CH 03 HWDue: 11:59pm on Sunday, September 25, 2016 You will receive no credit for items you complete after the assignment is due. Grading Policy Tactics Box 3.1 Subtracting Vectors Learning Goal: To practice Tactics Box 3.1 Subtracting vectors Vector subtraction has some similarities to the subtraction of two scalar quantities. With numbers, subtraction is the same as the addition of a negative number. For example, 5 − 3 is the same as 5 + (−3). Similarly, A⃗ − B⃗ = ⃗ ⃗ A + (−B) . We can use the rules for vector addition and the fact that −B⃗ is a vector opposite in direction to B⃗ to form rules for vector subtraction, as explained in this Tactics Box. TACTICS BOX 3.1 Subtracting vectors To subtract B⃗ from A⃗ perform these steps: 1. Draw A⃗ . 2. Place the tail of −B⃗ at the tip of A⃗ . 3. Draw an arrow from the tail of A⃗ to the tip of −B⃗ . This is vector A⃗ − B⃗ . Part A Find vector C ⃗ = ANSWER: ⃗ ⃗ A −B by following the steps in the Tactics Box above. Make certain to draw −B⃗ with the correct orientation. Part B Find vector F ⃗ = ⃗ ⃗ D−E by following the steps in the Tactics Box above. Make certain to draw −E⃗ with the correct orientation. ANSWER: Tactics Box 3.2 Finding the Acceleration Vector Learning Goal: To practice Tactics Box 3.2 Finding the acceleration vector. Suppose an object has an initial velocity v i⃗ at time ti and later, at time tf , has velocity v f⃗ . The fact that the velocity changes tells us the object undergoes an acceleration during the time interval Δt = tf − ti . From the definition of acceleration, a⃗ = vf⃗ −vi⃗ t f −t i = Δv ⃗ Δt , ⃗ we see that the acceleration vector points in the same direction as the vector Δv . This vector is the change in the velocity Δv ⃗ = v f⃗ − v i⃗ , so to know which way the acceleration vector points, we have to perform the vector subtraction v f⃗ − v i⃗ . This Tactics Box shows how to use vector subtraction to find the acceleration vector. t1 . The dots show the position of the object at three subsequent instants. To find the acceleration between velocity v i⃗ and velocity v f⃗ follow these steps: 1. Return to the original motion diagram. Draw Δv ⃗ = v f⃗ − v i⃗ = v f⃗ + (−v i⃗ ). Draw the vectors starting at the appropriate black dots. The location. Draw the velocity vector v f⃗ . ⃗ 4. This is the direction of a⃗ .TACTICS BOX 3. and length of the vectors will be graded. Draw a vector at the middle point in the direction of Δv ; label it a⃗ . (Tip for vector drawing tool: You may find it helpful to first draw −v i⃗ along v i⃗ . Then drag −v i⃗ to the tip of v f⃗ . Part A Below is a motion diagram for an object that moves along a curved path. ANSWER: Part B . and the final time interval. from the tip of v i⃗ to the tail of v i⃗ . Draw the acceleration vector a⃗ representing the change in average velocity of the object during the total time interval Δt = t 3 − t 1 .2 Finding the acceleration vector. The vectors v i⃗ and v f⃗ show the average velocity of the object for the initial time interval. respectively.) 3. Draw −v i⃗ at the tip of v f⃗ . Δt f = t 3 − t 2 . Δti = t2 − t1 . t2 . to get the correct length and angle. orientation. 2. and t3 . This is the average acceleration at the midpoint between v i⃗ and v f⃗ . Tactics Box 3. The sign of Ax is positive if A⃗ x points in the positive x direction; it is negative if A⃗ x points in the negative x direction.3 Determining the components of a vector 1. Part A What is the magnitude of the component vector A⃗ x shown in the figure? Express your answer in meters to two significant figures. denoted Ax and Ay .This question will be shown after you complete previous question(s).3 Determining the Components of a Vector Learning Goal: To practice Tactics Box 3. ANSWER: |A x | = m Part B What is the sign of the y component Ay of vector A⃗ shown in the figure? ANSWER: positive negative . This Tactics Box describes how to determine the x component and y component of vector A⃗ . 2. When a vector A⃗ is decomposed into component vectors A⃗ x and A⃗ y parallel to the coordinate axes. we can describe each component vector with a single number (a scalar) called the component.3 Determining the component of a vector. The absolute value |Ax | of the x component Ax is the magnitude of the component vector A⃗ x . TACTICS BOX 3. 