Physics Regular 1213 Williams Sports Physics: Vectors & Projectiles Packet Packets 5-6 / Chapters 3 & 4 1
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Describing angle directions (two methods) Name: No. Video Guide: Nova - Medieval Siege 1. King Edward Longshanks is best known for his role in this 1997 Mel Gibson epic:. 2. Sketch a trebuchet below, and describe it s basic principle of operation: 3. Trebuchets originated in this country:. 4. What dangerous tendency did the scale model of the trebuchet reveal? 5. How did medieval engineers correct this dangerous tendency? 6. What historical evidence exists that lead was used as the counterweight on Longshank s great trebuchet Warwolf? 7. List three benefits that the wheels on the base of a trebuchet offer to its operation: 3
Trig review/primer Be sure your calculator is in DEGREE MODE! Sin A = opposite/hypotenuse (SOH) Cos A = adjacent/hypotenuse (CAH) Tan A = opposite/adjacent (TOA) Trig is all about the ratios of the lengths of two of three of the sides of right triangles. There really is nothing more mysterious or illusive to it than that.that we ll ever do with it any way. It s all about ratios Look at the three identical triangles above. Notice the hypotenuse is the long side, since it s opposite the right triangle the biggest angle. The two shorter sides are given the names adjacent and opposite based on where they are relative to the angle the problem is about. If we look at ratios of the smaller sides, there are only two possibilities, 4/11 and 11/4, which are 0.364 and 2.75. Confirm the tangent function makes sense to you now The tangent function gives ratios of the short sides. Make sure you re in degree mode and find Tan (20 ) and Tan (70 ). Did you get 0.364 and 3.75? If you give a trig function an angle, your calculator gives you the ratio of sides. The kind of trig function you use determines the ratio of WHICH two sides. That s where SOH-CAH- TOA comes in. It helps you remember which trig function is for which ratio of sides. Look at the lengths and angles again and the numbers until you see where the angles and fractions (4/11 and 11/4) come from. Does it make sense? Inverse functions do the opposite & try the other two trig functions Sometimes you re not given an angle, but you re given a ratio-of-sides and asked to come up with the angle, that is you re asked to do the opposite function. You re calculator uses the inverse function to do this opposite operation. Prove it works to yourself. Find the inverse tangent of 4/11 (Tan -1 (0.364)). Do you get 20? Try the sin function now. Find the inverse sin of 4/11.7 (Sin -1 (0.342)). Do you get 20? Now find the inverse cos of 11/11.7 (Cos -1 (.940)). Do you get 20? Need a little more help? If you re still a little sketchy on this stuff, try drawing a perfect 5-12-13 right triangle (sides whose lengths are 5, 12, 13 work out to exact integers). Use this triangle to find the angles and see that you get the side correct. If you re still feeling insecure, bring your calculator and this page and come see your teacher for some extra help. I bet five minutes of one-on-one trig will help a lot. 4
Describing angle directions (two methods) Method 1: Relative to horizon If you were shooting a canon from a fortress at your enemy, you would adjust firing angle to get different distances. As the enemy got close, you would point your canon downward at them, below the horizon. In this way, you describe angles from 0 to 90 above or below the horizon. The canon at left is tilted upward and so is shooting St. Patricia above the horizon. I measured the angle to be about 21 relative to the horizontal line I drew. If she were shot from the canon at 45 mph (20 m/s), you could describe her velocity as: V = 20 m/s above the horizon. Method 2: Mapping method A more flexible method is to use the familiar N, S, E, W we re used to with maps. For the canon example above, you can see she s mostly going east (right), but she s also going a little north (up). As a matter of fact, she s tilted upward (northward) by 21, so you would describe her direction as 21 north of east. A second equally valid description would use the complementary angle: 69 east of north. Can you see both? Practice: Described each of the six (bolded) vectors below using the mapping method. Try to find two ways! 5
Vector math The story of Mary and Bob Mary and Bob decided they didn t like where they lived, Nowville. Mary wanted to move 500 km south because she thought the winters were too harsh. Bob didn t care about the winters so much, but he wanted to get out west because the skies were blue and he liked the stars at night. He wanted to move 500 km west. They had terrible arguments about where to move and were almost going to break up when they learned about vectors Mary came home one day and told Bob she had some exciting news, she was cleaning out the basement and found her old physics books and noticed stuff about vectors. They could both have what they wanted! Not only that, but they wouldn t have to be thousand miles from home, they would actually be much closer and could fly home on holidays to visit parents, friends and of course their old physics teacher. Bob and Mary got out a map and found that the perfect spot which was both 500 km west and 500 km south was in the middle of No man s land, which was infested with poisonous snakes, man-eating spiders and had really, really slow internet connection speeds. Bob was allergic to snakes and spiders and Mary s love of instant messaging made a fast internet connection a priority. Fortunately, there were two cities with good internet access and no snake or spider problems, Clearton and Hotsburgh. Help Bob and Mary by finding how far each city is (so they know how far flights will be) and how far south and how far west each is (so they know how clear skies are and how warm it will be). 1. Draw a resultant (overall) vector from Nowville to Clearton and do the same to Hotsbugh as well. 6
2. Resolve each vector into X and Y components: C x, C y, H x, H y. Anything left or down should be negative! 3. Describe the vectors C and H by finding the overall magnitude (flying distance) and direction. Use Pythagorean theorem to find flying distance and trig to describe angle direction. 4. What would be the best compromise for them to locate? 7
