Contents Section A Speed The Wave 1 Make Some Waves 2 Rates and Units 2 Speed Records 5 Reaction Time 6 In a Jiffy 8 Speed of Light 8 Distance in Space 10 Summary 12 Check Your Work 13 Additional Practice Answers to Check Your Work Student Activity Sheets Contents v
A Speed The Wave NW WEST SCOREBOARD NORTHWEST RAMP BROWNS BROWNS NORTHEAST RAMP EAST SCOREBOARD NE 695 ft The Cleveland Browns Stadium can seat over 73,200 people. Cleveland fans often create a wave people stand up lifting their arms and then quickly sit down. When one group sits down, an adjacent group takes over. The wave moves around the entire stadium. SW SE SO UTHWEST RAMP SOUTHEAST RAMP 933 ft 1. a. Estimate how many feet the wave travels one time around the stadium. Describe how you made your estimate. The stadium dimensions are 933 feet by 695 feet. b. How much time will it take the wave to go once around the stadium? Describe how you made your estimate. c. Use your estimates to find the average distance the wave travels in one second. You may want to use a ratio table for your calculations. Distance (in ft) Time (in sec) If a stadium wave travels 60 feet (ft) in 2 seconds (sec), it travels with an average speed of 30 feet per second, or 30 ft/sec. Section A: Speed 1
Make Some Waves For this activity, you will need: stop watch tape measure Discuss with a classmate how to find the average speed of a wave. Share your idea with the class. Decide on a plan to find the average speed of a wave in your classroom. Record the time (in seconds) and distance (in feet) for a wave to travel through your classroom. 2. a. Use the data from the activity to calculate the average speed of the wave in your class. b. Compare your class s wave with the Cleveland Stadium wave. Describe your findings. Rates and Units A rate is the ratio of two different measuring units. For example, you can express the rate of speed in miles per hour (mi/h), or in feet per second (ft/s). Using metric units, speed is usually expressed in kilometers per hour (km/h), or meters per second (m/s). 2 Revisiting Numbers
Speed A 3. What other rates do you know? Copy this table and complete it. Heart Rate Data Transfer Population Density Units Heartbeats/minute (bmp) Kilobytes/second (kb/s) Example My heart beats at a rate of 65 beats per minute. The speed of the download stream is 1,024 kilobytes per second. You may remember from the unit Ratio and Rates how you used ratio tables to express rates as a single number. Stuart and Lexa are researchers who analyze average speeds of large crowds. In a soccer stadium, they timed one wave taking 22 sec to travel 440 seats. Each seat had a total width of 2 ft. 4. a. Calculate the average speed of the wave in seats per second. b. Compare the average speed from this research to the speeds you found in problem 2b. How do they compare? Distance (in ft) 30 Time (in sec) Rita found 30 ft/s as the average speed of the wave in her class. She wants to know how fast this is in miles per hour. Here is how she started to solve the problem. 1 60 60 60 60 Rita: First, I multiplied by 60 to get the number of feet per 60 seconds, or per one minute. 5. a. Explain Rita s second step. b. Copy Rita s ratio table and calculate the missing numbers. Did you know? There are about 5,280 ft in 1 mi? c. Use this information to find the average speed of the wave in miles per hour (mi/h). Section A: Speed 3
A Speed You can use a similar technique to convert meters per second (m/s) into kilometers per hour (km/h). Kenny ran 60 m in 15 sec. 6. a. Copy this ratio table and use the arrows to show the steps you have to make to find Kenny s distance in one hour. Distance (in m) 60 Time (in sec) 15 b. What is Kenny s average speed in meters per hour? What is Kenny s average speed in km/h? c. Will Kenny really be able to run that distance in one hour? Explain. d. Henri ran 50 m in 12 sec. Is Henri s average speed higher or lower than Kenny s? Show your work. Maddie: I did the 5K run in 25 minutes. I wonder how fast I ran in kilometers per hour. 7. Calculate Maddie s average speed for the 5K run in km/h. 4 Revisiting Numbers To compare speeds, you may have to change the units. Changing kilometers per hour into miles per hour is easy if you have a speedometer like this one. 8. Explain why this speedometer is actually a double number line.