3. The y component Ay is determined similarly. overlap them. Vector Addition Ranking Task Six vectors (a⃗ through f ⃗ ) have the magnitudes and directions indicated in the figure.Part C This question will be shown after you complete previous question(s). Part A Rank the vector combinations on the basis of their magnitude. Part B ⃗ f + c ⃗ d Help ⃗ smallest . ANSWER: Reset ⃗ a⃗ + b a⃗ + c ⃗ ⃗ a⃗ + d a⃗ + e ⃗ largest The correct ranking cannot be determined. Rank from largest to smallest. To rank items as equivalent. You did not open hints for this part. labeled A⃗ through D⃗ . unless otherwise specified. ANSWER: . with coordinate axes x and y . overlap them.Rank the vector combinations on the basis of their angle. Part A What is the x component of A⃗ ? Express your answer to two significant figures. You did not open hints for this part. Rank from largest to smallest. Drawn on this grid are four vectors. measured counterclockwise from the positive x axis. The grid runs from 5 to 5 on both axes. Components of Vectors Shown is a 10 by 10 grid. ANSWER: Reset ⃗ a⃗ + b a⃗ + c ⃗ ⃗ a⃗ + d a⃗ + e ⃗ largest ⃗ f + c ⃗ d Help ⃗ smallest The correct ranking cannot be determined. This problem will ask you various questions about these vectors. All answers should be in decimal notation. All angle measures fall between 0∘ and 360∘ . To rank items as equivalent. Vectors parallel to the positive x axis have an angle of 0∘ . You did not open hints for this part. You did not open hints for this part. ANSWER: By = Part D What is the x component of C ⃗ ? Express your answer to the nearest integer. ANSWER: Cx = The following questions will ask you to give both components of vectors using the ordered pairs method. The answers below are all integers. By = Part F In ordered pair notation. ANSWER: . write down the components of vector B⃗ . In this method. followed by a comma. the components of A⃗ would be written 2.Ax = Part B What is the y component of A⃗ ? Express your answer to the nearest integer. Part E In ordered pair notation.5. ANSWER: Ay = Part C What is the y component of B⃗ ? Express your answer to the nearest integer.3 in ordered pair notation. Express your answers to the nearest integer. You did not open hints for this part. the x component is written first. For example. Express your answers to the nearest integer. write down the components of vector D⃗ . and then the y component. ANSWER: Bx . so estimate the components to the nearest whole number. x(t). sketch the shape of the corresponding motion graphs in Parts A to D. Part A Construct a possible graph for x position versus time. ANSWER: They have different components and are not the same vectors. You did not open hints for this part. One unit of time elapses between consecutive dots in the motion diagram. They have the same components but are not the same vectors. Dy = Part G What is true about B⃗ and D⃗ ? Choose from the pulldown list below.Dx . Use the indicated coordinate system. They are the same vectors. ANSWER: . Graphing Projectile Motion For the motion diagram given . ANSWER: Part C Construct a possible graph for the x velocity versus time. v x (t).Part B Construct a possible graph for the y position versus time. You did not open hints for this part. You did not open hints for this part. ANSWER: . y(t). Part D Construct a possible graph for the y velocity versus time. A ball thrown horizontally with speed v i = 25. ANSWER: PSS 3.0 m/s travels a horizontal distance of d = 51. From what height h was the ball thrown? PROBLEMSOLVING STRATEGY 3.1 Projectile motion problems.0 m before hitting the ground.1 Projectile motion problems . v y (t).1 Projectile motion problems Learning Goal: To practice ProblemSolving Strategy 3. You did not open hints for this part. ANSWER: Projectile motion is the motion of an object that moves in two dimensions under the influence of gravity and air resistance. Second. Define symbols and make a list of known values. SOLVE There are two sets of kinematics equations for projectile motion. Part B This question will be shown after you complete previous question(s). is reasonable. Part E This question will be shown after you complete previous question(s). Establish a coordinate system with the x axis horizontal and the y axis vertical. Projectile motion is made up of two independent motions: uniform motion at constant velocity in the horizontal direction and freefall motion in the vertical direction. You know that the horizontal acceleration will be zero and the vertical acceleration will be freefall acceleration: ax = 0 and ay = −g. you can find Δt by solving for the vertical or horizontal component of the motion and then using that value to complete the solution for the other component. they are connected through the time Δt that the object is in the air. PREPARE First. make appropriate simplifying assumptions. the motion will be the same. one for the horizontal component and one for the vertical: Horizontal x f = x i + (vx ) Δt i (vx ) f = (vx ) = constant i Vertical y f = y + (vy ) Δt − i i (vy ) f 1 2 2 g(Δt) = (vy ) − gΔt i Recognizing that Δt is the same for the horizontal and vertical components of the motion. make a list of known variables and identify what the problem is asking you to find. Part A When setting