Scale When we draw representations of vectors, they don t usually match the size of paper we re using exactly. To account for this and keep things proportional, we scale things so they fit properly. This is the same thing map makers do when they draw a map. They think about the size of paper for their map and the actual size of the place they re representing and declare a scale, like 1:10,000. Another convenient method commonly used is to publish a map length with the corresponding actual length, like drawing a 1 inch line and saying that it s equivalent to 75 miles. Say you wanted to represent a displacement vector for a trip from here to 450 miles east, in Ohio. This paper is about 8 inches wide, so 50 miles = 1 inch wouldn t quite fit. I d choose 100 miles = 1 inch. Try drawing that displacement vector below: Come up with appropriate scales to use and draw scaled vectors for each below. Declare the scale you choose! 1. 30 km, due North 2. 10 km, 45 o S of E 3. 20 km, 10 W of N 4. 20km, 10 N of W 8
Resolving resultants (finding X & Y components) Graphically: Draw horizontal (X) and vertical (Y) lines corresponding to the start and finish of your resultant vector. Measure them and use the provided scale to find the x component and y component. Up and right are positive and down and left are negative. Example: For the displacement vector at left, use a scale of 1 cm = 10 miles and graphically find the resultant, and the X and Y components. Use a compass to measure and hence describe the angle. Mathematically: Use the Pythagorean Theorem to find the magnitude of the resultant and trig to find the angle. Example: A canon is fired at the enemy such that the horizontal velocity component (V x ) is 50 m/s and the vertical velocity component is initially downward at 20 m/s (V y = - 20 m/s). Find the resultant. (54 m/s 22 below the horizon ) Adding vectors to find resultants 1. Resolve both vectors to be added into their X and Y components. Be careful with signs (left and down are negative) 2. Add X and Y components to find overall X and overall Y 3. Make a quick sketch to make sure you can visualize the direction of the resultant 4. Use trig and pythag. to find the resultant Example: A plane is flying in a northeast wind of 100 mph. Its airspeed is 150 mph 30 south of east. What s its groundspeed? Wind Plane Total (rel. to ground) X Y Resultant (ground speed) = 9
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Adding vectors, a realistic detailed example with solution: 11
Resolving vectors (finding the X and Y components) Adding vectors practice problems 12
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Name: No; Sailing the Seven Seas Navigating the vector straights http://mysite.verizon.net/vzeoacw1/velocity_composition.html Introduction: All motion is relative. When a boat moves north in a lake whose waters are still, the boat moves north. Now, put that boat in the Missouri river flowing east. Relative to the water, the boat is still moving north, but an observer sees the boat drifting west as well. The velocity of the boat to an off shore observer has both north and west components. This lab sim will show how vectors can help boat captains in moving water and airplane pilots in moving air. Part 1: Chugging Upstream & Downstream Adjust Water speed and Boat speed to 5 m/s and set the Boat direction to -180. Click on go and use your mouse to drag your boat to the center of the screen. 1. Assume the white screen area below the blue water is the shore. How fast is the boat moving relative to the shore? 2. What is the velocity of the Boat? Water? (Careful! + for East, - for West!) 3. What is the Sum of these two velocities? How does this value compare to the velocity of the boat to the shore? 4. Now set the Boat direction to 0. What is the velocity of the Boat? Water? Resultant Sum? Part 2: Michael Row the Boat Ashore! Problem: If the water velocity is +5 m/s (East is the + direction) and the boat speed is 10 m/s, in what Direction must the boat be directed to cause it to travel due NORTH in a direction perpendicular to the direction of the water? Set the Water speed to 5 m/s and the Boat speed to first to 6 m/s, and then to 10 m/s. Change the direction of the boat and determine when the black (resultant vector) is pointing perpendicular to the direction of the current. Direction(s) for 6 m/s and for 10 m/s. 14
Part 3: Go Home Gilligan! Adjust water speed and boat speed to 5 m/s; set the direction to 90. Drag your boat to the southwest corner of the screen and click go. 1. Freeze the computer using the freeze button when the boat is in the center of the screen. Notice the vectors on the boat. Draw a picture of the boat s velocity vectors and the resultant velocity vector. Label three items on your drawing: a) Water velocity vector b) Boat velocity vector c) Resultant (combined sum) vector. 2. Now PREDICT the positions of where on the opposite shore the boat will land when it is launched from the far southwest corner with a Boat speed of 3 m/s and water speeds of: 0, 1, 2, 4, & 8 m/s. Put 5 X marks on the diagram below for your predictions. Now go ahead and TEST your predictions and draw in where the boat actually lands on the opposite shore how close were your predictions? 15
3. Do you think the time to cross the shore depends on a) Boat speed only b) Water speed (stream speed) only c) Both a and b? 4. Do the following runs and use a partner to help you time. The partner should have a watch or be able to see the class clock second hand. d) Time the crossing for a Water speed of 3 m/s and Boat speeds of 1, 3, and 9 m/s starting the boat again in the Southwest corner of the screen and keeping the direction at 90. Write your times below. e) Time the crossing for a Boat speed of 3 m/s and Water speeds of 1, 3 and 9 m/s starting the boat again in the Southwest corner of the screen and keeping the direction at 90. Write your times below. Water Speed Boat Speed Time of crossing (seconds): 3 m/s 1 m/s 3 m/s 3 m/s 3 m/s 9 m/s 1 m/s 3 m/s 3 m/s 3 m/s 9 m/s 3 m/s Does the amount of time to make progress in the North direction depend on the east velocity or ONLY the north velocity? Conclusion: Summarize in your own words what this lab sim was about and how you would describe it to another student. 16