Speed A In most European countries, the speed limit on highways is 120 km/h, on county roads 80 km/h, and in towns 50 km/h. 9. How do these speed limits compare to those in the United States? Speed Records Here are some speed records for a variety of creatures. a A cheetah, or hunting leopard, is timed at 70 mi/h. b c The ostrich is a special sort of sprinter. It is a bird, but it does not fly. It can run 20 mi in 40 min. American quarter horses are the fastest horses in the world. They can cover a quarter-mile in less than 21 sec. d In the water, the speed record is held by the sailfish, which in a calm sea, can reach a speed of 100 m per 3.3 sec. e The Indian spine-tailed swift bird has repeatedly been clocked in level flight over a carefully measured two-mile track at 32.8 sec. f On September 14, 2002, Tim Montgomery of the United States set a world record in the 100 m, clocking 9.78 sec at the IAAF Grand Prix Final. 10. Order these speed records on the odometer on Student Activity Sheet 1. Show your work. Note that one of the speeds will not fit on the odometer. 11. Reflect What is the difference between an average speed and a maximum speed? Can these two be the same? Explain your answers. Try to include examples in your explanation. Section A: Speed 5
A Speed Reaction Time Digital timepieces improve accuracy. 12. a. What time is displayed on this timepiece? b. What does the 04 mean? c. How much more time needs to pass for the timepiece to read 1 hour? These hand-held timepieces are accurate; however, they depend on the reaction time of the humans using the timepiece. From the moment you see or hear a signal, it takes time for the signal to go to your brain, time for your brain to react, and finally time for your nerves and the muscles in your fingers to react. You will determine your personal reaction time using data collected from the following activity. For this activity, you need a centimeter ruler. You will work with a classmate. One student holds the ruler vertically, the other (the catcher) holds his or her thumb and pointer finger about 3 cm apart level with the 0 cm mark of the ruler. Without any signal, the person holding the ruler lets go. The catcher tries to react as quickly as possible and catches the ruler. Record the number of centimeters caught. The number of centimeters caught is a distance that will be used to calculate the catcher s reaction time. Do this experiment five times per person and record the distances in centimeters. 6 Revisiting Numbers
Speed A When you know the distance in centimeters, you can use a formula to calculate the reaction time in seconds. t sec 2 d 980 d is the distance you caught, in centimeters. Other formulas to find the reaction time are: t sec 0.002 d t sec d or t sec 0.045 d 490 13. a. Which of these three formulas gives the same result as the first formula? b. Use one of the correct formulas to calculate the five reaction times you recorded in the previous activity. Calculate your average reaction time. Australia s Cathy Freeman, a world-class athlete, had a reaction time of 0.223 sec in the women s 400 m final at the 1995 World Championships. Her reaction time was measured with an electronic device inside the starting block. This device recorded the interval between the starting shot and the first athlete leaving the blocks. 14. a. On Student Activity Sheet 1 fill in the missing times indicated by the blanks under the number line. 0 0.100 seconds b. How much longer does this number line need to be in order to locate 1 sec? c. Place Cathy Freeman s reaction time on the number line. Use an arrow to point to the location. d. Tests have confirmed that nobody can react in less than 0.110 of a second. Place this minimum reaction time and your reaction time from problem 13b on the number line. If necessary, extend the line. e. Explain why a false start is declared if the interval between the starting shot and the athlete leaving the block is less than 0.110 of a second. Section A: Speed 7
