Nelson Physics 12 PDF Past Paper
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Al Hirsch, Charles Stewart, David Martindale, Maurice Barry
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This Nelson Physics 12 document is a comprehensive learning resource for students studying physics in high school. It covers various physics concepts, including kinematics, dynamics, energy, momentum, and waves. The document includes problem-solving examples and emphasizes the independent learning and problem-solving skills essential for success in post-secondary studies.
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Nelson Physics 12 Program Goals 1 To ensure a thorough and comprehensible...
Nelson Physics 12 Program Goals 1 To ensure a thorough and comprehensible treatment of all content and process expectations in the SPH4U curriculum Complex physics concepts are presented in a clear, understandable fashion Important concepts, such as static equilibrium, are treated in greater depth than specified in the curriculum 2 To equip students with the independent learning, problem solving and research skills essential for success in post-secondary studies Self-contained and self-explanatory lessons A variety of self-evaluation and self-checking strategies A mature design similar to what students will experience with first year university/college texts The placement of lab activities at the end of chapters parallels the formal separation of theory and labs in university courses Extension and web link strategies provide opportunities to Physics 12 hone individual research and study skills 3 To acknowledge the science background and level of preparation of students entering this new course Specifically developed for the Physics, A wealth of diagnostic activities Grade 12, University Preparation Regular practice, assessment, and remediation opportunities (SPH4U) Curriculum Grade 11 skills reviewed next to its Grade 12 counterpart University of Waterloo Sir Isaac Newton (S.I.N.) contest questions provide an extended analysis of new concepts 4 To support busy teachers with the Authors implementation and administration of this new curriculum Al Hirsch Student lessons accommodate independent learning Charles Stewart David Martindale A comprehensive array of planning and assessment tools Maurice Barry reduce lesson preparation time Access to a variety of different instructional approaches Brief Table of Contents Unit 1: Forces and Motion: Dynamics Chapter 1: Kinematics Chapter 2: Dynamics Chapter 3: Circular Motion Unit 2: Energy and Momentum Chapter 4: Work and Energy Chapter 5: Momentum and Collisions Teacher’s Resource Chapter 6: Gravitational and Celestial Mechanics Computerized Test Bank Unit 3: Electric, Gravitational & Magnetic Field Chapter 7: Electric Charges and Electric Field Chapter 8: Magnetic Fields and Electomagnetism Unit 4: The Wave Nature of Light Chapter 9: Waves and Light Components Chapter 10: Wave Effects of Light Student Text (print and CD-ROM) 0-17-612146-3 Unit 5: Matter-Energy Interface Teacher’s Resource Binder (print, CD-ROM, and web- Chapter 11: Einstein’s Special Theory of Relativity based) Chapter 12: Waves, Photons and Matter 0-17-625954-6 Chapter 13: Radioactivity and Elementary Particles Solutions Manual 0-17-625955-4 Appendixes Lab and Study Masters Review 0-17-625956-2 Skills Computerized Assessment Bank (CD-ROM) Reference 0-17-625957-0 Answers Nelson Chemistry 12 Website www.science.nelson.com Glossary Index Evaluation Copies Available Spring 2002! 1120 Birchmount Road, Toronto, ON M1K 5G4 (416) 752-9448 or 1-800-268-2222 Fax (416) 752-8101 or 1-800-430-4445 E-mail: [email protected] Internet: www.nelson.com &) # ! # +, #& #& +234 #&, % )( - ) +( )" - !"#$% )& +( & ' ( - ! ) 4 ! 2 )* +,+ , - ! ) 4 ! 2 - ! ! #) - ! &- ! ! ' ( #) +234./"0 &1 ! * / #), # # +234 &' ( #, ; * ;/ : 7. ; " ! * *25 !./"0 ; # +" ! 6 7. +( !6 7. +( #; &"0 !