up projectile motion questions. draw a visual overview and establish a coordinate system. Although the horizontal and vertical motions of a projectile are independent. Last. PREPARE There are a number of steps that you should go through in setting up the solution to a projectile motion problem: Make simplifying assumptions. Identify what the problem is asking you to find. Part D This question will be shown after you complete previous question(s). and answers the question. which of the following characteristics of projectile motion are important to keep in mind? Check all that apply. ASSESS Check that your result has the correct units. Projectile Motion Ranking Task θ .We can solve projectile motion problems by considering the horizontal and vertical motions as separate but related problems. Draw a visual overview including a pictorial representation showing the beginning and ending points of the motion. Part C This question will be shown after you complete previous question(s). Whether the projectile is a car or a basketball. and measure its range and hang time (the amount of time in the air). use the PhET simulation Projectile Motion. For this problem. PhET Tutorial: Projectile Motion Learning Goal: To understand how the trajectory of an object depends on its initial velocity. In all cases. In each case. Part A Rank these throws based on the maximum height reached by the ball. This simulation allows you to fire an object from a cannon. Rank from largest to smallest.Six baseball throws are shown below. see its trajectory. the baseball is thrown from the same height H above the ground. Part B This question will be shown after you complete previous question(s). ANSWER: Reset largest Help smallest The correct ranking cannot be determined. To rank items as equivalent. the ball is thrown with speed v at an angle θ from the horizontal. . You did not open hints for this part. overlap them. Assume for the basis of these rankings that the effects of air resistance are negligible. and to understand how air resistance affects the trajectory. and g is the acceleration due to gravity. When you are done. click and drag on it or type in a numerical value (in degrees). Shoot the baseball straight upward (at an angle of 90∘ ) with an initial speed of 20 m/s. ANSWER: Part B This question will be shown after you complete previous question(s). How long does it take for the baseball to hit the ground? Express your answer with the appropriate units. Press Fire to launch an object. Leave Air Resistance unchecked. You can also adjust the speed. Play around with the simulation. mass.Start the simulation. v 0 is the initial speed. we will use an altitude of zero (sea level) and let the drag coefficient be automatically set when the object is chosen. You can choose the object by clicking on one of the objects in the scrolldown menu at top right (a cannonball is not among the choices). Part D This question will be shown after you complete previous question(s). For this tutorial. Part C This question will be shown after you complete previous question(s).2 m above the ground due to the wheels of the cannon). click Erase and select a baseball prior to beginning Part A. 0 where y0 = 0 is the initial position (which is 1. you will investigate purely vertical motion. Part E This question will be shown after you complete previous question(s). . The kinematics equation for vertical motion (ignoring air resistance) is given by 2 y(t) = y + v 0 t − (1/2)gt . and diameter of the object by typing in values. Part A First. To adjust the cannon barrel’s angle. Clicking Air Resistance displays settings for (1) the drag coefficient and (2) the altitude (which controls the air density). Drag the terms on the left to the appropriate blanks on the right to complete the sentences. you'll likely find that the beginner's slope has the smallest angle θ between the horizontal and the inclined surface. Part J This question will be shown after you complete previous question(s). Part K This question will be shown after you complete previous question(s).edu Conceptual Question 3. Part A Use the concept of acceleration on a ramp to explain why this is so. ANSWER: . Part G This question will be shown after you complete previous question(s). Part I This question will be shown after you complete previous question(s).Part F This question will be shown after you complete previous question(s). Part L This question will be shown after you complete previous question(s).11 If you go to a ski area.colorado. Part M This question will be shown after you complete previous question(s). Part H This question will be shown after you complete previous question(s). PhET Interactive Simulations University of Colorado http://phet. g cos θ higher The lower the angle of the slope. then you are accelerating at the lowest highest acceleration which is safe. you should keep your car's acceleration below some safe upper limit. therefore. the control will be better. our speed . if we take a turn with a radius which is four times smaller. the speed at the bottom of a slope will be . consequently. To turn safely. your car follows a path that is a segment of a circle. we have: . Now if the radius is reduced. . turn in a circle with a smaller radiushow should you adjust your speed? Drag the terms on the left to the appropriate blanks on the right to complete the sentences. 