A Speed In a Jiffy In 2002, Tim Montgomery of the United States set a world record in the 100 m, clocking 9.78 sec. His time was one hundredth of a second faster than the previous record of 9.79 sec set by Maurice Green in 1999. Suppose both athletes ran in the same 100-m race, at their own world record pace. Would the race be too close to call? To investigate, you will need to zoom in on the finish, the moment when Tim reached the finish line. 15 a. How much time (in sec) elapses between Tim s win and Maurice s finish? b. Make a guess. What is Maurice s distance from the finish line when Tim wins the race? Tim Montgomery Instead of guessing, you can calculate this distance in centimeters. This ratio table will help you with your calculations. Distance (in cm) 10,000 Time (in sec) 9.79 1 0.1 0.01 Maurice Green c. Explain the numbers 10,000 and 9.79 that are in the ratio table. d. Calculate the distance in the last column. What do you know now? e. If you were at the race, would you be able to tell who finished first? Explain your answer. Speed of Light The speed of light is 299,792,458 m/s. 16. a. How many kilometers does light travel per second? b. What is the speed of light in kilometers per hour? 8 Revisiting Numbers
Speed A The speed of light is often rounded to 300,000,000 m/s. This table shows the many different ways to write this number. Speed of Light 300,000,000 30,000,000 3,000,000 300,000 3 1 10 100 1,000 300,000,000 30,000,000 3,000,000 300,000 10 1 10 2 17. Copy the last row of the table in your notebook and fill in the missing numbers. You may remember writing very large numbers in scientific notation in the unit Facts and Factors. A number written in scientific notation is the product of a number between 1 and 10 and a power of 10. The first number is called the mantissa. 1,680,900, written in scientific notation is 1.68 10 6. Notice that the mantissa is rounded to two decimal places. A calculator may display this number as: 06 1.68 or as 1.68 E06 18. a. Write 43,986,000,000,000 in scientific notation. Round the mantissa to one decimal place. b. Write the speed of light in km/h from 16b in scientific notation. Round the mantissa to two decimal places. Section A: Speed 9
A Speed Distance in Space The average distance from Earth to the sun is about 1.5 10 8 kilometers. Neptune is 30 times farther away from the sun. 19. Write the distance from Neptune to the sun in scientific notation. Neptune Venus The distance from the sun to Venus is 0.72 times the distance from the sun to Earth. 20. Write the distance from Venus to the sun in scientific notation. Samantha and Jennifer are checking over their homework answers. They disagree on this problem: 10 2 10 3 Samantha claims the answer is 10 5, while Jennifer thinks that it has to be 10 6. 21. a. Who is right, Samantha or Jennifer? How would you explain it to the person who did the problem incorrectly? b. Explain why 10 4 10 4 10 4 10 4 10 4 10 4 10 4 10 4 10 4 10 4 10 5. Samantha and Jennifer disagree on this problem: 10 6 10 3 Samantha claims the answer is 10 2, while Jennifer thinks it has to be 10 3. c. Who is right, Samantha or Jennifer? How would you explain it to the person who did the problem incorrectly? Calculate the following problems without the use of a calculator. Write your answers in scientific notation. 22. a. Multiply 8 10 3 by 4 10 2. b. Divide 8 10 3 by 4 10 2. c. Add 8 10 3 and 4 10 2. d. Subtract 4 10 2 from 8 10 3. 23. If m and n are natural numbers, write this product, 10 m 10 n, as a power of ten. 10 Revisiting Numbers
Speed A Math History Different Number Systems Our number system uses ten digits, which is the same as the number of fingers we have. So we use a base-10 number system. Why do we have 24 hours in a day? Why are there 60 minutes in an hour and 60 seconds in a minute? Sippar Diyala Eshnunna AKKAD Kish BABYLONIA Babylon Borsippa Nippur Isin Adab Lagash Uruk Larsa Ur Eridu ARABIAN DESERT Euphrates 0 25 50 75 mi 50 km 0 100 SUMER Tigris HAWRS Susa Karkheh Dez Karun N ZAGROS MOUNTAINS W E S Persian Gulf To explain this, we have to go back in history 4,000 years, to the land between the Tigris and Euphrates Rivers, where the Sumerians lived. They also used their fingers to count things, but they did it differently. The Sumerians counted finger joints instead of fingers, and they used their thumb to do the counting. Their number system was a base-12 number system, which is why they divided a day into twelve parts. A thousand years later, the Babylonians, who lived in the same area as the Sumerians, used a base-60 number 1 4 system. It is not sure why they chose 23 56 7 8 60. One reason might be because 9 11 10 12 base-60 makes divisional operations Day Night easy since 60 is divisible by 2, 3, 4, 5, 6, 10, and 12. They divided a day into two times 12 hours because twelve fits nicely in their system. (12 3 4, and 60 3 4 5). Hours were further divided into 60 minutes, and the minutes were divided into 60 seconds. This system for time is still used today. For other divisions, they used their base-60 system. In our decimal system, the first decimal is tenths. In the base-60 system, the first fractional place is sixtieths. The first fractional place is called a minute, the second place is called a second. So that is why an hour has 60 minutes, and a minute has 60 seconds. The Babylonians not only measured time, they also studied astronomy and measured angles. They divided the heavens into twelve sectors, the time it takes the earth to complete one revolution around the sun. Each sector was 30, so a complete year took 360 (12 30). Each degree was further divided into 60 minutes, and each minute was divided into 60 seconds. 1 2 3 4 7 8 9 5 6 10 Section A: Speed 11