* * #; +234 )- 6 2- * +,+ #;, "$ / +* 8 + % + - ! &( ! ++ +* 8 +234 # 8 , # +234 #, )' ( )! 8 + ( 4 !* ( * ! #" +#. ! 0 SAMPLE problem 4 A golfer strikes a golf ball on level ground. The ball leaves the ground with an initial velocity of 42 m/s [32° above the horizontal]. The initial conditions are shown in Figure 12. +y If air resistance is negligible, determine the ball’s (a) horizontal range (assuming that it lands at the same level from which it started) +x (b) maximum height (c) horizontal displacement when it is 15 m above the ground vWi= 42 m/s viy =vWisin u Solution u (a) We begin by finding the horizontal and vertical components of the initial velocity. vix =vWicos u vix 5 v$icos v viy 5 v$isin v ay = 2g = 29.8 m/s2 5 (42 m/s)(cos 32°) 5 (42 m/s)(sin 32°) vix 5 36 m/s viy 5 22 m/s Figure 12 Initial conditions for Sample Problem Horizontally (constant vix ): 4. The golf tee is chosen as the ini- vix 5 36 m/s tial position, and the 1y direction is chosen as upward. Dx 5 ? Dt 5 ? Vertically (constant ay ): ay 5 2g 5 29.8 m/s2 Dy 5 0 viy 5 22 m/s Dt 5 ? vfy 5 222 m/s Since the horizontal motion has two unknowns and only one equation, we can use the vertical motion to solve for Dt: 1 Dy 5 viy Dt 1 }}ay (Dt )2 2 0 5 22 m/s Dt 2 4.9 m/s2 (Dt)2 0 5 Dt (22 m/s 2 4.9 m/s2 Dt) NEL Kinematics 47 Therefore, the ball was hit at Dt 5 0 and the ball lands at 22 m/s – 4.9 m/s2 Dt 5 0. Solving for Dt, we find that Dt 5 4.5 s, which we can use to find the horizontal range. Dx 5 vix Dt 5 (36 m/s)(4.5 s) Dx 5 1.6 3 102 m The horizontal range is 1.6 3 102 m. (b) To determine the maximum height, we start by noting that at the highest position, LEARNING TIP vfy 5 0 m/s. ( This also happens when an object thrown directly upward reaches the Applying Symmetry top of its flight.) The final vertical component of the velocity (222 m/s) has the same vfy2 5 viy2 1 2ay Dy magnitude as the initial vertical 0 5 viy2 1 2ay Dy component, since air resistance is negligible and the ground is level. viy2 Dy 5 } Recall that the same symmetry 22ay occurs for an object thrown (22 m/s)2 directly upward. 5 }} 22(29.8 m/s2) Dy 5 25 m The maximum height is 25 m. (c) To find the horizontal displacement when Dy 5 15 m, we must find the time interval Dt between the start of the motion and when Dy 5 15 m. We can apply the quadratic formula: 1 Dy 5 viy Dt 1 }} ay (Dt)2 2 15 m 5 22 m/s Dt 2 4.9 m/s2 (Dt)2 4.9 m/s2 (Dt)2 2 22 m/s Dt 1 15 m 5 0 Using the quadratic formula, 2b 6 Ïbw 2 2 4w ac Dt 5 }} 2a where a 5 4.9 m/s2, b 5 222 m/s, and c 5 15 m 2(222 m/s) 6 Ïw (222 mw/s)2 2w 4(4.9 w m/s2)(w 15 m) 5 }}}}}} 2 2(4.9 m/s ) Dt 5 3.7 s or 0.84 s Thus, the ball is 15 m above the ground twice: when rising and when descending. We can determine the corresponding horizontal positions: Dxup 5 vix Dt Dxdown 5 vix Dt 5 (36 m/s)(0.84 s) 5 (36 m/s)(3.7 s) Dxup 5 3.0 3 101 m Dxdown 5 1.3 3 102 m The horizontal position of the ball is either 3.0 3 101 m or 1.3 3 102 m when it is 15 m above ground. 48 Chapter 1 NEL Section 1.4 As you learned in the solution to Sample Problem 4, the range of a projectile can be +y found by applying the kinematics equations step by step. We can also derive a general equa- tion for the horizontal range Dx of a projectile, given the initial velocity and the angle +x of launch. What, for example, happens when a projectile lands at the same level from which it began (Dy 5 0), as shown in Figure 13? For the horizontal range, the motion is found using the equation Dx 5 vix Dt, where the only known variable is vix. To find the other variable, Dt, we use the vertical motion: vWi viy 1 Dy 5 viy Dt 1 }}ay (Dt ) 2 u 2 vix where Dy 5 0 because we are considering the situation where the final level is the same ∆x as the initial level. Figure 13 viy 5 vi sin v Initial conditions for deriving the ay 5 2g horizontal range of a projectile in 1 terms of launch angle and initial 0 5 vi sin vDt 2 }}g(Dt )2 velocity 2 1 2 1 0 5 Dt vi sin v 2 }}g(Dt) 2 Therefore, either Dt 5 0 (on takeoff) or TRY THIS activity 1 vi sin v 2 }}gDt 5 0 (on landing). 