# t y = v yi t +1 2 gt 2 Note: These equations are really just d = vt and x = v x v y = vsin! = vcos! v x v 2 = v x 2 + v y

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1 Chapter 3: Vectors & Projectile Motion NAME: Text: Chapter 3 Think and Explain: 1, 2, 6-10 Think and Solve: 1a, 2-6 Vocabulary: vector, scalar, resultant vector, component vector, projectile, horizontal component of velocity, vertical component of velocity, range, satellite Equations: x = v x t y = v yi t +1 2 gt 2 Note: These equations are really just d = vt and v y = gt + v yi d i + 2 = v t 1 2 at! v 2 = v x 2 + v y 2 v x = vcos! v y = vsin! Constants: g = ±10 m/s 2 Key Objectives: Concepts! Distinguish between a vector quantity and a scalar quantity.! Distinguish between a component vector and a resultant vector.! Identify the initial horizontal and vertical components of velocity for a projectile launched horizontally.! State which velocity component changes over time and which component of velocity remains the same.! Identify the velocity and acceleration at the highest point for a projectile launched at an angle on a level surface.! Recognize and be able to sketch the motion graphs for a projectile. (x vs. t, y vs. t, v x vs. t, v y vs.t)! State the launch angle that will yield maximum range.! State the relationship between launch angles that will yield the same range.! Given the paths of three projectiles, be able to describe the motion qualitatively and determine which projectile has the greatest time in air, horizontal velocity, initial vertical velocity, etc.! Relate the motion of a satellite to the motion of a projectile. Problem Solving! Add vectors graphically using the tip to tail method.! Find the magnitude of a resultant vector using the Pythagorean Theorem when given two component vectors at right angles to one another.! Find the components of a vector given its magnitude and direction (angle.)! Solve vector word problems. (River problems, airplane/wind speed problems.)! Set up table and fill in given information for a horizontal projectile problem and solve for missing values.! Set up table and fill in given information for a projectile launched at an angle problem and solve for missing values.! Find the final velocity of a projectile when it hits the ground

2 CP Physics Chapter 3- Vectors and Projectile Motion Date Class Reading Homework Lab 3-0: Where s the X? Vector Concepts (Graphical) Finish Worksheet Right Triangle Trig Review Finish Worksheet Vector Concepts (Trig) Vector Word Problems Horizontal Projectile (Cliff 3-4 Cliff Problems WS Problems) Notes and Examples Vector Quiz (30 points) RQ 8-11 More Cliff Problems Finish WS Lab 3-1: Projectile Challenge Lab Lab 3-2: Projectile Motion Finish Lab Projectile Motion Concept 3-5 Sheet Projectile Motion Concept Sheet (cont.) Lab 3-3: Range Lab Projectile Motion Problems Finish WS WS Review Projectile Motion Exam

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5 Vectors Vector Concepts (Graphical) NAME: Quantities that have both a magnitude (tells how much) and a direction. Examples are displacement, velocity, acceleration and force. Resultant The sum of two or more vectors. You can think of this as the one vector that can replace the two or more vectors you have added up. 1. Some vectors are shown in the picture below. For each problem, sketch how you would add the given vectors using the Tip to Tail method. a.! A +! B b. A B C D E F G! A +! D c.! D +! G d.! B +! C e.! C +! E f.! F +! G g.! E +! F 2. What is the resultant displacement for these pairs of displacement vectors? a. 20 m West + 15 m South b. 12 km East + 10 km North East Scale: cm = m Magnitude: Direction: Scale: cm = km Magnitude: Direction: 3. The resultant velocity of a plane is the sum of its velocity in the air and the velocity of the air itself. What is the resultant velocity of a plane flying with a velocity of 100 m/s due East in air that has a velocity of 60 m/s North? Scale: cm = m/s Magnitude: Direction: side 1