Vector Map Lab Purpose: Add vectors using a map. This will require knowledge of maps, vectors, including using scales and simple trig to describe a vector using map notation: Scales Using a scale enables people to make reasonable-sized models. For example, a tiny atom can be scaled up to model it, or Europe can be scaled down so it can be modeled on a single sheet of paper. Open the vector program and load the map of the USA. Find how many horizontal pixels are equivalent to 500 miles using the scale provided on the lower left of the map. Depending on screen resolution this may vary, but by clicking on the 0 and the 500 mile hash marks on my computer I got it as 153 pixels = 500 miles. I can use this to convert between pixels and miles exactly the same way as I can use 1 foot = 12 inches to convert feet into inches. Your conversion factor: 500 miles = pixels Sample measurement A lot of people enjoy visiting Orlando on vacation. Do you know how many miles it takes to fly there? Let s come up with the distance and the direction and describe a vector from Chicago to Orlando. When I did this I got the following values: Chicago: (610, 416); Orlando (748, 149) Based on this I can see that Orlando is 748 610 = 138 pixels East (451 miles East) And it s 149 416 = -267 pixels North (negative, so 873 miles South) Using pythagorus to find the magnitude of R = 983, Find the angle using inverse Tangent: Tan -1 (873/451) = 63 Therefore the resultant from Chicago to Orlando is: 983 miles 63 South of East Adding two vectors 1. Bob in Chicago picks up his friend in St. Louis on the way to see a football game in Kansas City. Use vector addition to add both legs of his trip from Chicago to Kansas City Chicago to St. Louis ΔX ΔY Sketch/Scratch St. Louis to KC Total Result vector for entire trip: 17
More adding vectors 2. Mary the movie fan is looking for cheap air fare to Los Angeles to see her favorite Hollywood stars. If she connects through Seattle, she will save a lot of money. Find the vector describing her overall trip by adding together both legs of her journey below. Chicago to Seattle ΔX ΔY Sketch/Scratch Seattle to LA Total Resultant vector for entire trip: Resolving a vector 3. Resolve the vector from San Diego into its two components and find the total straight-line distance East miles North miles Straight-line miles 4. Name the vector you just resolved using map notation 18
Getting local Load the Hinsdale map and answer the following questions. Remember, there are 8 blocks in a mile, so use this fact to find a conversion factor. Conversion factor: 1 mile = pixels 5. My favorite pizza is Lou Malnati s. There is a red dot showing where it s at on the map. If you re not sure which red dot it is, look it up on the internet, it s in Western Springs.you should know how to do this! Find the resultant vector describing a trip from school to Lou Malnati s. 6. Pretend you take a trip from 55 th and Willow Springs road to Plainfield and Wolf road in the following two parts: First you go to 55 th and Plainfield. Second, you go to your final destination. 55 th & Willow Springs to Plainfield & 55th ΔX ΔY Sketch/Scratch Plainfield & 55 th to Painfield & Wolf Total Result vector for entire trip: 19
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Physics practice problems (use 4-steps) 1. You roll a steel ball off of a black physics table where the ball is going 0.5 m/s. If your table top is 0.75 m high, then how far from the table does the ball land? (0.20 m) 2. Little John is about to be hung. You want to shoot the noose from around Little John s neck. The noose is (horizontally) 15 m from you. If you aim 20 degrees above horizontal and shoot a 30 m/s arrow from a height of 1.5 m, how high does your arrow strike the rope? (5.4 m) 3. Some hand guns have muzzle velocities as high at 1000 m/s. If you shoot horizontally at a target 50 m away, then how far will the bullet drop? (1.2 cm) 21
4. HARD PROBLEM! I once saw Sammy Sosa hit a ball past the second house after Waveland. I think it was about 150 m from home plate. To get that much distance, I think he must have hit it at about a 45 angle. It seemed surreal, like the ball was in the air forever. Oh, you need to know this. If the ball was hit at a 45 angle and went 150 m horizontally, it went half that distance vertically. How long was the ball really in the air? (5.53 seconds ) 5. A boat crosses the Missouri river. If the river flows 3 m/s 30 south of east and the boat s water speed is due west at 5 m/s, what s the overall velocity of the boat? (add two vectors to find resultant) (2.8 m/s, 32 S of W) 22
Name: No: Projectile Motion A Do it Yourself Intro Lab Objective: The objective of this lab is to begin to qualitatively and quantitatively describe projectile motion. At the conclusion of the lab (which should occur before the bell rings) you should be able to: 1. Describe the path of a projectile 2. State the relationship between the horizontal velocity and the time of flight Materials: Physics Ball, Stopwatch, Pencil, Brain Procedure: Be sure to answer procedure questions clearly and completely using complete sentences! 1. Roll you physics ball off the edge of the table. 2. Through what part of the ball s motion is it considered to be a projectile? 3. Roll your physics ball of the edge of the table at three different speeds: slow, medium and fast. On the picture below, sketch the path of each of these balls. (Your picture should have three paths. Make sure you indicate which initial velocity is associated with which path.) 23
4. Using your stopwatch, determine how long the fast, the medium, and the slow ball are in the air. Record your data below: Horizontal Speed Trial 0 (dropped) Slow Moderate Fast Time to hit floor 1 Time to hit floor 2 Time to hit floor 3 Average Time 5. Compare the average time the ball was in the air for the fast, the medium, and the slow ball. (Are they the same? Are they different? Are they significantly different? Before you answer, think about the greatest sources of error in your experiment.) 6. Explain the relationship between the initial horizontal velocity, and the time that the ball is in the air. Support your explanation using the data you collected. (You collected it, now use it like a scientist!!) 7. Predict: Two identical balls are released from a platform simultaneously. One is dropped perfectly vertically, the other shot out perfectly horizontally. Using your knowledge of projectile motion, which one hits the ground first? Explain: 24