2 Comparing Horizontal Range As a class or in a large group, set up a table using these titles: Solving the latter equation for Dt gives Launch Angle, Time of Flight, Maximum Height, and Horizontal 2vi sin v Range. Complete the table for a projectile that has an initial Dt 5 }} g velocity of magnitude 25.00 m/s and lands at the same level from which it was launched. Perform the calculations using four Now we return to the horizontal motion: significant digits, using every third degree from 3° to 87° (i.e., 3°, 6°, 9°, … 81°, 84°, 87°). Write conclusions about maximizing Dx 5 vix Dt height and horizontal range. 5 (vi cos v)Dt 2vi sin v 5 vi cos v 1 } g 2 vi2 LAB EXERCISE 1.4.1 Dx 5 }g} 2sin v cos v Hang Time in Football (p. 58) “Hang time” in sports is the time Since 2sin v cos v 5 sin 2v (as shown in the trigonometric identities in Appendix A), the interval between the launch of a ball horizontal range is and the landing or catching of the vi2 ball. In football, when a punt is D x 5 }g} sin 2v needed, the punter tries to maximize the hang time of the ball to give his where vi is the magnitude of the initial velocity of a projectile launched at an angle v to teammates time to race downfield to the horizontal. Note that this equation applies only if Dy 5 0. tackle the punt receiver. Of course, All of the previous discussion and examples of projectile motion have assumed that at the same time the punter tries to maximize the horizontal range to air resistance is negligible. This is close to the true situation in cases involving relatively give his team better field position. dense objects moving at low speeds, such as a shot used in shot put competition. However, Write your hypothesis and predic- for many situations, air resistance cannot be ignored. When air resistance is considered, tions to the following questions, and the analysis of projectile motion becomes more complex and is beyond the intention then explore these concepts further of this text. The concept of “hang time” in certain sports, especially football, is impor- by conducting the lab exercise. tant and is explored in Lab Exercise 1.4.1 in the Lab Activities Section at the end of this (a) What factors affect the hang time of a punted football? How do chapter. they affect hang time? (b) What launch angle of a punt maximizes the hang time of a football? NEL Kinematics 49 Practice Understanding Concepts Answers 8. A field hockey ball is struck and undergoes projectile motion. Air resistance is negligible. 9. (a) 1.2 3 103 m (a) What is the vertical component of velocity at the top of the flight? (b) What is the acceleration at the top of the flight? (b) 32 s (c) How does the rise time compare to the fall time if the ball lands at the same level (c) 4.9 3 103 m from which it was struck? (d) 2.2 3 102 m/s [45° below 9. A cannon is set at an angle of 45° above the horizontal. A cannonball leaves the the horizontal] muzzle with a speed of 2.2 3 102 m/s. Air resistance is negligible. Determine the 10. (a) 2.4 s cannonball’s (b) 22 m (a) maximum height (b) time of flight (c) 18 m/s [60° below the (c) horizontal range (to the same vertical horizontal] level) vi 42° (d) velocity at impact 10. A medieval prince trapped in a castle wraps a message around a rock and throws it from the top of the castle wall with an initial 9.5 m velocity of 12 m/s [42° above the horizontal]. The rock lands just on the far side of the castle’s moat, at a level 9.5 m below the initial level (Figure 14). Determine the rock’s moat (a) time of flight (b) width of the moat (c) velocity at impact Figure 14 The situation for question 10 SUMMARY Projectile Motion A projectile is an object moving through the air in a curved trajectory with no propulsion system. Projectile motion is motion with a constant horizontal velocity combined with a constant