6 Vector Concepts (Graphical) NAME: 4. Imagine you have two vectors, one of magnitude 3 and the other of magnitude 4, but you don t know anything about their actual directions. a. What is the largest possible resultant you could have? How would you add them up? b. What is the smallest possible resultant you could have? How would you add them up? *c. If their vector sum had a magnitude of 5, what is true about the vectors? 5. Find the horizontal and vertical components of each given vector. They are all drawn to scale. Scale: 1 cm = 5 units (m, km, or m/s) a. b. c. 15 m 20 m/s 20 km x: y: x: y: v x : v y : d. e. f. 15 m/s 100 m/s 20 km v x : v y : v x : v y : x: y: side 2

7 Vector Concepts (Trig) NAME: The diagrams below shows right triangles, one representing position and the other representing velocity. For each picture, what is the Pythagorean Theorem? r! x y What is the sine of θ? What is the cosine of θ? v! v x v y In terms of the hypoteneuse and the angles shown above, what are x & y and v x & v y? x = y = v x = v y = Questions 1. Use the Pythagorean Theorem to find the speeds of the following velocity vectors: a. v 8 m/s 5 m/s v b. 5 m/s 20 m/s c. 9 m/s v 4 m/s d. v 30 m/s 10 m/s 2. You are given the horizontal and vertical components of different velocity vectors. Find the resultant speed: a. v x = 7 m/s v y = 5 m/s v = b. v x = 15 m/s v y = 8 m/s v = c. v x = 20 m/s v y = 25 m/s v = d. v x = 10 m/s v y = -15 m/s v = 3. For each of the triangles shown, calculate the sides of the right triangles, given the hypoteneuse and angle: x a º x y b. 25º 40 y c. 7 40º x y d. x 15º 20 y Side 1

8 Vector Concepts (Trig) 4. Calculate the components of each of the velocities shown: NAME: 25 m/s 15º 8 m/s 40 m/s a. 20º b. 10 m/s c. 70º d. 90º 5. Calculate the components of the given velocities: a. A ball is kicked with a velocity of 30 m/s at an angle of 35º above the horizontal. v x = m/s v y = m/s b. A pen is tossed with an initial velocity of 5 m/s at an angle of 65º above the horizontal. v x = m/s v y = m/s c. A projectile hits the ground with a velocity of 25 m/s at an angle of 40º below the horizontal. v x = m/s v y = m/s d. A block of ice slides off a roof with an initial velocity of 9 m/s at an angle of 30º below the horizontal. v x = m/s v y = m/s e. A ball rolls horizontally off a table with a speed of 8 m/s. v x = m/s v y = m/s f. A soccer ball in the air has a velocity of 32 m/s at an angle of 25º above the horizontal. v x = m/s v y = m/s g. A pen is thrown straight up in the air with an initial velocity of 18 m/s. v x = m/s v y = m/s h. A bullet is fired with an initial velocity of 400 m/s at an angle of 15º above the horizontal. v x = m/s v y = m/s Side 2

9 Vector Word Problems NAME: 1. The pilot of a plane points his airplane due South and flies with an airspeed of 120 m/s. Simultaneously, there is a steady wind blowing due West with a constant speed of 40 m/s. a. Make a sketch that shows how to find the resultant velocity of the plane. Roughly in what direction is the resultant velocity? b. What is the resultant speed of the airplane? b. After one hour, how far away is the plane from its starting point? 2. A swimmer is able to swim with a speed of 5 m/s in a pool (this is her water speed.) This same swimmer goes swimming in a river which has a current flowing to the East with a constant speed of 3 m/s. Assume her water speed is always 5 m/s. a. What would be her resultant velocity if she tries to swim due East with the current? (Include a vector sketch.) b. What would be her resultant velocity if she were to try to swim due West against the current? (Include a vector sketch.) c. What would be her resultant velocity if she points herself due North straight across the river? (Include a vector sketch.) 3. A plane is flying due North at 80 m/s. There is a cross wind of 30 m/s that is blowing due East. a. Draw a vector diagram showing how these velocities add. Roughly in what direction is the resultant velocity? b. How fast is the plane flying with respect to the ground? side 1

