Unit 1 Uniform Velocity & Position-Time Graphs

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1 Name: Unit 1 Uniform Velocity & Position-Time Graphs Hr: Grading: Show all work, keeping it neat and organized. Show equations used and include units in all work. Vocabulary Distance: how far something travels Displacement: how far something travels in a given direction Notice that these two terms are very similar. Distance is an example of a scalar quantity. That is, it has magnitude but not direction. Displacement is a vector quantity because it has both magnitude and direction. Helpful Hints Displacements smaller than a meter may be expressed in units of centimeters (cm) or even millimeters (mm). Displacements much larger than a meter may be expressed in kilometers (km). You should become familiar with these common prefixes and their meanings. The SI (Système International) unit for distance and displacement is the meter (m). Vocabulary Speed: how fast something is moving distance traveled speed= time or v = d t Velocity: how fast something is moving in a given direction velocity= displacement elapsed time or v = d t = d f-d i t f -t i where d f and t f are the final position and time respectively, and d i and t i are the initial position and time. The symbol (delta) means change in so Δd is the change in position, or the displacement, while t is the elapsed time. A d by itself (no in front of it) refers to the total distance traveled, regardless of direction. Because speed is given without regard to direction, it is a scalar quantity. Velocity is given with a direction, so it is a vector. Helpful Hints The SI unit for both speed and velocity is the meter per second (m/s). When traveling in any moving vehicle, you rarely maintain the same velocity throughout an entire trip. If you did, you would travel in a straight line at constant speed. Instead, speed and direction usually vary while traveling. If you begin and end at the same location but you travel for a great distance in getting there (for example, when you travel in a circle), you have a measurable average speed. However, since your total displacement is zero, your average velocity is also zero. Solved Examples Example 1: Benjamin watches a thunderstorm from his window. He sees the flash of a lightning bolt and begins counting the seconds until he hears the clap of thunder 10.0 s later. Assume the speed of sound in air is m/s. How far away was the lightning bolt a) in m? b) in km?

2 Note: the speed of light, 3.00 x 10 8 m/s, is considerably faster than the speed of sound. That is why you see the lightning flash so much earlier than you hear the clap of thunder. In actuality, the lightning and thunder clap occur simultaneously. a) Given: v = m/s t = 10.0 s Unknown: d =? Original equation: v= d t d = vt = (340.0 m/s)(10.0 s) = 3400 m b) For numbers this large you may wish to express the final answer in km rather than in m. Because kilo means 1000, then km means m m ( 1 km ) = 3.4 km 1000 m The lightning bolt is 3.4 km away, or just over 2 miles. Example 2: On May 29, 1988 Rick Mears won the Indianapolis 500 in 3.45 h. What was his average speed during the 500.-mile race? Given: d = 500. miles t = 3.45 h Unknown: v =? Original equation: v = d t v = d t = 500. miles 3.45 h = 145 mph Example 3: The slowest animal ever discovered was a crab found in the Red Sea. It traveled with an average speed of 5.70 km/y. How long would it take this crab to travel 100. km? Given: d = 100. miles v = 5.70 km/y Unknown: t =? Original equation: v = d t t = d v 100. km = = 17.5 y A long time! 5.70 km/y Example 4: Tiffany, who is opening in a new Broadway show, has some limo trouble in the city. With only 8.0 minutes until curtain time, she hails a cab and they speed off to the theater down a m-long one-way street at 25 m/s. At the end of the street the cab driver waits at a traffic light for 1.5 minutes and then turns north onto a m-long traffic filled avenue on which he is able to travel at a speed of only 10.0 m/s. Finally, this brings them to the theater. a) Does Tiffany arrive before the theater lights dim? b) Draw a distance-time graph of the situation. a) Segment 1: (one-way street) Given: d = m v = 25 m/s

3 Unknown: t =? Original equation: v = d t t = d v = m 25 m/s = 40. s Segment 2: (traffic light) Given: t = 1.5 min 1.5 min( 60 s 1 min ) = 90. s Segment 3: (traffic-filled avenue) Given: d = m v = 10.0 m/s Unknown: t =? Original equation: v = d t t = d v = m 10.0 m/s = 170. s Total time = 40. s s s = 300. s 300. s( 1 min ) = 5.00 min 60 s Yes. She not only makes it to the show on time, but she even has 3.00 minutes to put on her costume and make-up. b) The motion of the cab can be described by the following graph: In segment 1, the distance of m was covered in a fairly short amount of time, which means that the cab was traveling quickly. This high speed can be seen as a steep slope on the graph. In segment 2, the cab was at rest. Notice that even though the cab did not move, time continued on, resulting in a horizontal line on the graph. In segment 3, the distance of m was covered in a much longer amount of time so the cab was traveling slowly. This low speed is indicated by a slope that is not very steep. Remember that all graphs should have titles and axes should be labeled with correct units.

4 Exercises (Show ALL work! Please include units and circle answers!) 1) Hans stands at the rim of the Grand Canyon and yodels down to the bottom. He hears his yodel echo back from the canyon floor 5.20 s later. Assume that the speed of sound in air is m/s. How deep is the canyon at this location? 2) The horse racing record for a 1.50 mile track is shared by two horses: Fiddle Isle, who ran the race in 143 s on March 21, 1970, and John Henry, who ran the same distance in an equal time on March 16, What were the horses average speeds in a) miles/s? b) miles/h?

5 3) For a long time it was a dream of many runners to break the 4-minute mile. Now quite a few runners have achieved what once seemed an impossible goal. On July 2, 1988, Steve Cram of Great Britain ran a mile in 3.81 minutes. During this amazing run, what was Steve Cram s average speed in a) miles/hr? b) meters/s? 4) Los Angeles sits west of the San Andreas fault and is moving at an average velocity of 6 cm/year northwest toward San Francisco, which sits east of the fault. If the separation between these two cities is 590 km, how many years into the future might these two cities join at the same latitude and be neighbors?

6 5) A train travels 100 km/hr for 0.52 hr, then 50 km/hr for the next 0.24 hr, and finally 125 km/hr for the last 0.65 hr. What is the average speed of the train for this trip? 6) It is now 10:29 a.m., but when the bell rings at 10:30 a.m. Suzette will be late for French class for the third time this week. She must get from one side of the school to the other by hurrying down three different hallways. She runs down the first hallway, a distance of 35.0 m, at a speed of 3.50 m/s. The second hallway is filled with students, and she covers its 48.0-m length at an average speed of 1.20 m/s. The final hallway is empty, and Suzette sprints its 60.0-m length at a speed of 5.00 m/s. a) Does Suzette make it to class on time or does she get detention for being late again? b) (On the next page) Draw a position vs. time graph of the situation. Include a title and axes labels with units.

7 Draw your graph in the space below: 7) Calculate the slope of the following graphs. Be sure to include units and show work. a) b) c) Position (km) Position (m) 6 4 Mass (kg) Time (hrs) Time (s) Time (wks)

8 8) Refer to the graph Motion of a Person 80 Position (m) a) Describe the trip of this person. Make something up that matches the graph. Time (s) b) At what time is the person going the fastest? Calculate this instantaneous speed. c) How fast is the person going at time 25 seconds? How can you tell? d) What is the average speed for the entire trip?

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