vertical acceleration. The horizontal and vertical motions of a projectile are independent of each other except they have a common time. Projectile motion problems can be solved by applying the constant velocity equa- tion for the horizontal component of the motion and the constant acceleration equations for the vertical component of the motion. Section 1.4 Questions Understanding Concepts 3. A projectile launched horizontally moves 16 m in the 1. What is the vertical acceleration of a projectile on its way horizontal plane while falling 1.5 m in the vertical plane. up, at the top of its trajectory, and on its way down? Determine the projectile’s initial velocity. 2. (a) For a projectile with the launch point lower than the 4. A tennis player serves a ball horizontally, giving it a speed landing point, in what part of the flight is the magni- of 24 m/s from a height of 2.5 m. The player is 12 m from tude of the velocity at a maximum? a minimum? the net. The top of the net is 0.90 m above the court sur- (b) In what part of the flight is the magnitude of the face. The ball clears the net and lands on the other side. velocity at a maximum, and in what part is it at a Air resistance is negligible. minimum, for a projectile with the launch point higher than the landing point? 50 Chapter 1 NEL Section 1.4 (a) For how long is the ball airborne? (b) What is the horizontal displacement? (c) What is the velocity at impact? (d) By what distance does the ball clear the net? 5. A child throws a ball onto the roof of a house, then catches it with a baseball glove 1.0 m above the ground, as in Figure 15. The ball leaves the roof with a speed of 3.2 m/s. (a) For how long is the ball airborne after leaving the roof? (b) What is the horizontal distance from the glove to the edge of the roof? (c) What is the velocity of the ball just before it lands in Figure 16 the glove? Projectile motion in the garden 10. Describe how you would build and test a device made of simple, inexpensive materials to demonstrate that two coins 33 ° launched simultaneously from the same level, one launched horizontally and the other dropped vertically, land at the same instant. Making Connections 6.2 m 11. In real-life situations, projectile motion is often more com- plex than what has been presented in this section. For example, to determine the horizontal range of a shot in shot put competitions, the following equation is used: Dx 5 Dx1 1 Dx2 1 Dx3 2vi2 sin v cos v i vi sin v Ïw v 2 sin2w v 1 2w g Dy Dx 5 0.30 m 1 }} 1 }}}} Figure 15 g g where 0.30 m is the average distance the athlete’s hand 6. For a projectile that lands at the same level from which it goes beyond the starting line, vi is the magnitude of the ini- starts, state another launch angle above the horizontal that tial velocity, v is the angle of launch above the horizontal, would result in the same range as a projectile launched at Dy is the height above the ground where the shot leaves an angle of 36°, 16°, and 45.6°. Air resistance is negligible. the hand, and g is the magnitude of the acceleration due to 7. During World War I, the German army bombarded Paris gravity (Figure 17). with a huge gun referred to, by the Allied Forces, as “Big (a) Determine the range of a shot released 2.2 m above Bertha.” Assume that Big Bertha fired shells with an initial the ground with an initial velocity of 13 m/s [42° above velocity of 1.1 3 103 m/s [45° above the horizontal]. the horizontal]. (a) How long was each shell airborne, if the launch point (b) Compare your answer in (a) to the world record for the was at the same level as the landing point? shot put (currently about 23.1 m). (b) Determine the maximum horizontal range of each shell. (c) Why do you think the equation given here differs from (c) Determine the maximum height of each shell. the equation for horizontal range derived in this 8. An astronaut on the Moon, where g $ = 1.6 m/s2, strikes a section? golf