t 4.0 sec. 2. A bicyclist travels 60.0 kilometers in 3.5 hours. What is the cyclist s average speed?

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1 Name: Skill Sheet 3.1 Speed Problems This skill sheet will allow you to practice solving speed problems. To determine the speed of an object, you need to know the distance traveled and the time taken to travel that distance. However, by rearranging the formula for speed, v = d/t, you can also determine the distance traveled or the time it took for the object to travel that distance, if you know the speed. For example, Equation Gives you If you know v = d/t speed time and distance d = v t distance speed and time t = d/v time distance and speed 1. Solving problems Solve the following problems using the speed equation. The first problem is done for you. 1. What is the speed of a cheetah that travels meters in 4.0 seconds? speed = d -- = m = t 4.0 sec m sec 2. A bicyclist travels 60.0 kilometers in 3.5 hours. What is the cyclist s average speed? 3. What is the average speed of a car that traveled miles in 5.5 hours? 4. How much time would it take for the sound of thunder to travel 1,500 meters if sound travels at a speed of 330 m/sec? 5. How much time would it take for an airplane to reach its destination if it traveled at an average speed of 790 kilometers/hour for a distance of 4,700 kilometers? 1

2 Skill Sheet 3.1 Speed Problems 6. How far can a person run in 15 minutes if he or she runs at an average speed of 16 km/hr? (HINT: Remember to convert minutes to hours) 7. A snail can move approximately 0.30 meters per minute. How many meters can the snail cover in 15 minutes? 2. Unit conversion So far we have been mostly using the metric system for our problems. Now we will convert to the English system of measurement. Remember that there are 1,609 meters in one mile. Do not forget to include all units and cancel appropriately. These questions refer to problems in Part In problem 1.1, what is the cheetah s speed in miles/hour? m sec 2. In problem 1.5, what is the airplane s speed in miles/ hour? mile 1,609 m 3, 600 sec hour = 63 miles hour 3. In problem 1.6, what is the runner s distance traveled in miles? 4. You know that there are 1,609 meters in a mile. The number of feet in a mile is 5,280. Use these equalities to answer the following problems: a. How many centimeters equals one inch? b. What is the speed of the snail in problem 1.7 in inches per minute? 2

3 Name: Skill Sheet 3.2 Making Line Graphs Graphs allow you to present data in a form that is easily and quickly understood. Graphs are especially good for describing changing data. Here is how a line graph is made. 1. Examine your data Graph data consist of data pairs. Each data pair represents two variables. The independent variable will be plotted on the x- (horizontal) axis and the dependent variable will be plotted on the y-(vertical) axis. It s much easier to find the dependent than the independent variable. Test for the dependent variable by asking yourself which one depends on the other. For example, in a data set of money earned for hours worked, the dependent variable is money earned. That is because the money depends on the hours worked. Use this test to determine the dependent variable in any data set. Next, determine the numerical range between the smallest and largest value for each variable. If you started working with zero hours and worked 60 hours to earn this money, the range for hours would be 60. During this time, if you started with $50 and finished working with $320, the range for dollars would be 270. Be sure to calculate the range for the independent and depend variables separately. 2. Examine your graph Check the space that you will use for your graph. If you are using a piece of graph paper, allow some space on the side and bottom for labels and other information. Now count the number of lines right from the y-axis to nearly the edge of the space. This is the maximum graph space that you have for the independent variable. Repeat this process for the dependent variable by counting the number of lines up from the x-axis to nearly the top of the space. This is the maximum graph space that you have for the dependent variable. 3. Determine the graph scale You are now going to set the scale for the independent and dependent variables. It is important that you calculate the scales separately. The independent and dependent variables usually have different scales. We know that the dependent variable, money, has a range of $270. Now imagine that the graph space has a maximum of 20 lines on the y-axis before it nearly runs off the page. The first line (the x-axis) will be labeled $50. What will the next higher line be labeled? The point here is that if the value is too small, some of the money data will run out of the graph space. But if the value is too large, the plotted line will be small and hard to read. Divide the number of lines into the range to find a starting value. Increase the scale to an easier-to-use scale if necessary. $ lines = $13.50/line Increase to an easier scale: $15/line 1