10 Vector Word Problems NAME: river flow 4. A 50 meter wide river is flowing at 5 m/s to the left, as shown in the diagram above. A person in a kayak always rows with a water speed of 8 m/s. a. If the kayaker points straight across, what is the final speed of the kayaker? (Include a vector sketch.) b. What would be the maximum possible speed of the kayaker (and in what direction should they point?) c. What would be the slowest possible speed of the kayaker (and in what direction should they point?) *d. How long would it take the kayaker to cross from part a? (Hint: what is the component of the velocity straight across the river?) **e. In what direction should they point so that their resultant velocity is straight across the river? (Include a vector sketch.) 5. A 50 meter wide river is flowing at 5 m/s to the left, as shown in the diagram above. A person in a kayak always rows with a water speed of 3 m/s. a. What would be the maximum possible speed of the kayaker (and in what direction should they point?) b. What would be the slowest possible speed of the kayaker (and in what direction should they point?) *c. If the kayaker tries to kayak heads straight across the river, how long would it take the kayaker to cross? Answers: 1. a) ~SW (71.6º S of W) b) m/s c) 455,000 m (=455 km) 2. a) 8 m/s E b) 2 m/s W c) 5.83 m/s 3. a) ~NE (69.4º N of E) b) 85.4 m/s 4. a) 9.43 m/s b) 13 m/s W c) 3 m/s E d) 6.25 s e) ~NE (51.3º N of E) 5. a) 8 m/s W b) 2 m/s W - but they point E c) 16.7 s side 2

11 Cliff Problems NAME: 1. A ball rolls off the edge of a table. It has an initial horizontal velocity of 3 m/s and is in the air for 0.75 seconds before hitting the floor. a. How high is the table? b. How far away (horizontally) from the edge of the table does the ball land? c. What are the horizontal and vertical components of the ball s velocity when it lands? d. How fast is the ball going when it lands? 2. The Coyote is chasing the Road Runner when the Road Runner suddenly stops at the edge of a convenient cliff. The Coyote, traveling with a speed of 15 m/s, does not stop and goes flying off the edge of the cliff, which is 100 meters high. a. How long is the Coyote in the air? b. Where does the Coyote land? c. What are the horizontal and vertical components of the Coyote s velocity when he lands? d. How fast is the Coyote going when he lands? 3. A car full of bad guys goes off the edge of a cliff. If the cliff was 75 meters high, and the car landed 60 meters away from the edge of the cliff, calculate the following: a. The total time the car was in the air. b. The initial velocity of the car. (Give the components.) c. The final velocity of the car just as it hits the ground. (Give the components.) d. The final speed of the car just as it hits the ground. Answers: 1. a) 2.81 m b) 2.25 m c) v x = 3 m/s & v y = 7.5 m/s d) 8.1 m/s 2. a) 4.47 s b) 67.1 m c) v x = 15 m/s & v y = 44.7 m/s d) 47.2 m/s 3. a) 3.87 s b) v x = 15.5 m/s & v y = 0 m/s c) v x = 15.5 m/s & v y = 38.7 m/s d) 41.7 m/s

