to looking for good lab technique and solid analysis, we are also looking for a well-written report where you explain things clearly. 3
The athletic fields across the river are probably the best place to do this lab. However, Soldiers Field Road wraps around the athletic fields and the cars will make it harder to hear the sound, so the person doing the timing should be as far away from the cars as possible, all other things being equal. (Don t do the lab inside the stadium you ll need a longer distance than that provides, and you ll probably get yelled at for being on the field anyway.) Another possible location is the open space along the river, but the cars are even more of an issue there. And people will give you funny looks. In any event, do NOT do this lab in the Yard or along a street. The pedestrians who aren t tripping over your string will be calling the police about the excessive sounds you re making. ANALYSIS The Lab Companion handout on the course webpage has a good introduction to error analysis. (This was written for another physics class, but it works perfectly well here too. You can ignore the class-specific parts on page 3.) As noted at the bottom of page 5, errors are not necessarily things that arise due to mistakes that you make. Rather, they are the unavoidable uncertainties involved in any measurement. Review of the standard deviation: In order to draw the best-fit line mentioned above, we ll need to get a handle on the errors in your best guesses for what the true times (travel times plus your reaction times) are at the different distances. And in order to get a handle on these errors, we ll need to talk about standard deviations a bit. Assuming that you have a consistent average for your reaction time, there is a definite true value for the average total time of travel plus reaction. However, you will undoubtedly not measure this true value, because your reaction time will inevitably vary from one measurement to the next in a random way. This will cause a spread in your measured values. It turns out that most of the random probability distributions that come up in physics and math can be approximated by a Gaussian function (a bell curve ), at least in the limit of a large number of trials. So we ll assume here that the random fluctuations in your measurements are distributed in a Gaussian way. We won t have time to get into the details of the theory behind Gaussians, so we ll just invoke a few results here. The standard deviation of a Gaussian probability distribution is defined on page 10 of the Lab Companion by (with a slight change in notation here) Ni=1 (t i t) 2 σ = N 1, (1) where t is the average of the N time measurements at a given distance. The significance of the standard deviation is that any t value you measure has a 68% chance of lying between t σ and t + σ. In other words, if the curve below gives the probability, P (t), of measuring a certain value of t, then the shaded area is 68% of the total area under the curve. 4
P(t) t = t 0 σ t The standard deviation σ is also called the uncertainty, or the error, in the distribution. The Gaussian curve happens to be given by the function, P (t) = e (t t 0) 2 /2σ 2 σ 2π, (2) where t 0 is the true value. But we won t need this explicit form here. As an example, let s say you make N = 10 time measurements, with the results (in units of seconds):.66,.67,.63,.64,.68,.63,.67,.61,.65,.66 (3) Of course, the distribution of these 10 values doesn t look much like a Gaussian distribution, but presumably if you took more measurements and drew a histogram of the times, it would start to look like a bell curve. The average of the above 10 times is t =.65, so eq. (1) gives the standard deviation as (.01) 2 +(.02) 2 +(.02) 2 +(.01) 2 +(.03) 2 +(.02) 2 +(.02) 2 +(.04) 2 +(0) 2 +(.01) 2 σ = 9.022. (4) Now, 10 isn t such a large number, so we can t be too sure about this standard deviation, 1 but assuming that it s basically correct, we can say that if you continue to make many more measurements, then about 68% of them will fall within the range.65 ±.022. Given your data, what is the best guess for the true value of the time? It can be shown that best guess is simply the average, t. This is completely reasonable, of course, and you would probably just have assumed it anyway. But now consider the more difficult question: What is the uncertainty (that is, the standard deviation) in this guess? That is, if you make the statement, I am 68% sure that the true value of the time lies within the range.65 ± σ, then what is σ? A partial answer is that σ is not equal to the standard deviation of the distribution, σ. This can be seen by the following argument. If someone else comes along and makes 1000 time measurements (instead of your 10), then she will obtain the same (or nearly the same) standard deviation σ that you obtained. However, she will able to say that the true value of the time is much closer to her average that you can say with your 10 measurements, simply because it is much less likely for a Gaussian of 1000 points to be significantly shifted away from the true value, because a larger number of unlikely events would have to occur for this shift to exist. It can be shown (and this is by no means 1 There s a way of getting the error on the standard deviation (that is, the error on the error), but we won t get into that here. 5