2 a = v r If you are traveling at the greatest speed which is safe. lower Conceptual Question 3. for v .1 Part A Draw the vector D⃗ = ⃗ ⃗ A +B . this tends to the acceleration above safe values. 2 a = v /r If we solve the formula for centripetal acceleration. ANSWER: Reset reduce To make a tighter turn with a smaller radius. your speed should be . Part A If you want to make a "tighter" turn that is. To bring it back down. . and.17 When you go around a corner in your car. you need to Help your speed.Reset g sin θ The acceleration down the slope is given by Help if the friction is neglected. we need to in 2 times increase increased reduced − − − v = √ a/r in 4 times Problem 3. − − v = √ar So. the the acceleration along the ramp. ANSWER: Part B Draw the vector C ⃗ = ANSWER: ⃗ ⃗ A −B . . 45 ∘ left of −yaxis). ANSWER: = d m . Enter the x and y components of the vector separated by a comma. m/s Part A How far will it coast before starting to roll back down? Express your answer using two significant figures. Enter the x and y components of the vector separated by a comma. cm/s Express your answer using two significant figures. ANSWER: v || = m/s Problem 3. Enter the x and y components of the vector separated by a comma. ANSWER: a⃗ x = m/s2 Problem 3.9 A cannon tilted upward at θ = 33 ∘ fires a cannonball with a speed of 95 . m/s Part A At that instant.0 ∘ slope. what is the component of the cannonball’s velocity parallel to the ground? Express your answer using two significant figures. Express your answer using two significant figures.12 Part A Find the x and ycomponents of the vector d ⃗ = (7. ANSWER: cm/s = v ⃗ Part C Find the x and ycomponents of the vector a⃗ = (12 2 m/s .0 km . 32 ∘ left of +yaxis). ANSWER: d km ⃗ = Part B ⃗ Find the x and ycomponents of the vector v = (9.0 .17 A car traveling at 24 runs out of gas while traveling up a 7. −xdirection). Express your answer using two significant figures.Problem 3. what is the speed of the first ball? Express your answer using two significant figures. what is its final speed in m/s? Express your answer using two significant figures. Assume that the track begins with a 57f tlong (1 m = 3.28 f t) section tilted 16 ∘ below horizontal. young participants build nonmotorized cars with very lowfriction wheels. Cars race by rolling down a hill.21 Anita is running to the right at 5. Balls 1 and 2 thrown toward her at 10 m/s Part A According to Anita.18 In the Soapbox Derby in . as shown in the figure. ANSWER: v m/s = Problem 3. ANSWER: = a Part B If a car starts from rest and undergoes this acceleration for the full l. Part A What is the maximum possible acceleration of a car moving down this stretch of track? Express your answer to two significant figures and include the appropriate units.Problem 3. ANSWER: by friends standing on the ground.0 . m/s . 25 A boat takes 4.23 Anita is running to the right at 5 m/s . with what speed was ball 1 thrown? Express your answer to two significant figures and include the appropriate units. ANSWER: v2 = Problem 3. then 6.0 Part A How fast is the river flowing? Express your answer using two significant figures. Part A According to her friends. ANSWER: h to return.v1 m/s = Part B According to Anita. ANSWER: v1 = Part B According to her friends. . According to Anita. what is the speed of the second ball? Express your answer using two significant figures. ANSWER: v2 m/s = Problem 3. with what speed was ball 2 thrown? Express your answer to two significant figures and include the appropriate units. as shown in . Balls 1 and 2 are thrown toward her by friends standing on the ground. both balls are approaching her at 18 m/s .0 h to travel 33 km down a river. 