12 More Cliff Problems NAME: 4. A ball is shot horizontally from a window. It has an initial horizontal velocity of 4 m/s and is in the air for 1.35 seconds before hitting the ground. a. How high is the window? b. How far away (horizontally) from the edge of the building does the ball land? c. What are the horizontal and vertical components of the ball s velocity when it lands? d. How fast is the ball going when it lands? 5. The Coyote is chasing the Road Runner when the Road Runner suddenly stops at the edge of a convenient cliff. The Coyote, traveling with a speed of 25 m/s, does not stop and goes flying off the edge of the cliff, which is 200 meters high. a. How long is the Coyote in the air? b. Where does the Coyote land? c. What are the horizontal and vertical components of the Coyote s velocity when he lands? d. How fast is the Coyote going when he lands? 6. A plane is flying across a level field and is 150 meters off the ground. When the plane is directly over point A, it releases a package, which then falls to the ground, and lands at point B, which is 500 meters away from point A. Calculate the following: a. The total time the package was in the air. b. The initial velocity of the package. (Give the components.) c. The final velocity of the package just as it hits the ground. (Give the components.) d. The final speed of the package just as it hits the ground. Answers: 4. a) 9.1 m b) 5,4 m c) v x = 4 m/s & v y = 13.5 m/s d) 14.1 m/s 5. a) 6.32 s b) m c) v x = 25 m/s & v y = 63.2 m/s d) 68 m/s 6. a) 5.48 s b) v x = 91.3 m/s & v y = 0 m/s c) v x = 91.3 m/s & v y = 54.8 m/s d) m/s

13 Lab 3-1: Projectile Challenge NAME: Part I: Calculate how fast the launcher shoots projectile Shoot launcher straight up and measure the maximum height of the ball. Maximum height = m Use v 2 f = v 2 i + 2 g d where g = -10 m/s 2 Vi = Part II: Calculate how far away to place the cup. Vi = Vx = X = Side 1

14 Part III: Optional Bonus (+1) Lab 3-1: Projectile Challenge NAME: Combine d v f + v i 2 2 = and v f = vi + gt to show that v f = vi + 2ad t 2 Part IV: Optional Bonus (+1) Side 2

15 Lab 3-2: Projectile Motion NAME Purpose: 1. To examine the motion of a projectile through the use of a camcorder. 2. To produce position and velocity graphs of a projectile s motion for horizontal and vertical components. 3. To analyze the motion of the projectile. Procedure: 1. Two students will be video-taped tossing a tennis ball back and forth. This video will be converted into a small computer file, which will be analyzed using Logger Pro. 2. Make sure that the LabPro is NOT plugged into the computer. Open up Logger Pro. Under Insert, choose Movie... Choose the correct movie. It will open up in the middle of the screen of Logger Pro. 3. Enable video analysis by clicking on the box on the bottom right of the movie that looks like the button to the left. 4. Set the scale of the movie by clicking on the Set Scale button (upper right corner), then clicking and dragging across the length of the meter stick on the wall. 5. Set the origin by clicking on the Set Origin button (upper right corner), and then clicking on the first position of the tennis ball. 6. Now to record the actual position of the tennis ball for each frame of the movie, click on the Add Point button (upper right corner.) Carefully center the mouse on the tennis ball, and click. Logger Pro will record the x and y coordinates of the mouse click, and the movie will automatically go the next frame. Do this for each frame of the movie. 7. To clean up the window, under Page, choose Auto Arrange. You should now see the position vs. time graphs on the main screen. 8. To add the velocity vs. time graphs, under Insert, choose Graph. A floating window will appear with a new graph in it. Again, under Page, choose Auto Arrange. 9. To add the second velocity graph, click on the axis label (probably X Velocity ) and then choose More... in the pop-up window that appears. Make sure both X Velocity and Y Velocity are checked off and then click OK. 10. Sketch what the graphs look like in the space below. Make sure you label each graph. three of the graphs should be lines; write down the slopes for those under neath the graph. 11. Answer the questions on the other side. Graphs from Logger Pro slope = slope = slope = slope = side 1