4 Skill Sheet 3.2 Making Line Graphs Now label the y-axis at $15 per line. It is not necessary to label each line; perhaps each fourth line as a multiple of $60. Repeat this process for the independent variable. 1. Use the above information and the graphic to help you make a graph of the following data: (0, $50), (10, $95), (20, $140), (30, $185), (40, $230), (50, $275), (60, $320). Use your own graph paper or the graph paper on the last page of this skill sheet. 2. Use the data and the graph to determine the amount earned per hour during the 60 hours of work time. 4. Determine the independent and dependent variables Two variables are listed in each row of the first two columns of the table below. Identify the independent and dependent variable in each data pair. Rewrite the data pair under the correct heading in the next two columns of the table. The first data pair is done for you. Data pair not necessarily in order Independent Dependent Temperature Hours of heating Hours of heating Temperature Reaction time Number of people in a family Stream flow rate Tree age Alcohol consumed Cost per week for groceries Amount of rainfall Average height 5. Find the data range Calculate the data range for each variable: Lowest value Highest value Range

5 Skill Sheet 3.2 Making Line Graphs 6. Set the graph scale Using the variable range and the number of lines, calculate the scale for an axis and then determine an easy-touse scale. Write the easy-to-use scale in the column with the heading Adjusted scale. Range Number of lines Range Number of lines Calculated scale Adjusted scale

6 Skill Sheet 3.2 Making Line Graphs Graph paper 4

7 Name: Skill Sheet 3.3A Analyzing Graphs of Motion Without Numbers Graphs change columns of figures into images that are easy to interpret. Position-time and speed-time graphs describe the movement of objects. Here are stories for you to tell as graphs and a graph for you to tell as a story. Both will sharpen your graph interpreting skills. 1. Position-time graphs Data Remember the The Three Little Pigs? The wolf started from his house. Traveled to the straw house. Stayed to blow it down and eat dinner. Traveled to the stick house. Again stayed, blew it down, and ate seconds. Traveled to the brick house. Died in the stew pot at the brick house. The wolf started at his house, and the graph starts at the origin. Each time the wolf moves farther from his house, the line moves upward with passing time. At each pig s house, the line continues to the right but neither rises nor falls, indicating that the wolf has stopped moving relative to his starting point. We can deduce that the pigs houses are generally in a line away from the wolf s house and that the brick house was the farthest away. How would the line look if the brick house were on the way back to the wolf s house? Remember that position refers to the starting point in this case, the wolf s house. 2. Speed-time graphs A speed-time graph displays the speed of an object over time and is based on position-time data. You know that speed is the relationship between distance and time, R = D/T. Look at the first part of the wolf s trip. The line rises steadily to a new position and a new time. It would be easy to calculate a speed for this first leg. What if the wolf traveled this first leg faster? The new line would rise to the same position, but it would take less time. That would make the new line steeper. Here is the speed-time graph for the wolf: The wolf moved at the same speed toward his first two visits. His third trip was slightly slower. Except for this slight difference, the wolf was either at one speed or stopped. That is why this graph is so angular. 1

8 Skill Sheet 3.3A Analyzing Graphs of Motion Without Numbers 3. Stories for you to tell as graphs Read each of the following stories. Then sketch in the line for a position-time graph and a speed-time graph. 1. Little Red Riding Hood. Graph Red Riding Hood's movements. Data: Little Red Riding Hood set out for Grandmother s cottage at a good walking pace. She stopped briefly to talk to the wolf. She walked a bit slower because they were talking as they walked to the wild flowers. She stopped to pick flowers for quite a while. Realizing she was late, Red Riding Hood ran the rest of the way to Grandmother s cottage. 2. The Tortoise and the Hare. Use two lines to graph both the tortoise and the hare. Data: The tortoise and the hare began their race from the combined start-finish line. Quickly outdistancing the tortoise, the hare ran off at a moderate speed. The tortoise took off at a slow but steady speed. The hare, with an enormous lead, stopped for a short nap. With a start, the hare awoke and realized that he had been sleeping for a long time. The hare raced off toward the finish at top speed. Before the hare could catch up, the tortoise s steady pace won the race with an hour to spare. 2