obvious) that the uncertainty in the best guess (which is the average, t) for the true value of the time is equal to σ = σ, (5) N where N is the number of measurements you used to obtain the average. This result has the desired property of decreasing with N. This uncertainty σ is also known as the standard deviation of the mean, for the following reason. If you do your 10 measurements and write down the average, and then do another 10 measurements and write down their average, and keep on doing this, then it can be shown that these averages that you write down will form a Gaussian distribution with a standard deviation of σ = σ/ N, with N = 10. In other words, the probability distribution of these averages will look like the solid curve below (the dotted curve is simply the probability distribution of obtaining a value of t on any given single measurement). P(t ) (st dev = σ/ 10 ) P(t) (st dev = σ) t = t 0 t So if N = 10, then your uncertainty in the true value of the time is σ/ N = (.022)/ 10 =.007. In other words, you can be about 68% confident that the true value lies between.643 and.657. On the other hand, the person who had done the 1000 measurements would obtain an uncertainty in the true value equal to (.022)/ 1000 =.0007. So if her average happened to come out to be.6400 (although it undoubtedly wouldn t be exactly equal to your average), then she could be about 68% confident that the true value of the time lies between.6393 and.6407. Best-fit line: To determine the best-fit line to your data, you should look at the procedure explained on the Mathworld website: www.mathworld.wolfram.com/leastsquaresfitting.html The treatment there doesn t incorporate errors, so we ll need to make one modification: Eq. (4) on the webpage becomes (in slightly new notation) m R 2 [t i (a + bx i )] 2 (a, b) = σ 2, (6) i=1 i where σ i is the uncertainty in your best guess (given above in eq. (5)) for the true value of the time for the ith set of data points (there are m different sets, corresponding to the m distances you use). These σ i s are the error bars that you should draw on your graph. The division by σ 2 i in eq. (6) has the effect of forcing a + bx i to be closer to t i, the smaller σ i is. This is what we expect, because σ i is your uncertainty in the value of 6
t i. 2 Eqs. (7) and (8) on the webpage then become ( m ) ( 1 m a + b a ( m ) x i + b ) x i ( m x 2 ) i = = m t i m i=1 x i t i σ 2 i, (7). (8) These are two equations (formidable looking, but they re really not so bad) in the two unknowns, a and b, so you can easily solve for these. Although you can probably obtain a good answer for the best-fit line by just eyeballing your graph, the above method is good to know. Note: You must calculate the values of a and b with this method, and NOT with Excel. But feel free to plot your graph using a spreadsheet (and then draw in the error bars by hand). Bounds on the slope: As far as the bounds on the slope (and also the t-intercept) of your line go, you can take a less technical approach than the above one: Just draw the lines of maximum and minimum slope that are consistent with all your error bars. The resulting bounds on the slope will give roughly the bounds on the speed of sound. Likewise for the t-intercept and your reaction time. Of course, it may be the case for your data that there doesn t exist a line that passes through all of your error bars, in which case you won t be able to draw lines of max and min slope that are consistent with the bars. This isn t a disaster the error bars only represent a 68% confidence interval, after all. Just pick the bounds on the lines that seem roughly consistent with your error bars. Final note: You will inevitably not obtain the true value of v = (331 + 0.6 T C ) m/s from your best-fit line. The goal of this lab is not to get the right answer for the speed of sound. Rather, the goal is to get a handle on your errors, which arise from measurement uncertainty, and to then show that the true value is consistent with your errors, or at least close. 2 Note that it wouldn t make sense to just have the original σ 2 i for the ith set of data points in the denominator, because if one set had 10 data points and another had 1000 data points, then they would both have (roughly) the same σ, which would mean that the better bounds on the true value in the 1000-point case wouldn t be taken into consideration. The use of σ properly incorporates the improved bounds on t for larger sets of data points. 7