16 Lab 3-2: Projectile Motion NAME Questions: 1. The graph of horizontal position verses time is a straight line. What is the slope of the line, and what does the slope represent? 2. The graph of horizontal velocity verses time may be a little scattered, but should be basically horizontal. How do you interpret this graph, taking into account the graph of horizontal position verses time? Was there any acceleration? 3. The graph of vertical position verses time is a curve what does this graph tell you about the motion of the projectile? (It should look like a graph from an earlier lab, if that helps to interpret the graph.) 4. The graph of vertical velocity verses time is a straight line. What is the slope of the line, and what does the slope represent? 5. Let s summarize these results: a. What happens to the horizontal velocity (v x ) of a projectile while in the air? b. What happens to the vertical velocity (v y ) of a projectile while in the air? c. What is the magnitude of the acceleration of a projectile? d. In which direction does a projectile accelerate? 6. For an object that is caught at the same height from which it was thrown and ignoring air resistance a. what is true about the time needed to go up compared to the time needed to go down? b. what is true about the initial horizontal velocity compared to the final horizontal velocity? c. what is true about the initial vertical velocity compared to the final vertical velocity? d. what is its velocity at its maximum height? e. what is its acceleration initially, at its maximum height and finally? side 2

17 Projectile Motion Concept Sheet NAME: Projectile motion is a combination of two separate motions: constant speed horizontally and constant acceleration due to gravity vertically. On this sheet, you will calculate what happens to the components of a projectile's velocity and position, and then graph the positions, much as you did on some previous concept sheets. For this problem, we have a projectile launched upward with an initial horizontal velocity of 20 m/s and an initial vertical velocity of 30 m/s. Answer the following questions first: 1. What is the actual initial speed of the projectile? 2. What happens to the horizontal component of the velocity as the projectile flies through the air? 3. What happens to the vertical component of the projectile as it flies through the air? 4. At the projectile s maximum height, what is the horizontal component of its velocity? 5. At the projectile s maximum height, what is the vertical component of its velocity? Now to fill out the chart on the other side by completing the following: 6. Fill out the column for the horizontal velocity (V x ) at each point in time. Explain how you filled the chart out, or show your calculations here. 7. Fill out the column for the vertical velocity (V y ) at each point in time. Explain how you filled the chart out, or show your calculations here. 8. Fill out the column for the horizontal position (X) at each point in time. Explain how you filled the chart out, or show your calculations here. 9. Fill out the column for the vertical position (Y) at each point in time. Explain how you filled the chart out, or show your calculations here. side 1

18 Projectile Motion Concept Sheet NAME: Time (s) V x (m/s) Velocity V y (m/s) X (m) Position Y (m) Mark each of the positions of the projectile (X,Y) on the coordinate shown below. Label each position t= with the appropriate time. The first position is already done for you. 11. At each position, draw vectors to represent both components of the velocity. Use the scale of 1 square = 10 m/s. The first position is already done for you m/s 20 m/s side 2

19 Projectile Motion Concept Sheet NAME: Questions: 1. Imagine that you did the same thing for a projectile with an initial V x of 10 m/s and V y of 30 m/s. a. What would be different? b. What would be the same? c. How long would the projectile be in the air? d. What would be the maximum height of this projectile? e. How far away would the projectile land? 2. Imagine that you did the same thing for a projectile with an initial V x of 30 m/s and V y of 30 m/s. a. What would be different? b. What would be the same? c. How long would the projectile be in the air? d. What would be the maximum height of this projectile? e. How far away would the projectile land? 3. If you wanted the projectile to go higher, a. what should you change? Explain. b. would this affect the time in the air? Explain. c. would this affect how far away the projectile landed? Explain. side 3

21 Lab 3-3: Projectile Range NAME Purpose: 1. To experimentally determine the initial launch angle that will give the maximum range of a projectile with a given initial speed. 2. To experimentally determine the relationship between angles that give the same range of a projectile with a given initial speed. 3. To use your experimental results to predict the landing position of a projectile for a given angle and to predict the angle to get a given landing position. Materials: 1 projectile launcher 1 paper strip 1 carbon paper 1 meter stick 1 c-clamp launcher paper strip, taped to table Procedure: 1. Clamp the projectile launcher to the end of your lab bench so that it will launch the ball bearing down your lab bench from the level of the table top. (Use the guide on the side of the launcher to see the initial launch position.) 2. Tape a strip of paper to the lab table so that the ball bearing will land on it. 3. As best you can, fire the projectile and record the range for 5º intervals, from 80º to 10º. You can assume that the angle of 90º and 0º will have a range of 0 cm. Fire the projectile to see about where it lands, place the carbon paper at that spot, and relaunch the projectile to measure its range. Try 3 launches per angle. (Angles less than 15º can be hard to do.) 4. Measure the distances to the average landing spot for each angle, and record in the data table. 5. Make a graph of Range vs. Angle. Make sure axes are labeled and your graph has a title. Data: Launch Angle (º) Range (cm) Launch Angle (º) Range (cm) Launch Angle (º) Range (cm) Answer questions on other side. side 1

