THE FALL OF JAVERT Could Inspector Javert have survived the fall? Mathalicious 2013 lesson guide At the end of the popular musical Les Misérables, a dejected Inspector Javert throws himself off a bridge and into the River Seine. As he falls, he sings and sings and sings. According to the song, he falls for a full eight seconds! In this lesson students use quadratic functions and information about how objects fall (i.e. gravity) to determine how high Javert s bridge must have been. Then, they use linear functions to figure out how fast he was traveling when he hit the water, and whether we can believe anything that Broadway says er, sings. Primary Objectives Model the motion of a falling object with a quadratic function Use the duration of Javert s fall to determine how far he fell Develop a (linear) model for speed vs. time Determine Javert s speed when he hit the water, and discuss whether or not he could have survived Content Standards (CCSS) Mathematical Practices (CCMP) Materials Algebra Functions CED.1 BF.1a, IF.6, IF.7a MP.3, MP.4 Student handout LCD projector Computer speakers Before Beginning Students should be able to distinguish between linear functions (e.g. y = 2x + 4) and non- linear functions (e.g. y = x 2 ). They should also have experience graphing linear functions.
2 Preview & Guiding Questions Students listen to a song from the musical Les Misérables in which the French police officer Javert commits suicide by jumping off a Parisian bridge. As he falls, he sings one final note. Have students estimate the length of this note; the answer is around eight seconds. Next, students need to realize that eight seconds is a long time to fall. You can try to give them a feel for what s reasonable by asking them how long they think it would take to fall a certain distance, or how far they think they would fall in a certain amount of time. Once they believe that eight seconds seems fairly long, ask them what this must mean about the bridge Javert fell from. The answer: it must be really tall. But just how tall, exactly? This is where the main lesson begins. Note: In case you (or your students) wonder why Javert jumped off the bridge in the first place, when the show opens, Javert is a jailor and enemy of the prisoner Jean Valjean. After serving 19 years for stealing bread, Valjean is released. Many years later, during the Paris Uprising, revolutionaries sentence Javert to death and Jean Valjean offers to take him outside and shoot him. But instead, Jean Valjean fires his gun into the air and tells Javert to flee. Javert, however, is unwilling to live in the debt of a thief, and so he finds a bridge and takes his life. Around how long is Javert s final note? Does it seem like Javert fell for a long time, or a short amount of time? How long do you think you could hold a note if you fell one foot? Ten feet? How long do you think you d fall in one second? Two seconds? How tall do you think the bridge must have been? Act One In Act One, students watch an animation of three objects falling and graph the data. They use this to predict the distances fallen after 4 and 5 seconds, and they use their reasoning to determine the height of Inspector Javert s bridge. The big takeaway here is that the distances objects fall can be modeled with quadratic functions. Using this model, students discover that in order for Javert to have sung for eight seconds as he fell, the bridge must have been around 1024 feet high, which is absurd. (To put this in perspective, the height of the Eiffel Tower is 1063 feet, while the height of the Golden Gate Bridge in San Francisco is only 220 feet.) Act Two In Act One students focused on distance as a function of time, d(t). In Act Two they ll look at speed as a function of time, s(t). While this is often taught in physics classes and can involve calculus, we can actually begin to explore speed using basic algebra. And that s exactly what students will do here. By looking at how far an object falls each second, students will compute average speeds over one- second intervals, and will compare these results to the continuous speed function s(t) = 32t (measured in feet per second). They can then use this equation to calculate Javert s speed when he hit the water and make a determination about whether or not the fall could ve actually taken his life.