9 Skill Sheet 3.3A Analyzing Graphs of Motion Without Numbers 3. The Skyrocket. Graph the altitude of the rocket. Data: The skyrocket was placed on the launcher. As the rocket motor burned, the rocket flew faster and faster into the sky. The motor burned out; although the rocket began to slow, it continued to coast ever higher. Eventually, the rocket stopped for a split second before it began to fall back to Earth. Gravity pulled the rocket faster and faster toward Earth until a parachute popped out, slowing its descent. The descent ended as the rocket landed gently on the ground. 4. A story to be told from a graph Tim, a student at Cumberland Junior High, was determined to ask Caroline for a movie date. Here are the graphs of his movements from his house to Caroline s. You write the story. 3

10 Name: Skill Sheet 3.3B Analyzing Graphs of Motion With Numbers Speed can be calculated from position-time graphs and distance can be calculated from speed-time graphs. Both calculations rely on the familiar speed equation: R = D/T. 1. Calculating speed from a position-time graph This graph shows position and time for a sailboat starting from its home port as it sailed to a distant island. By studying the line, you can see that the sailboat traveled 10 miles in 2 hours. The speed equation allows us to calculate that the vessel speed during this time was 5 miles per hour. R R = = D T 10 miles 2 hours R = 5 miles/hour, read as 5 miles per hour This result can now be transferred to a speed-time graph. Remember that this speed was measured during the first two hours. The line showing vessel speed is horizontal because the speed was constant during the two-hour period. 2. Calculating distance from a speed-time graph Here is the speed-time graph of the same sailboat later in the voyage. Between the second and third hours, the wind freshened and the sailboat increased its speed to 7 miles per hour. The speed remained 7 miles per hour to the end of the voyage. How far did the sailboat go during this time? We will first calculate the distance traveled between the third and sixth hours. 1

11 Skill Sheet 3.3B Analyzing Graphs of Motion With Numbers On a speed-time graph, distance is equal to the area between the baseline and the plotted line. You know that the area of a rectangle is found with the equation: A = L W. Similarly, multiplying the speed from the y-axis by the time on the x-axis produces distance. Notice how the labels cancel to produce miles: speed time = distance 7 miles/hour ( 6 hours 3 hours) = distance 7 miles/hour 3 hours = distance = 21 miles Now that we have seen how distance is calculated, we can consider the distance covered between hours 2 and 3. The easiest way to visualize this problem is to think in geometric terms. Find the area of the rectangle labeled 1st problem, then find the area of the triangle above, and add the two areas. Area of triangle A Geometry formula Area of rectangle B Geometry formula The area of a triangle is one-half the area of a rectangle. time speed = distance 2 ( 7 miles/hour 5 miles/hour) speed time = distance ( 3 hours 2 hours) = distance = 1 mile 2 5 miles/hour ( 3 hours 2 hours) = distance = 5 miles Add the two areas Area A + Area B = distance 1 miles + 5 mile = distance = 6 miles We can now take the distances found for both sections of the speed graph to complete our position-time graph: 2

12 Skill Sheet 3.3B Analyzing Graphs of Motion With Numbers 3. Practice: Finding speed from position-time graphs For each position-time graph, calculate and plot speed on the speed-time graph to the right. 1. The bicycle trip through hilly country. 2. A walk in the park. 3. Strolling up and down the supermarket aisles. 3

13 Skill Sheet 3.3B Analyzing Graphs of Motion With Numbers 4. Practice: Finding distance from speed-time graphs For each speed-time graph, calculate and plot the distance on the position-time graph to the right. For this practice, assume that movement is always away from the starting position. 1. The honey bee among the flowers. 2. Rover runs the street. 3. The amoeba. 4