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Name: No:: The Buick Launcher A Projectile Lab Simulation Objective: Investigate the effects of initial velocity, launch angle and air resistance on the path of a projectile. Materials: computer, internet, pencil, paper, brain Procedure: 1. Open internet explorer and go to the following web address: http://www.colorado.edu/physics/phet/ 2. Click on Simulations at the top of the page 3. Click on Motion in the window on the left of the screen 4. Click on the Projectile Motion simulation 5. Familiarize yourself with the workings of this Java Applet. Try launching an adult human into the target area! Notice that you have access to a tape measure. You can move it around the screen, extend the tape out, and tilt it at any angle. You can zoom in and out If you mess up your simulation, say by launching a Buick at 1000m/s straight up, just click refresh at the top of your browser window. Answer the following questions in the space provided. Be sure the box for air resistance is unchecked (i.e. we are neglecting the effects of air resistance during this portion of the simulation). When measuring the maximum range and maximum elevation of your projectile, do so from the thin black reference line extending out from the cannon like so: Part I: Initial Velocity 1. Set your cannon to fire at an angle of 60 degrees. You can fire whatever projectile you want. 2. Note that the initial velocity that you set is the speed at which the projectile leaves the barrel of the cannon. 3. Measure the range along the horizontal black reference line rather than to the ground and use the Measuring Tape to measure the Maximum Elevation. Don t rely on the numbers at the top of the screen since they tell you what happens to the projectile as it returns to the ground. 27
Complete the following data table Initial Velocity Maximum Range Maximum Elevation Final Velocity (just before hitting the ground) Total Time (in the air) (m/s) (m) (m) (m/s) (s) 10 20 30 40 50 1. How do you know the final velocity of the projectile? 2. When the initial velocity of the projectile is increased, what other values increase? 3. Your cannon is aimed 10 degrees above the horizontal. Choose a single initial velocity from the table, and calculate the initial vertical, and the initial horizontal velocity. Show your work below. 4. Sketch the path of three projectile shot at the same angle, but at different starting speeds. Indicate which speed corresponds to which path. 28
Part II: Launch Angle Complete the following data table. Set your initial velocity to 20m/s for all of these trials. Again be sure to measure range along the horizontal black reference line. Angle of Maximum Maximum End Velocity Total Time Launch Range Elevation (degrees) (meters) (meters) (m/s) (s) 10 20 30 40 45 50 60 70 80 90 1. Which launch angle gives the biggest maximum range? 2. What angle reaches the highest maximum elevation? 3. What angle gives the longest hang time? 4. List three pairs of angles for which the maximum range was very similar: a. and b. and c. and 5. What is the sum of each pair of angles? 6. List three additional pairs of angles for which the maximum range would be the same that aren t already included in your data table: a. and b. and c. and 7. Launch a projectile at 56 with an initial speed of 50m/s. Use the simulation to determine the maximum range: 8. What other angle would give the same range: 9. Verify your answer to question 8 with the simulation. 10. What is the sum of 56 and your answer to question 8: 29
Part III: The Dreaded Air Resistance!!! 1. With the air resistance box UNCHECKED determine if mass of the projectile affect its motion. Does it? 2. Does the mass of the projectile affect its motion with air resistance checked? Complete the following data table: Air Resistance Initial Velocity Angle Maximum Range Maximum Elevation Total Time (m/s) (degrees) (meters) (meters) (s) Checked 30 35 Unchecked 30 35 Checked 30 65 Unchecked 30 65 Checked 60 35 Unchecked 60 35 Checked 60 65 Unchecked 60 65 Checked 60 90 Unchecked 60 90 3. How does air resistance effect each of the following: Maximum range: Maximum Height: Hang time: 4. Would you expect the final velocity to be bigger or smaller than the initial velocity? Why? 5. What is the drag coefficient? 6. List two objects you would expect to have very large drag coefficients: a. b. Summarize the effects of initial velocity, launch angle, and air resistance on projectile motion in the space below: 30
Names of group members: Projectile Bull s Eye Challenge (AKA Ball in a cup ) Objective: Accurately estimate where a steel ball will land using your physics knowledge. Groups: 3 to 4 people of your choosing Procedure: 1. Your group will get a unique position on a wooden ramps from which a steel ball will begin rolling downward and then across a flat table. In other words, your group may roll the ball from a mark half way up the ramp and the group next to you might roll it from ¾ th of the way up the ramp. 2. You get three trials with up to three timers at your disposal. The ball is not allowed to leave the table and you must find another way to estimate the ball speed. Note: Do you want an average speed, or do you want to know how fast the ball is rolling on the horizontal table? 3. Take note of where the ball appears it will leave the table. This will help you get the best possible left/right target position. 4. Once your calculations are done, sign up on the board that you are ready to do your challenge launch 5. You only get one launch. It s your job to let go of the ball smoothly and to pre-locate the target. Your target location will be marked on the target and points are awarded for how close to the Bull s Eye you get. 6. NO PASSES. Sign up early to make sure you re not late. If you run late and don t have time to do the challenge launch, turn in your calculations. By default, running out of time to try a Bull s Eye challenge means you will finish last in the Bull s Eye competition. Calculations: Show your work below. This should include your calculation of, Avg. speed and horizontal distance from the table edge you expect the ball to land. Avg. speed = m/s Equation (by itself, AKA step 3 ) used to find distance from table edge: Meters from table edge the ball is expected to land = m 31
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SHOOTIN MONKEYS A conservationist needs to capture a monkey to relocate it to an area where it will be safe. To do this, the conservationist must shoot a tranquilizer dart in the monkey. The instant the conservationist pulls the trigger, the monkey will see him and let go of his branch. Should the shooter aim high (gravity will pull on the dart, making the dart land lower than sited)? Aim low (since the monkey will be dropping)? Aim at the monkey? Assume the monkey is 30 degrees above horizontal relative to the shooter and that s where the shooter aims (at the monkey). In order to see if this approach hits the monkey, we must: 1. Determine how long it will take the projectile to reach the monkey 2. Determine where the projectile is when it gets to the monkey 3. Determine where the monkey is when the projectile gets to the monkey. 5.0 m/s m 2.0 m 1. How long will it take the projectile to reach the monkey s horizontal position? 33