3 Acte Une: Le Misérable 1 You can predict how far a body a basketball, anvil, etc. will fall over time. Ignoring air resistance, after one second a body will have fallen 16 feet; after two seconds, 64 feet; after three seconds, 144 feet. Plot the data below. How far will a body have fallen after four seconds? Five seconds? Distance (ft) 400 300 256 ] 400 ] Method 1: Time 0 s 1 s 2 s 3 s 4 s 5 s Distance 0 ft 16 ft 64 ft 144 ft 256 ft 400 ft 200 144 ] 1 st difference +16 +48 +80 +112 +144 2 nd difference +32 +32 +32 +32 100 0 64 ] 16 ] 0 1 2 3 4 5 Time (s) Method 2: 0, 16, 64, and 144 are all perfect squares: 0 2, 4 2, 8 2, and 12 2. Continuing this pattern, the next two distances should be 16 2 = 256 ft after 4 seconds, and 20 2 = 400 ft after 5 seconds. As you can see, a falling object will fall farther and farther with each second. This means that it isn t falling at a constant rate. This should naturally lead students to wonder, How much farther is it falling each second? This is a great question, and is an important first step in figuring out how far an object will fall after any amount of time. To answer it, students might first figure out how far an object travels during each one- second interval 16 feet during the first second, 48 feet during the second second (no pun intended), and so on. From here, you can ask them how much farther an object travels each second. The goal is to get students to realize that the distance an object travels each second keeps increasing by 32 feet (from 16 to 48 feet between the first and second seconds, then from 48 to 80 feet between the second and third). Students can continue this pattern to plot the other points. Note: some may notice that all the distances are perfect squares, and continue the pattern based on this fact. Does an object travel the same distance during each one- second interval? Since the change in distance is not constant, what does this mean about the function? Can it be linear? How far does the object travel during each one- second interval? How much farther does the object travel during each one- second interval? Do you notice anything about the types of numbers that show up as distances? Continuing your pattern, what will happens after four seconds? Five? Why isn t the distance an object falls in a second the same from one second to the next? (Gravity is always pulling the object down, and when you pull on something in the direction it s moving, it will move faster.) In most situations, do you think all objects really fall at the same rate? (No. Because of air resistance, some objects (e.g. a feather) will typically fall more slowly than others (e.g. a piano).)
4 2 Write an equation to predict how far an object will have fallen after t seconds. After 1 second, an object has fallen 4 2 feet. After 2 seconds, 8 2 feet. 3 seconds: 12 2 feet. To get the distance, we can multiply the time by 4, then square the product. In other words, let t = time object has fallen in seconds, d = distance in feet. Then d = (4t) 2, d = 16t 2. If students haven t yet realized it, now s the time to make sure they see that the distances they ve plotted are all perfect squares. It will make coming up with the equation much easier. Once they see this pattern, have them rewrite the distances as squares (16 = 4 2, 64 = 8 2, 144 = 12 2 ). The rule for finding distance then becomes much clearer: to find the distance an object has fallen, we take the time, multiply it by four, and square the result. Students need to be careful with the order of operations when writing down their functions. Some may write down the equation d = 4t 2 instead of d = (4t) 2. After coming up with their equations, be sure that they check their work by comparing values from their table to values that come out of their equation. What do you notice about the total distances traveled after each second: 0 ft, 16 ft, 64 ft, 144 ft? What are the distances if you write them as squares? What s a rule that takes 1 to 4 2, 2 to 8 2, 3 to 12 2, etc.? Based on your rule, how far will an object have traveled after t seconds? Does your equation give you the correct distances for t = 0, 1, 2, 3, 4, and 5? What s the difference between 4t 2 and (4t) 2? Can you write an equation to predict how long it will take an object to fall a distance d? (t = d/4.) If you double the length of time an object falls, what happens to the distance it travels? (It quadruples. If d = 16t 2 and we double t, then 16(2t) 2 = 16(4t 2 ) = 64t 2 = 4d.) How long would an object have fallen after a minute, and do you think this is reasonable? (Based on the equation, the object will have fallen 57,600 feet, or over 10 miles. This seems pretty far even airplanes don t fly that high into the sky, and it would take more than a minute to fall from one! The issue here is that we ve ignored air resistance, which will cause an object to fall more slowly than our model suggests.)