22 Lab 3-3: Projectile Range NAME Questions: 1. Based on your data and graph, what is the relationship for launch angles that will have the same range? 2. Which angle will give the maximum range? 3. Would your results (questions 1 and 2) have worked if the projectile were fired off a cliff? Explain. 4. From your graph, predict where the projectile will land when fired with the initial angle given to you by your teacher. Place the target (from your teacher) at that location and call your teacher over to test it. 5. From your graph, predict the angle at which you need to launch the projectile so that it hits with the range given to you by your teacher. Place the target (from your teacher) at that location and call your teacher over to test it. side 2

23 Projectile Motion Problems NAME: 1. A student tosses an eraser to his friend. The initial horizontal velocity of the eraser was 4.5 m/s and the initial vertical velocity was 5.36 m/s. The friend catches the eraser at the same level from which it was tossed. a. How long was the eraser in the air? b. How far apart were the two friends? c. What was the maximum height of the eraser? d. What were the components of the velocity at the top of its flight? 2. A kangaroo is jumping across a field in the outback. The kangaroo jumps with an initial horizontal velocity of 8 m/s and an initial vertical velocity of 5 m/s. a. What was the initial speed of the kangaroo? b. How long was the kangaroo in the air? c. What was the maximum height of the kangaroo? d. What was the horizontal distance of the kangaroo s jump? 3. Mary throws a ball to Suzy, who is standing 25 meters away. Suzy catches the ball from the same height at which it was thrown. If the ball was in the air for 4 seconds, calculate the following: a. Horizontal velocity. b. Maximum height of the ball. c. Initial vertical velocity. d. What happens to the components of the velocity and the acceleration as the ball flies through the air? side 1

24 Projectile Motion Problems NAME: 4. Larry tosses a volleyball to his wife, Lise, who catches it at the same height from which it was tossed. The volleyball has an initial velocity of 15 m/s at an angle of 30º above the horizontal. a. What are the components of the initial velocity? b. How many seconds does it take the volleyball to reach its maximum height? c. How far apart are Lise and Larry? d. What was the acceleration of the volleyball after 1 second? Give the magnitude and direction. *5. An astronaut on the moon tosses a rock with an initial velocity of 3 m/s at an angle of 35º above the horizontal. The acceleration due to gravity on the moon is 1.7 m/s 2. a. What were the components of the initial velocity of the rock? b. How long was the rock in the air? c. What was the maximum height of the rock? d. What was the horizontal distance traveled by the rock? Answers: 1. a) 1.07 s b) 4.82 m c) 1.44 m d) v x = 4.5 m/s & v y = 0 m/s 2. a) v = 9.43 m/s b) 1.0 s c) 1.25 m d) 8 m 3. a) 6.25 m/s b) 20 m c) 20 m/s up d) v x = constant = 6.25 m/s & acceleration = constant = 10 m/s 2 down & v y starts positive 20 m/s (up) decreases to 0 m/s at top and continues to decrease to -20 m/s (down) when finally caught 4. a) v x = 13 m/s & v y = 7.5 m/s b) 0.75 s c) 19.5 m d) acceleration = gravity = 10 m/s 2 down 5. a) v x = 2.46 m/s & v y = 1.72 m/s b) 2.02 s c) 0.87 m d) 4.97 m side 2

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