2. How high in the air will the projectile be when it reaches the monkey s position? 3. How high in the air will the monkey be when the projectile reaches his horizontal position? Conclusion about shootin monkeys: 34
Lab: Range of a Stomp Rocket Intro: Our goal is to be able to predict where a stomp rocket will land given its muzzle speed and its angle of launch. ***This lab must be performed with launcher aimed at all times away from other people. If you disobey this rule, you and your group will receive a zero for the lab and a detention. Procedure: 1. Aim the rocket straight up and time its entire flight. total flight time- seconds peak flight time- seconds 2. Use a protractor to set your stomp rocket s launch angle at 45 degrees. 3. Launch the rocket and time its flight. total flight time- seconds 4. Determine the range of the rocket experimentally by pacing it out. 5. Use a protractor to set your stomp rocket s launch angle at 45 degrees. Launch the rocket and time its flight. 6. Change the angle of the launcher and launch the rocket, measuring the range to the nearest.1 meters each time. Do this for four distinctly different angles between 0 and 90 degrees. Try each angle twice and average the numbers. angle tried 90 45 Time 1 Time 2 Ave time Range 1 (paces) Range2 (paces) Average range (paces) 35
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Impetus, Parabolas, & Projectile Motion (4:54) 1. Whose famous 3 Laws of Motion explain the trajectories (paths) of any projectile? 2. Galileo first explained the proper trajectory of a projectile in terms of what two components of motion? 3. The secret to understanding projectile motion is to realize that two separate motions are occurring at the same time independent of each other. In which direction is there acceleration, and in which direction is the motion constant? Why does one component of projectile motion experience acceleration while the other component of motion remain constant? 4. The medieval explanation of projectile motion was based on the incorrect idea of impetus (not to be confused with impotence!) a projectile would travel through the air when given an initial impetus; sketch the path of a projectile fired from a cannon and explain what happened when the object ran out of impetus. 5. Galileo properly realized the proper path of a projectile is a parabola and it s horizontal motion is not due to impetus but to inertia an object s natural tendency to maintain its state of motion; sketch the proper path of a projectile fired from a cannon. Inertia, Galileo s Sailboat, & Flying Peanuts (3:00) 6. When a ball is dropped from a tower and allowed to fall and bounce, what does its path really look like? Why doesn t a ball dropped from a tower on the spinning surface of the earth end up in the next city as the earth spins out from underneath it? 7. What analogy did Galileo offer to his contemporaries to explain how a falling ball would move horizontally with the spinning earth? Aristotle believed that objects would naturally fall towards the center of the earth and come to rest what would happen if a ball was dropped from the mast of a moving sailboat according to Aristotle? What really happens? 37
8. If you accidentally dropped a bag of peanuts while flying on an airplane traveling at 500 mph, where would the peanuts land? How come the peanuts don t fly towards the back of the plane at 500 mph like deadly bullets when dropped? What if you poured your bag of peanuts outside the plane s window? Newton s Cannon, Zero g, & Lunar Orbits (2:24) 9. If g = -32 ft/s 2, how far does a cannonball in 1 sec? Show all your work below! 10. If an apple will free fall towards the surface of the earth, how come the moon doesn t fall and strike the surface of the earth? 11. Why is zero g a misnomer for objects in orbit around the moon? What really is happening to astronauts and objects inside a space craft orbiting around the moon? 38
Projectile Motion Problems 1) A stone is thrown horizontally at a speed of 10 m/s from the top of a cliff 78.4 m high. a) How long does it take the stone to reach the bottom of the cliff? 4 sec b) How far from the base of the cliff does the stone strike the ground? 40 m 2. A steel projectile is shot horizontally at 20.0 m/s from the top of a 49.0 m high tower. How far from the base of the tower does the projectile hit the ground? d h = 63.25 m 3. A steel ball with a constant velocity of 0.800 m/s rolls off the edge of a table. The table is 0.950 m high. How far from the edge of the table does the ball land? d h = 0.352 m 4. A person standing on a cliff throws a stone with a horizontal velocity of 15.0 m/s and the stone hits the ground 47 m from the base of the cliff. How far does the stone fall? How high is the cliff? d v = -48 m 5. A projectile is launched horizontally from the top of a building with a velocity of 12.7 m/s. At which height is the projectile launched if the projectile lands 15.0 meters from the side of the building? d v = - 6.8 m 6. An arrow is fired horizontally with a speed of 89 m/s directly at the bull s-eye of a target 60.0 m away. When it is fired, the arrow is 1.0 m above the ground. How far short of the target does it strike the ground? 20 m short! 7. A projectile is fired at such an angle from the horizontal that the vertical component of velocity is 49 m/s. The horizontal component of its velocity is 61 m/s. (a) How long does the projectile remain in the air? 10 sec (b) What horizontal distance does it travel? 610 m (c) What is the initial velocity of the projectile? 78.2 m/s at an angle of 38.8 8. A projectile is fired with a velocity of 196 m/s at an angle of 60 with the horizontal. Find: (a) the vertical velocity and the horizontal velocity of the projectile v v = 169.7 m/s; v h = 98 m/s (b) the time the projectile is in the air t total = 34.6 sec (c) the horizontal distance the projectile travels d h = 3390.8 m (d) the maximum height the projectile reaches d v = 1469.3 m 9. A B-52 bomber is flying horizontally at a speed of 250 mi/hr at an altitude of 8000 ft and is attempting to bomb an enemy ammunition factory. When should the B-52 release its bomb before reaching the target, right above the target, or past the target? Sketch the flight path and trajectory of the released bomb as it travels to its target where is the B-52 in relation to the bomb when it explodes? 10. While standing on an open bed of a truck moving at a constant velocity of 22 m/s, an archer sees a duck flying directly overhead. The archer shoot an arrow at the duck and misses. The arrow leaves the bow with a vertical velocity of 98 m/s. (a) For what time interval does the arrow remain in the air? t total = 20 sec (b) Where does the arrow finally land? (back in the truck!) (c) What horizontal distance does the arrow travel while it is in the air? d h = 440 m. 11. A golf ball is hit at an angle of 45 with the horizontal. If the initial velocity of the ball is 52 m/s, how far will it travel horizontally before striking the ground? t total = 7.51 sec / d h = 276.4 m 39