A Speed Rates A rate is the ratio of two different measuring units, written as a single number. Examples of rates are: Resting heart rate, such as 85 bpm Speed of a car on a highway, such as 55 mi/h or 88 km/h Data transfer rate, such as 1,024 kb/s A ratio table is a helpful tool for finding a rate. Suppose you traveled 150 km in 2.5 hours. A ratio table can help you determine your average speed in km/h. Units Distance (in km) 150 Time (in hr) 2 5 300 60 2.5 5 1 2 5 Traveling 150 km in 2.5 hours, what is the average rate of speed? Answer: 60 km/h Sometimes you have to convert the measuring units of a rate. How fast is 5 m/s in kilometers per hour? 60 60 Distance (in m) 5 300 18,000 Time (in sec) 1 60 3,600 How fast is 5 m/s in kilometers per hour? Answer: 18 km/h Now convert: 60 60 18,000 m 18 km, and 3,600 sec 1 hr. 12 Revisiting Numbers
How fast is 60 km/h in meters per minute? First convert: 60 km 60,000 m, and 1 hr 60 min. 60 Distance (in m) Time (in min) 60,000 1,000 60 1 How fast is 60 km/h in meters per minute? Answer: 1,000 m/min 60 Scientific Notation Any positive number written in scientific notation is a product of two factors: a number between 1 and 10 (the mantissa) and a power of 10. 28,600,000 written in scientific notation is 2.86 10 7. A calculator may display this number as: 07 2.86 or as 2.86 E07 You may round the mantissa to one decimal place: 2.9 10 7. 1. Helen hiked 18 km in 4 1 hours. What was her average rate of speed 2 in km/h? On January 31, 2005, the International Programs Center of the U.S. Census Bureau estimated the world population to be 6,415,905,543 people. 2. a. What is the meaning of the six in this number? b. Write this number in scientific notation rounding the mantissa to one decimal place. Section A: Speed 13
A Speed The earth makes one complete revolution on its axis in 24 hours. The rotational speed begins at zero at either geographical pole and increases as you head toward the equator. 3. Calculate the earth s rotational speed at the equator in miles per hour. The circumference of the earth at the equator is about 2.5 10 4 miles. In one year, the earth travels about 5.8 10 8 miles as it orbits around the sun. 4. Calculate the earth s average orbital speed around the sun in miles per hour. 5. Mercury s average orbital speed around the sun is about 48 km/sec. Is Mercury s orbital speed faster or slower than the orbital speed of the earth? Explain your answer. 6. Calculate the following problems without the use of a calculator. Write your final answer using scientific notation. a. Multiply 3 10 4 by 2 10 2. b. Divide 2 10 10 by 10 6. c. Add 2 10 3 and 4.5 10 2. d. Subtract 6 10 2 from 2 10 3. 14 Revisiting Numbers
When timekeepers used hand-held stopwatches, it was very difficult to rank evenly matched competitors. At the 1960 Olympic games in Rome, Australia s John Devitt and America s Lance Larson finished neck-and-neck in the final of the 100-m freestyle swimming events. All three timekeepers for Devitt s lane clocked him at 55.2 sec. Larson was clocked at 55.0, 55.1, and 55.1 sec. The judges placed Devitt as the winner. The official time for both swimmers was recorded as 55.2 sec. 7. a. Is this fair? Explain your reasoning using your knowledge about reaction time. b. Suppose Larson swam 100 m in 55.2 sec and Devitt finished 0.1 sec before Larson. What is Larson s distance (in cm) from the wall when Devitt finished the race? Would this have been visible? Write 236.7 10 4 as a number. Why is 236.7 10 4 not written in scientific notation? Section A: Speed 15