5 3 In Les Misérables, the officer Javert is searching for his former prisoner, Valjean, during an uprising in early 19 th century Paris. After failing to capture Valjean, he stands on the edge of a bridge and jumps into the river below. a. Listen to Javert sing as he falls. Based on the length of his note, how high must the bridge have been, and does this seem reasonable? b. Estimate the height of a bridge in 19 th century Paris. Based on this, how long should Javert s fall have taken? Javert falls for around 8 seconds. This means the bridge must have been 16(8) 2 = 1,024 feet tall. A building story is around 10 feet. A 1,024 ft bridge would be taller than a 100- story skyscraper. This doesn t seem reasonable. Based on Google Images ( bridge in Paris ), a more realistic height is probably around 30-40 feet. 30 feet 40 feet 30 = 16t 2 40 = 16t 2 1.875 = t 2 2.5 = t 2 1.37 t 1.58 t Here, students can immediately use their equation to figure out how far Javert fell during his eight- second fall. However, students may have a hard time interpreting their answer just how far is 1,024 feet, anyway? There are a few ways to help contextualize this height. For example, a building story is typically 10 feet, which means that Javert would have had to jump from the equivalent of a 100- story building. To make the absurdity of their answer even more apparent, remind them that Les Misérables takes place during the early 19 th Century, well before the era of the skyscraper. In fact, for an eight- second fall to be reasonable, Javert would have had to jump of a bridge as tall as the Eiffel Tower! (It s around 1,063 feet tall.) This is clearly ridiculous. There s no way Javert could have jumped from such an extreme height. To close out Act One, students will come up with a more reasonable height (either using examples provided or by conducting their own research). In the end, Javert must only have fallen for a fraction of the time required by his final note. Still, though even if Javert had fallen for a full eight seconds, could he have survived? Or was this plan to take his life bound to fail? Students will turn their attention towards this question in Act Two! If we know how long Javert fell, why do we also know how far he fell? Around how tall is a building story? Based on this, does the bridge s height seem reasonable? Do you think it s reasonable that there was a 1000- foot bridge in Paris during the 19 th Century?! Does anyone know how tall the Eiffel Tower is? Anybody want to take a guess? How tall do you think a typical bridge in 19 th century France might have been? Based on your estimated bridge height, how long would Javert s fall have lasted? If Javert fell for eight seconds, do you think he could ve survived? Why or why not? When calculating the time, why do we take the positive square root and ignore the negative square root? (This model only works for nonnegative values of t, since Javert begins to fall at t = 0. Before that, he isn t falling, and so our model won t describe his motion.) Would air resistance increase or decrease the duration of his fall? (Increase, since the air would be pushing up against him and the downward force of gravity. However, even accounting for air resistance, it s unlikely that his fall would ve lasted anywhere near 8 seconds.)
6 Acte Deux: La Vitesse 4 We already know how far an object falls, and we measure distance in feet. Now let s explore how this distance changes over time. Here, we ll measure speed in feet per second. Use the table below to calculate how fast an object falls over each one- second interval and graph your results. Then, does your graph seem reasonable? Why or why not? Time 0 s 1 s 2 s 3 s 4 s 5 s Distance 0 ft 16 ft 64 ft 144 ft 256 ft 400 ft Speed Over Interval 16 ft over 1 st s 48 ft over 2 nd s 80 ft over 3 rd s 112 ft over 4 th s 144 ft over 5 th s Speed (ft/s) 160 140 120 100 80 60 40 20 0 0 1 2 3 4 5 Time (s) This graph (in blue) doesn t seem reasonable. It suggests that when we drop an object, its speed is 16 ft/s. But after one second its speed suddenly (wham!) becomes 48 ft/s. This would look really strange. Could Javert have survived? To answer this, we need to know how hard he must ve hit the water, i.e. his speed. To start, students calculate speeds over each one- second interval. This may seem like a good idea, but their graph will be a step function, which doesn t make sense. For example, the graph suggests that a falling object s speed immediately triples after one second! Of course, this isn t what happens as objects fall, their speeds increase smoothly. Once students realize that their graph is lacking, they ll be ready to explore a more accurate model. Note: students graph the green line in the next question. For now, they will only consider the blue step function. How far does an object fall in the 1 st second? What is its speed over the interval, & how will the graph look? How far does an object fall in the 2 nd second? What is its speed over that interval? Based on the graph, what happens to an object s speed after one second? Does this make sense? From your own experiences, what do you think the graph of speed vs. time should look like, and why? What does your graph represent? (The average speed of a falling object over one- second intervals.) If this model were right, how would a falling object move? (Its motion would be really jerky, not smooth.)