Projectile Practice Problems (an admirable alliteration) 1. An Indian is riding on a horse due north at 12 m/s. He spots game and aims his bow and arrow at it. His arrow is pointed 30 degrees E of N and it shoot an arrow at 50 m/s. What is the velocity of the arrow relative to the ground? 2. How fast must Roger throw his bottle of steroids at his friend Andy? Roger releases the bottle horizontally from a height of 2.0 m and it strikes Andy in the stomach at a height of 1.0 m. Andy is 8.0 m from Roger. 3. Paul kicks a football with a hang time of 4.0 seconds when he kicks it the farthest he can. How far can Paul kick a football? 4. Using your information from the previous problem, assuming Paul kicks just as hard (same speed), but this time at a 60 degree angle (to get a big hang time). How far would Paul kick this time? 40
Projectile Practice Problems II 1. An airplane flies into the Jetstream. The velocity of the plane within the air is 200 m/s 60 W of N and the Jetstream velocity is 80 m/s due South. What is the velocity of the plane relative to the ground? 2. Cara rolls a ball down a ramp onto a level (horizontal) table. The table is 1.0 m tall and the ball is carefully timed and found to go 1.40 m in 0.70 seconds. How far from the edge of the table top will the ball land? 3. Alfonso throws a ball to Jermaine. If Alfonso throws the ball 30 above horizontal and at a speed of 29.4 m/s then how far is Jermaine from Alfonso? 4. Randy Johnson s fastball was once timed at a record 102 mph (45.6 m/s). If there was no air resistance (a big IF, but let s say it anyway), then could Randy throw a ball from home plate to the center field wall (about 125 m away is typical)? (I want you to calculate how far he COULD throw) 41
Vector & projectile practice 1. Find the resultant for the displacement resulting from driving 20 km as shown below. Remember that the resultant is a vector and describe it as such (including units!) 8 km due east 2. A boat is crossing within the waters of a river moving at 6 m/s. The direction of the river and the velocity of boat relative to the river is shown below. Find the resultant velocity that would describe the motion of the boat relative to an observer on shore. 6 m/s 15 N. of E. 9 m/s 30 S. of E. Don t forget to do back side! 42
3. Chief Illiniwek shoots an arrow horizontally from a height of 1.5 m. If his arrow goes 60 m before hitting the ground. How fast was the arrow shot? 4. If a major league pitcher throws a 90 mph fastball horizontally (40 m/s), and home plate is 18.4 m away, then how high must he release the ball to make it to home plate on the fly? 43
Projectile motion, Vector addition (finding resultants), Angle projectile strikes ground Do as many as you can. Sign up on the board for help if you want me to help you 1. Two people are playing catch on a luxury liner moving in calm waters. If someone throws a ball 30 E of N at 12 m/s and the boat is moving due South at 9 m/s then what vector describes the velocity of the ball? 2. A ball is thrown horizontally from a height of 1.2 m toward first base from 15 m away. The first baseman has to dig it out of the dirt. How fast was the ball thrown? 3. A golfer standing on a 4 m tall platform hits a ball horizontally at 40 m/s. How far will the ball travel in the air? 4. On an alien planet, a dart is thrown horizontally and lands on a target 2.5 m away. The dart was released from a height of 2.0 m and the target is 1.4 m above the ground. If the dart was thrown at 10 m/s then what is the acceleration vector on this planet? 44
5. Assume a javelin design requires an angle of at least 30 in order to stick in the ground (smaller angles will bounce). If a javelin strikes the ground with v x = 30 m/s and v y = -25 m/s then a. At what angle will the javelin strike the ground? b. One of these velocity components represents final velocity and the other represents constant velocity, which one represents which? 6. A projectile is launched at an angle such that v i is +29.4 m/s. If v x is + 10 m/s, then how far does the projectile land from its launch point? (hint: by symmetry you know that if v i is + 29.4 m/s on the way up, what must v f be on the way DOWN) 7. A projectile is launched at an angle such that v f is -19.6 m/s. If v x is + 12 m/s, then how far does the projectile land from its launch point? (hint: see hint in previous problem!) 8. Billy got a new remote control plane and he decides to try it outside on a warm, windy day. The direction of the wind is 20 N of E and it s blowing at 10 m/s. His airplane is pointed due west and is flying at 15 m/s. What is the velocity vector of Billy s plane relative to the ground? 9. A soldier is moving in a tank due North at 16 m/s. He throws a grenade manually from the moving tank at 20 m/s and 30 W of N. What vector describes the velocity of his grenade? 45
Sports Physics- Vectors and Projectiles Review 1. What is a projectile? 2. What is a vector? 3. What is a scalar? 4. Give two examples of vectors. 5. Give two examples of scalars. 6. What is the resultant? 7. What two things do you have to tell when you report the resultant? 8. What is the resultant if you end up where you started? 9. How do you draw the angle 30 degrees north of west? 10. If you use the graphical method, what two tools do you use? 11. What tools do you use if you try the component method? 12. How can you find the horizontal and vertical components of the initial velocity given the initial speed and the direction it s being projected? 13. What is the acceleration in the x direction for a projectile? 14. What is the acceleration in the y direction for a projectile? 15. What are some examples of hidden data? 16. What is the only equation which has v ix in it? 17. What is the difference between a spring board and a platform diving board? 18. If the object starts and ends on the ground, what data do we know? 19. If you have an equation with two unknowns in it, what do you do? 20. What is the vertical acceleration of a projectile at all points in its flight? 