Additional Practice Section A Speed Speed of Sound Lena saw a flash of lightening and heard the thunder three seconds later. She knows a rule to estimate the storm s distance one mile for every five seconds. 1. What estimate will Lena get for the storm s distance? To understand why this works, you have to know that there is a big difference between the speed of light and the speed of sound. Sound travels through air with a speed of about 760 mi/h. Light travels almost 10 6 times faster than sound. 2. a. What is the speed of light in mi/h? Write your answer as a numeral. b. Use the speed of sound to calculate the distance in miles that sound can travel in three seconds. c. Is Lena s rule reasonable? Explain. Peter is more familiar with metric units. He uses this rule one kilometer for every three seconds. 3. Which rule is more accurate, Lena s rule or Peter s rule? Explain. 4. Calculate the speed of sound in feet per second. When you stand in front of a rock wall and you clap your hands, you may hear an echo. The sound waves move from your hand to the wall, then reflect off the wall and travel the same distance back to you. Kim heard an echo a half a second after she clapped her hands. 5. How far was Kim standing from the wall? You may want to use the speed of sound you calculated in problem 4. 54 Revisiting Numbers
You can hear an echo when the time the sound travels from you to the wall and back again is more than 0.05 seconds. 6. What is the minimum distance you need to stand from the wall to hear an echo? Additional Practice 55
Section A Speed 1. Helen s average speed was 4 km/h. You may have used a ratio table to find your answer. 2 9 Distance (in km) 18 36 4 Time (in hr) 1 4 2 9 1 2. a. The six means 6,000,000,000, which is six billion. b. 6.4 10 9 3. The speed of the earth at the equator is about 1,042 mi/h. Note that it doesn t make sense to write this answer with decimals. You may have used the following strategy. The earth completes one revolution in one day, or 24 hours. At the equator, the distance around the earth is about 2.5 10 4 miles, which is 25,000 miles. To find the speed in mi/h you can use a ratio table. 2 9 Distance (in mi) 25,000 24 1,042 Time (in hr) 24 1 4. The average speed of the earth around the sun is 6.6 10 4 mi/h, or 66,000 mi/h. You may have used the following strategy. The earth travels 5.8 10 8 miles in 365 days. One day 24 hours, so 365 days 8,760 hours. 1,000 8.76 24 Distance (in mi) 5.8 10 8 5.8 10 5 0.66 10 5 or 66,000 Time (in hr) 8,760 8.76 1 1,000 8.76 60 Revisiting Numbers
Answers to Check Your Work 5. Mercury s average orbital speed is faster than Earth s average orbital speed. Your strategy may be different from this strategy: Mercury travels 48 km/s. One hour 3,600 seconds, so that is my target. 60 60 Distance (in km) 48 2,880 172,800 Time (in sec) 1 60 3,600 60 60 Mercury s average speed is about 17 10 4 km/h. Earth s average speed is about 6.6 10 4 mi/h. Comparing 6.6 miles and 17 km, 17 km is more than 6.6 miles. Mercury travels faster around the sun. 6. Here are the answers written in scientific notation along with one sample solution strategy. a. 6 10 6 (3 10 4 ) (2 10 2 ) 3 10 4 2 10 2 (3 2) (10 4 10 2 ) 6 10 6 b. 2 10 4 (2 10 10 ) 10 6 2 (10 10 10 6 ) 2 10 4 c. 2.45 10 3 (2 10 3 ) (4.5 10 2 ) 2,000 450 2,450 Writing 2,450 in scientific notation is 2.45 10 3. d. 1.4 10 3 (2 10 3 ) (6 10 2 ) 2,000 600 1,400 Writing 1,400 in scientific notation is 1.4 10 3. Answers to Check Your Work 61
Answers to Check Your Work 7. a. Your opinion might vary from these. I do not think it is fair because it is unlikely that all of Larson s timekeepers had a faster reaction time than all of Devitt s timekeepers. It could be fair since the race was timed manually, and the timekeepers would have had different reaction times. I think it is fair because the swimmers were too close to distinguish actual finishing times. More than likely, the judges decided that John Devitt finished first, so they had to award him the race in spite of the recorded times. b. Larson was about 18 cm behind, which is more than the length of a hand. This distance is visible, but it occurs in 0.1 second. Here is one strategy. When Devitt finished, Larson had to swim 0.1 second. Larson swam 100 meters in 55.2 seconds. Using 100 m 10,000 cm to set up this ratio table: 55.2 10 Distance (in cm) 10,000 181.2 18 Time (in sec) 55.2 1 0.1 55.2 10 Larson was about 18 cm behind, which is more than the length of a hand. In less than 0.1 seconds, I can understand why the manual timers had difficulty reacting to this visible distance. 62 Revisiting Numbers
Name Student Activity Sheet 1 Use with Revisiting Numbers, pages 5 and 7. 10. Encyclopædia Britannica, Inc. This page may be reproduced for classroom use. 14. 0 0.100 seconds Student Activity Sheet 1 Revisiting Numbers 79