7 5 The speed of a falling body can be modeled by the equation s(t) = 32t. Draw this on the axes above. a. How is this graph different than the first one, and what does it tell us about how a body really falls? At 1 second, the ball is falling at a rate of 32 ft/s. At 2 seconds, it s falling at a rate of 64 ft/s. What this means is that the ball is falling faster and faster as time goes on, i.e. it s speeding up. The fact that the graph is a straight line means the speed is changing at a constant rate. Each second the speed increases by an additional 32 ft/s. b. Based on the song, how fast was Javert traveling when he hit the water? Could he have survived? In the song, Javert falls for 8 seconds. This implies a speed of 32(8) = 256 feet per second, or: 256 ft 921,600 ft 174.5 mi = 1 s 1 h h 175 mph is really fast, so Javert probably wouldn t have survived this fall. (Then again, the bridge probably wasn t 1024 ft high in the first place!) Speed equation in hand, students are now ready to explore a more accurate model for Javert s fall. If students are comfortable with linear functions, they can graph the equation directly. Otherwise, they can plot points for example, after one second, the speed of a falling object must be 32 ft/s, so the point (1, 32) must be on the graph. Once their graph is complete they ll see that it s a straight line. In other words, speed is a linear function of time; every second that passes, the speed increases by a constant 32 feet per second. Roughly speaking, as time passes, the object falls faster as faster. Note: the line they graph should intersect the midpoint of each step in their step function from the previous question. If you re looking to challenge your students, you can ask them why these intersections appear at the midpoints. Returning to Javert, students may have trouble contextualizing the speed they come up with, since the units are in feet per second. To address this, students might want to convert this to miles per hour. They can also think in terms of models they may have used in Act One. For instance, if 10 feet represent a story, then when Javert hit the water, he would ve been traveling at a rate of around 25 stories per second. Regardless of how they think about it, the answer is the same: there s no way Javert could have survived. Sacrebleu! Based on the equation, what s the object s speed after one second? Two seconds? What does it mean that the graph of speed is a straight line? Do you think 256 feet per second is fast? Why or why not? Do you think Javert could have survived? Why or why not? Why does the speed equation pass through the midpoint of each step? (These are points where the speed is equal to the average speed on one of the one- second intervals. For example, when t = 1/2, the object s speed is 16 ft/s, which is also the average speed of the object during the first second of its fall.) What do you think is the tallest bridge someone could fall from and survive? (Answers will vary, but certainly anything over 100 feet seems to be pushing it. At that height it would take around 2.5 seconds to fall, and you d land with a speed of 80 ft/s, or around 54.5 miles per hour.) What does it mean to accelerate, and what s the acceleration of a falling object? (Acceleration measures how quickly an object s velocity increases. For falling objects, we see that the velocity increases by 32 ft/s, so the acceleration is 32 ft/s per second! That 32 in the speed equation is telling us about the acceleration.) In reality, do you think speed is actually linear? (No, because of air resistance. Eventually a falling object will reach a terminal velocity at that point, it will stop accelerating and will fall at a constant speed.)