21. What is the vertical acceleration of a projectile at the peak of its flight? 22. If you know the time of flight of a punted football and its initial horizontal speed, how could you find the range? 23. If you know the angle of the punt and the initial speed along that angle, how could you find the range? 24. For the above question, how could you find the maximum height achieved? 25. When can you say peak time is half the total flight time? 26. When is y = 0? 27. When is v iy = 0? 28. When is v fy = 0? 29. If you know v ix how can you find v fx? 30. How do you add vectors: head to head, tail to tail, or head to tail? 31. Does the resultant arrow go from the beginning of the first direction to the end of the last direction, or vice versa? 32. If a cannonball is fired horizontally off a cliff, what do you need to know to determine where it lands? 33. Why did we fire the stomp rockets straight up for the first trial? 34. What data did you need before you could predict where the ball bearing should land in the projectile challenge? 35. How can you use hang time to determine vertical jump? 36. Does vertical motion affect horizontal motion? 37. Does horizontal motion affect vertical motion? 38. If you know the x and y components of the initial velocity, how can you get the launch angle and launch speed? 46
Option 1: Physics of Sports Project: projectile launcher! Idea: You are a carpenter during the Middle Ages. The King requests all carpenters to build him a miniature scale model of a boulder launcher to knock down fortress walls. The King sets up a competition to find the most accurate and powerful launcher. Your team must build from scratch a device that will launch a small & safe ping pong projectile into the air to hit a target (about 1/2 x ½ x 1/2 m) at distance set by you. Your score depends both on accuracy and total distance, so you want to get it to go far but still hit the target! Read the grade rubric & rules! Rules: 1. It s a trebuchet not a catapult! So no elastics, springs, etc. A falling weight makes it fly! 2. No kits. They re not very good. You will have better ideas! Total cost should be < $40 3. Keep it small: - no larger than 50 x 50 x 50cm in size when in ready-to-fire position 4. Do it safely- -- follow all school rules. If in doubt, discuss with your teacher before building! 5. Work together! Teams up to 4. Your grade can be lowered by your teammates if they agree you did less than your share 6. Have a data table. Vary one factor (falling weight, angle, string length etc) and see how that affects your target distance. 7. No cheating. Once you announce your target distance, no changes. Launcher must stay on start line. 8. Keep it real. You re allowed one touch to release your trebuchet. The falling weight must be the only power source. 9. Don t forget the data! Measure hang time and target distance for your best projectile launch. 10. Do the calculations: for your best projectile launch, find it s horizontal velocity (V x ), initial vertical velocity (V Y ), max height ( y), total launching velocity (V T ), and angle of launch ( ). Scoring (counts at 25% of grade along w/calculations, workmanship, spirit) Score = Target Distance 2* actual distance- target distance + 10% bonus if hit Ex.: you shoot for 6m. it only goes 4m. no bonus. Score= 6-2 6-4 +0 = 2 Ex : you shoot for 4m and hit it! Bonus! Score= 4-4-4 + 0 1*4 = 44 47
Projectile Physics Project: Grade Sheet Trebuchet Name: Group Members: Scoring Score = target distance 2* actual distance- target distance + 10% bonus if hit Ex.: you shoot for 6m. it only goes 4m. no bonus. Score= 6-2 6-4 +0 = 2 Accuracy & Precision 10 pts You will do three launches in the physics room from at least 2.5 m away. You will be given the exact distance at launch time (you need to know how to adjust to different target lengths). The target will be a bucket (or something similar, trash can, etc.). You will get NO warm-up launches after being given your target distance. Accuracy points are as follows: 10 pts hit target on the fly all three launches 9 pts hit target on the fly once and come within 1 m on other two 8 pts come within 1 m of hitting the target at least twice 7 pts shoot ping pong at least 1 m all three times 6 pts shot ping pong at least 1 m once Workmanship 10 pts 10 pts rugged, durable, well-decorated, solidly built (not fragile), steady and does not wobble easily 0-9 pts and less: lacking some/all of the above characteristics to varying degrees Physics 10 pts Turn in the following sheets: 1. Hand drawn sketch (this is not art class, but must be reasonable) of your launcher including name of launcher and names of participants. 2. Data table showing various trial runs including the factor that was varied. Trial runs are numbered to make referncing them easy. You should have at least 10 trial runs where your adjusted variable (see #6 in rules) was changed to see its effect. 3. Hang time calculation (assume no air resitance). This will include measuring, relative to the floor, the height of release point and maximum height for one of your typical trial lanuches. 4. Using the same trial as above, find V x. Spirit/Rules/Effort 10 pts 1. Time given in class was used for this project and/or physics: All members used time in class wisely to work on project (did not over socialize, work on Spanish, say I did it at home, etc.) 2. All members contributed to project.as agreed upon by fellow group members (your group is satisfied with your contribution) 3. Group was positive and got along with each other and rest of class. 4. Launcher had a cool name, theme and so did group TOTAL POINTS : (40 possible) Grade Percentage: Comments: 48