00 m/s above the floor. m Part A How long will it take the ball to hit the floor? Express your answer with the appropriate units. water in the Niagara River is moving horizontally at 4. After moving over the edge. ANSWER: t = s Part B How far from a point on the floor directly below the edge of the bench will the ball land? Express your answer with the appropriate units.25 rolls off a bench 1.vw = km/h Problem 3. ANSWER: Δt = Part B How far does the water move horizontally during this time? Express your answer to two significant figures and include the appropriate units. how much time does it take for the water to go from the top of the falls to the bottom? Express your answer to two significant figures and include the appropriate units.33 A gray kangaroo can bound across a flat stretch of ground with each jump carrying it 9. .30 On a day when the water is flowing relatively gently.5 m/s before shooting over Niagara Falls.28 A ball with a horizontal speed of 1.0 from the takeoff point. Part A If we ignore air resistance. ANSWER: = d Problem 3. m Part A If the kangaroo leaves the ground at a 21 ∘ angle. ANSWER: = d m Problem 3. the water drops 53 m to the water below. what is its takeoff speed ? Express your answer to two significant figures and include the appropriate units. .42 Entrance and exit ramps for freeways in are often circular stretches of road. in what is now referred to as "Brady's Leap.63 In 1780.ANSWER: v = Part B What is its horizontal speed? Express your answer to two significant figures and include the appropriate units. circular turn can attain a centripetal acceleration 1. you will experience a constant acceleration. Continental Army escaped certain death from his enemies by running over the edge of the cliff above Ohio's Cuyahoga River in .1 m). It was reported that he leapt 22 f t (≈ 6. Part A If the falcon is flying at 20 . Suppose you drive through an entrance ramp at a modest speed and your acceleration is 3.5 m/s2 . Part A What will be the acceleration if you triple your speed? Express your answer to two significant figures and include the appropriate units.7 m) across while falling 20 f t (≈ 6. ANSWER: vx = Problem 3. ANSWER: = r m Problem 3.S. what is the radius of the turn? m/s Express your answer using two significant figures. He landed safely on the far side of the river.5 times the freefall acceleration. ANSWER: = a Problem 3.43 A peregrine falcon in a tight. which is confined at that spot to a gorge. As you go around one at a constant speed." Captain Sam Brady of the U. ANSWER: Δx = Part B The pilot looks down at the weight after she drops it. the obtained speed is greater than the worldrecord. dropping heavy weights (for which air resistance can be ignored) from their lowflying planes and scoring points for having the weights land close to a target. No. A plane 55 m above the ground is flying directly toward a target at 51 m/s . is it reasonable to expect Brady to be able to run fast enough to achieve Brady's leap? ANSWER: Yes. Where is the plane located at the instant the weight hits the ground? ANSWER: . Part A At what distance from the target should the pilot drop the weight? Express your answer to two significant figures and include the appropriate units. Yes. Given this.66 Smallplane pilots regularly compete in "message drop" competitions. ANSWER: v = Part B The worldrecord time for the 100 m dash is approximately 10 s. the obtained speed is almost equal to the worldrecord. the obtained speed is almost equal to the worldrecord. the obtained speed is less than the worldrecord. No.Part A What is the minimum speed with which he’d need to run off the edge of the cliff to make it safely to the far side of the river? Express your answer to two significant figures and include the appropriate units. Problem 3. ANSWER: xmax = Problem 3. a rider in one swing is moving at 32 m/s with respect to the ground in a 50mdiameter circle.Suppose you could train a dolphin to launch itself out of the water at this same speed but at an angle.0%. how fast do the riders move with respect to each other? Express your answer to two significant figures and include the appropriate units. It's actually two swings moving in opposite directions. You received 0 out of a possible total of 10 points. Part A What is the acceleration. that riders experience? Express your answer using two significant figures. but in the exact opposite direction. in m/s2 . The rider in the other swing is moving in a similar circle at the same speed. not yet over the target directly over the target past the target not enough information to determine Problem 3.76 The "Screaming Swing" is a carnival ride that is not surprisingly a giant swing.0 m straight up from the surface of the wateran impressive feat. as they pass each other. ANSWER: = a m/s2 Part B What is the acceleration. At the bottom of its arc.68 Trained dolphins are capable of a vertical leap of 7. ANSWER: = a g Part C At the bottom of the ride. ANSWER: v = Score Summary: Your score on this assignment is 0. that riders experience? Express your answer using two significant figures. in units of g. . Part A What maximum horizontal range could the dolphin achieve? Express your answer to two significant figures and include the appropriate units.