Distance, Rate and Time



Reality Math

Joseph Sulock, University of North Carolina at Asheville

Dot Sulock, University of North Carolina at Asheville

Unit 95. Distance, Rate, and Time in Sports

If you drive 60 mph for 3 hours, you will clearly go 180 miles.

Distance = Rate [pic] Time, at least if the units match.

1. How far will you go if you are driving 60 mph for (a) 2.25 hours (b) 2 hours and 15 minutes (c) [pic] hours (d) 2 hours and 25 minutes (e) 2.15 hours?

Your first three answers should be the same, (d) is farther, and (e) is less far! If you have driven 150 miles in 2 hours, how fast did you go? Well, miles per hour is another example of a rate such as people per square mile. Per means divide. So miles per hour is [pic] and is sometimes even written miles/hour. (150 miles)/(2 hours) = 75 mph.

2. What speed were you averaging if you have driven 150 miles in (a) 2.5 hours

(b) [pic] hours (c) 2 hours 30 minutes (d) 2 hours and 25 minutes (e) 2 hours and 15 minutes?

The first three answers are the same again, (d) was slower, (e) was faster. How long will it take you to go 200 miles if you are traveling 50 mph? Looking at the units is interesting: [pic]

It seems sensible to divide 200 miles by 50 mph and get 4 hours. Anytime you divide miles by miles per hour you get hours, as it shows above. What if it doesn’t come out so evenly. How long would it take you to drive 200 miles going 70 mph?

(200 miles)/(70 mph) = 2.88 hours, not an ordinary way of expressing time. How many hours and minutes is this? Well, clearly 2 hours and 0.88 hours = 0.88(60 minutes) = 53 minutes. 2 hours and 53 minutes or 2:53 as it is sometimes written.

3. How long would it take you to go 200 miles if you were driving (a) 25 mph

(b) 75 mph (c) 60 mph (d) 65 mph. Give your answers in hours and minutes.

1. The Indianapolis 500

The Indianapolis 500-Mile Race, more commonly known as the Indi 500, may well be the most famous motor-racing event. The event is nationally televised and attendance is around 400,000. To put this in perspective, the number of spectators at the Super Bowl is unlikely to exceed 100,000.

As one might expect, average speeds have increased over time. The 1930 winner, Billy Arnold, averaged 100.45 miles per hour (the first time a driver had an average speed in excess of 100 mph). It took him

500 miles/100.45 mph

= 4.98 hours

= 4 hours and 0.98(60 minutes)

= 4 hours and 58.8 minutes

= 4 hours and 58 minutes and 0.8(60 seconds)

= 4 hours and 58 minutes and 48 seconds

4:58:48

How to round is an issue here. Since the 500 miles has three significant digits (we hope it really is 500 miles and not 545 miles rounded down to 500 miles or 455 miles rounded up to 500 miles) the original 4.98 hours was appropriately rounded off to 3 significant digits.) See Appendix B for more information about significant digits.

In 2009, the winner was Helio Castroneves with an average speed of 150.32 mph.

4. How long did it take Castroneves to finish the race? Express your answer in the hours:minutes:seconds format.

5. Castroneves’ average speed was what percent higher than Arnold’s average speed?

How far behind Castroneves would you be if you were averaging 145 mph? This is an interesting question with two different types of answer. How far behind in time? How far behind in distance?

6. (a) Determine how long it would take you to go the 500 miles, expressing your answer in hrs:min:sec. (b) Compare to Castroneves’ time. How many minutes and seconds were you behind?

What distance were you behind? If you are driving 145 mph, how far would you go in 7 minutes and 12 seconds? It will be tricky to get the time in hours, but the time must be in hours if the speed is miles per hour. 7 minutes + (12/60) minutes = 7.2 minutes = (7.2/60) hours = 0.12 hours. In 0.12 hours you would go 0.12 hours x 145 mph = 17.4 miles. Since the track is 2.500 miles long, 17.4 miles would be almost 7 laps behind! Slow!!!

Danica Patrick, , finished third in 2009, the best finish ever for a female driver. Patrick finished fourth as a rookie driver in 2005. Check out her website!

7. Assuming Danica’s average speed was 149 miles per hour, how far behind Castrovenes was she? (a) in time? (b) in miles? (c) in laps? 1 mile = 1760 yards

Though this is an American race, its popularity is worldwide. Let’s imagine you have an international friend, Tiago, who has an interest in auto racing and you want him to better understand the race and facts surrounding the race. An important communication issue is that nearly all countries outside of the United States use the metric system. And you are not likely to enlighten Tiago discussing the race in terms of miles and miles per hour. He is much more likely to relate to kilometers and kilometers per mile.

One meter equals 1.0936 yards, a messy relationship for sure. Let’s simplify our lives a bit and assume that 1 meter [pic] 1.1 yards. In other words, we will assume that a meter is 10 percent longer than a yard.

8. (a) Which is bigger, a meter or a yard? (b) Convert 1 mile = 1760 yards to meters (c) Will there be more meters than yards in a mile? Why or why not?

Your answer to (b) should be quite close to the actual distance of 1609.364 meters.

1 kilometer = 1,000 meters

9. (a) One mile is about how many kilometers? (b) How many kilometers is the Indi 500?

The speed record (as of 2009) was 186 miles per hour set by Arie Luyendyk in 1990.

10. What is the record in kilometers per hour?

11. (a) How many kilometers was Danica Patrick behind the winner Castroneves? (b) How many meters?

2. Reaction Times

The time it takes an athlete to react and decide what to do is crucial in many sports. For example, if you are Albert Pujols how much time do you have to decide whether to swing at a pitch thrown at 90 mph? If you are the opponent of tennis star Venus Williams, how much time do you have to return a serve that is likely to be moving at 105 mph?