Physics of Sports Project (Option 2): Digital Video Analysis of Sports Science Goal / Description: This project is like the lab done in class, but now it s personal! The goal is to apply physics to your own sports video to help coach and compare yourself to a world class athlete. You ll turn in a report with numbers analyzed off the video and give coaching advice based on your research and numbers taken from professional athletes. But you must follow the steps below to do it right and make it fun! Lights, Camera, Action (raw video clip due by Monday, March 23, 2009 via flash card, disc (no emails that large!)) 1. Pick an athlete in your group to perform a projectile stunt. The stunt should last about 1-5 seconds and have a trajectory. 2. Find a digital camera or digital camcorder to use in class only on video day. (Bring your own props & equipment!) 3. Point the camera so it s 90 degrees (perpendicular) to the athlete s motion (this avoids angle distortion of the distances!) 4. Have a background object of known height so you can later scale the video to the correct distances. 5. It s your responsibility to make a file that will project on the screen for everyone to see. You will use Tracker for this. I have a $120 camera that makes perfectly fine AVI video files that I have used with Tracker dozens of times without fail. I can try to help you with video capture and other technical problems, but it s not my responsibility to make sure your project works. If you re having technical difficulties, be prepared to spend some time after school, waiting along with other students for help. With that said, if you meet your deadlines and show me that your project works on MY computer and the room 187 projector ahead of time, there will be no penalty if it fails to work on presentation day. 6. Find some video footage of a professional athlete doing a similar motion to analyze digitize if possible for analysis so you can compare your athlete to that of a pro! Video Analysis: To the whole class on presentation day 1. Show both videos 2. Use Tracker to find meaningful motion variables that affect performance of the sport being analyzed 3. Provide a consultation on how the amateur might change something to perform more like the pro, or better in some way include how Tracker data helps make this clear. Try to provide at least 3 suggestions and provide a detailed analysis on at least one of those suggestions. 4. Do a physics NIFTY calculation pertaining to the analysis (confirm Tracker data, do a what if scenario regarding your suggest amateur s change, etc.) 5. Show research as to your coach s tips (at least three sources with citations) 49
Grading: Category Description Pts Preparation Project time provided was spent wisely in class on physics. Team showed spirit and got along. All members contributed in a mutually satisfactory way. Raw video clip was handed in on time and if problems were found, students worked diligently to solve them taking responsibility for finding a solution. 10 pts Presentation Presentation went smoothly and did not exceed 5 minutes 10 pts Analysis was meaningful, audience was involved and interested, positive energy and comments were made throughout Video files were provided the day before and tested on YOUR TEACHER S computer and projector (not a similar computer belonging to another teacher or a projector that should work the same). Physics Physics was used to solve the problem 10 pts Meaningful physics was done (not trite) No physics errors were made Report Paper work was neat and organized (in the same order as follows!) and included 10 pts 1. Athlete improvement summary a. Three meaningful suggestions for improvement were made b. At least one of the suggestions included a detailed analysis including motion analysis data to support it c. Three sources were cited regarding coaching tips d. A professional athlete was used as a comparison 2. A NIFTY calculation was done pertaining to analysis (confirm Tracker data, do a what if calculation or other related item) Total 40 pts 50
Vocabulary: previous vocabulary scale above (or below) the horizon pythagorean theorem ("use pythagorus") soh-cah-toa parabolic path resolve resultant magnitude direction horizontal component (of a vector) vertical component (of a vector) horizon ground speed air speed wind speed range (Δx) Unit 03 Vocabulary and Equations Vectors & Projectile Motion Symbols: Δ, x, v, t, Δx, Δv, Δt, a, f, i Equations & constants: You get these on test: Δx Δv v = a = Δt Δt Δx = v 0 Δt + ½ at 2 v = v 0 + a Δt (v means v f ), 51 v f 2 = v i 2 + 2a Δx 60 mph = 27 m/s; 60 seconds = 1 min.; 60 min = 1hr. You don't get these on the test: v x = v cosɵ Δx = v x t v i = v sinɵ Δy = v y t Tneom chart Unit Objectives - Williams 1. I understand all the vocabulary & math of this unit and all demos, videos, equations, and class assignments 2. I remember objectives & vocabulary from previous units. 3. I can scale things, use scales with representations like maps, and use arrows and length to represent vectors correctly 4. I can add and subtract vectors both mathematically and graphically using head-to-tail or parallelograms 5. I can use vector addition to find relative motions such as boats relative to shore or plane relative to ground 6. I can apply trig to resolve vectors and come up with resultants always using "degree mode" 7. I know which measurement types are vectors & which are scalars and the product of vector and scalar is a vector 8. I understand and apply trig to the individual problem and how it is depicted. For example: v x = v cosɵ is true only where Ɵ represents the angle relative to the horizon. 9. I know projectile motion assumptions, can contrast it with terminal velocity/freefall & know conceptually how air resistance or external forces beyond gravity alone would change the motion 10. I know for projectiles a y and v x are constant, a x is zero, v f = -v i, and max height is when v y = 0, but it doesn't "stop" 11. I know and can apply independence of X and Y motion to solve numeric and conceptual problems 12. I am aware of a projectile's constant X motion and constantly changing Y motion with time tying them together 13. I can make, explain, understand & use initial & final motion triangles for a projectile showing v, v i, v x,ɵ 14. I know angles that maximize range, hang time, maximum height, etc. & complementary angles have the same ranges 15. I know and can apply both the mapping and horizon methods for vectors 16. I can solve horizontal projectile problems 17. I can solve problems involving projectiles launched at angles DuPage ROE Objectives 101. I can distinguish between scalar and vector quantities. 102. I can differentiate between accelerated and constant velocity motion. 103. I can describe and analyze motion based on graphs, numeric data, words, and diagrams. 104. I can differentiate between speeding up, slowing down, and change in direction, based on the direction of velocity and acceleration. 105. I can recognize the independence of X and Y variables in 2-dimension problems. 106. I can determine the range of a horizontally launched projectile given initial launch conditions. 107. I can justify that if the only force acting on an object is gravity, it will have the same constant downward acceleration regardless of mass, velocity or position. 108. I can apply the various kinematics equations in one and two dimension.