Let’s begin with baseball and address the earlier question: How much time does Pujols have? Remember that if you are driving 200 miles at 50 mph it will take 200 miles/50 mph = 4 hours. Dividing both sides of D = R x T by R gives D/R = T

We have been given “R” ,what about “D”? The pitcher’s mound is 60 feet and six inches from home plate though the actual distance the pitch travels will not be for two reasons. First, the pitcher will release the ball (when it leaves the hand) closer to home plate. Second, the batter can move around a bit in the “batter’s box.” A reasonable assumption is that with these two adjustments, the “distance” is 55 feet.

Can you see a more important issue that we need to address? “Rate” is in miles per hour, while distance is in “feet.” We need to convert either the feet to miles or the miles to feet. Given the relatively small distance we are working with (unlike the Indi 500), it makes sense to use feet instead of miles. After all, a distance of 55 feet equals only .01042 miles, a difficult number to relate to! 1 mile = 5280 feet

12. A 90 mph pitch is moving at a rate of how many (a) feet per hour? (b) feet per second?

Your answer to (a) should be very big and the answer to (b) is obviously less than the answer to (a) and turns out to be a rather sensible number.

So how much time does Pujols have to decide what to do? Since T = D/R, he has 55 feet/(132 feet per second) or a mere 0.4167 seconds to react!

An 80 mile-per-hour fastball is considered slow by major-league standards. Batters are likely to get many hits off a pitcher with this speed of a fastball!

13. (a) Assuming a pitch of 80 mph, how much time does the batter have to react? (b) How much extra time compared to a 90 mph pitch does the batter have to react?

14. Venus Williams serves a tennis ball at 105 mph. If her opponent is 80 feet away, how much time does the opponent have to decide what to do?

3. Track

Nearly all track events (including American) are measured in meters. An exception is the mile run, one of the oldest events and arguably the most famous.

The table below shows the world record for men and women at various distances as of 2009.

Men’s

|Event |Time |Athlete (Country) |Site |Date |

|100 meters |9.69 |Usain Bolt (Jamaica) |Beijing, China |August 16, 2008 |

|200 |19.3 |Usain Bolt (Jamaica) |Beijing, China |August 20, 2008 |

|400 |43.18 |Michael Johnson (US) |Seville, Spain |August 26, 1999 |

|800 |01:41.1 |Wilson Kipketer (Denmark) |Cologne, Germany |August 24, 1997 |

|1000 |02:12.0 |Noah Ngeny (Kenya) |Rieti, Italy |September 5, 1999 |

|1500 |03:26.0 |Hicham El Guerrouj (Morocco) |Rome, Italy |July 14, 1998 |

|Mile |03:43.1 |Hicham El Guerrouj (Morocco) |Rome, Italy |July 7, 1999 |

|3000 |07:20.7 |Daniel Komen (Kenya) |Rieti, Italy |September 1, 1996 |

|5000 |12:37.4 |Kenenisa Bekele (Ethiopia) |Hengelo, Netherlands |May 31, 2004 |

|10,000 |26:17.5 |Kenenisa Bekele (Ethiopia) |Brussels, Belgium |August 26, 2005 |

Women’s

|Event |Time |Athlete (Country) |Site |Date |

|100 meters |10.49 |Florence Griffith Joyner (US) |Indianapolis, Indiana |July 16, 1988 |

|200 |21.34 |Florence Griffith Joyner (US) |Seoul, South Korea |September 29, 1988 |

|400 |47.6 |Marita Koch (East Germany) |Canberra, Australia |October 6, 1985 |

|800 |01:53.3 |Jarmila Kratochvílová (Czechoslovakia) |Munich, Germany |July 26, 1983 |

|1000 |02:29.0 |Svetlana Masterkova (Russia) |Brussels, Belgium |August 23, 1996 |

|1500 |03:50.5 |Yunxia Qu (China) |Beijing, China |September 11, 1993 |

|Mile |04:12.6 |Svetlana Masterkova (Russia) |Zürich, Switzerland |August 14, 1996 |

|3000 |08:06.1 |Junxia Wang (China) |Beijing, China |September 13, 1993 |

|5000 |14:11.2 |Tirunesh Dibaba (Ethiopia) |Oslo, Norway |June 6, 2008 |

|10,000 |29:31.8 |Junxia Wang (China) |Beijing, China |September 8, 1993 |

15. Usain Bolt ran the 100 meter dash in 9.69 seconds. If he could keep running at this speed for a mile, what would his time for a mile be?

16. The winner of the men’s 100 meter dash is often called “The World’s Fastest Human.” At what speed (in miles per hour) did Usain Bolt run this race?

17. In 1973 Secretariat set the Kentucky Derby record by running the 2 km course in 1 minute 59.2 seconds. Running 2 km in 2 minutes would be how many mph?

18. Florence Griffin holds the women’s 100 meter record. What was her speed (in miles per hour) for this race?

4. Luge at the 2010 Olympics in Vancouver, British Columbia

19. Felix Loch (Germany) had a time of 48.171 seconds to win the gold medal. He finished 0.679 seconds ahead of David Moeller (Germany), who won the silver medal. The course was 4507 feet long. (a) What speed in mph did Loch average?

(b) How many feet behind Loch would Moeller have been if they were racing together?

5. Safe Driving

Let’s finish this module with an example that could save your life! Experts recommend that drivers allow a two-second reaction time from the car in front. In other words, a safe driver will allow a two-second “following distance.”

20. If you follow this guideline and are driving at 60 miles per hour, what is the minimum distance (in feet) that you should be from the car in front of you?

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