ball giving the ball a velocity of 32 m/s [35° above the Moon’s horizontal]. The ball lands in a crater floor that is ∆x 15 m below the level where it was struck. Determine (a) the maximum height of the ball ∆x1 ∆x3 (b) the time of flight of the ball ∆y ∆x2 (c) the horizontal range of the ball starting line Applying Inquiry Skills Figure 17 9. A garden hose is held with its nozzle horizontally above the ground (Figure 16). The flowing water follows projectile motion. Given a metre stick and a calculator, describe how you would determine the speed of the water coming out of the nozzle. NEL Kinematics 51 Frames of Reference 1.5 and Relative Velocity frame of reference coordinate Air shows provide elements of both excitement and danger. When high-speed airplanes system relative to which motion is fly in constant formation (Figure 1), observers on the ground see them moving at high observed velocity. Seen from the cockpit, however, all the planes appear to have zero velocity. Observers on the ground are in one frame of reference, while the pilots are in the plane’s frame of reference. A frame of reference is a coordinate system relative to which motion is described or observed. The most common frame of reference that we use as a stationary, or fixed, frame of reference is Earth or the ground. In the examples of motion presented in the previous sections, all objects were assumed to be moving relative to the frame of reference of Earth. Sometimes, however, other frames are chosen for convenience. For example, to ana- lyze the motion of the planets of the solar system, the Sun’s frame of reference is used. If we observe a spot near the rim of a rolling wheel, the wheel or the centre of the wheel is the most convenient frame of reference, as in Figure 2. (a) (b) Figure 1 The Canadian Forces Snowbirds fly at velocities of between 400 and 600 km/h (relative to the ground), but when they are flying in forma- tion, as shown here, the velocity of one plane relative to another is zero. Figure 2 (a) The motion of a spot near the rim of a rolling wheel is simple if viewed from the frame of reference of the wheel’s centre. (b) The motion of the spot is much more complex when viewed from Earth’s frame of reference. DID YOU KNOW ? Viewing the Solar System The velocity of an object relative to a specific frame of reference is called relative It is easy to visualize planets velocity. We have not used this term previously because we were considering motion revolving around the Sun, using the relative to one frame of reference at a time. Now we will explore situations involving at Sun as the frame of reference. least two frames of reference. Such situations occur for passengers walking about in a Ancient astronomers, however, used Earth’s frame of reference to try to moving train, for watercraft travelling on a flowing river, and for the Snowbirds or other explain the observed motion of the aircraft flying when there is wind blowing relative to the ground. planets, but had to invent forces To analyze relative velocity in more than one frame of reference, we use the symbol for that do not exist. For example, when relative velocity, $v, with two subscripts in capital letters. The first subscript represents the watching the motion of a planet object whose velocity is stated relative to the object represented by the second subscript. beyond Earth (such as Mars) against the background of the stars, In other words, the second subscript is the frame of reference. the planet appears to reverse direc- For example, if P is a plane travelling at 490 km/h [W] relative to Earth’s frame of tion from time to time, much like a reference, E, then v$PE 5 490 km/h [W]. If we consider another frame of reference, such flattened “S” pattern. In fact, the as the wind or air, A, affecting the plane’s motion, then $vPA is the velocity of the plane rel- planet doesn’t reverse directions; it ative to the air and v$AE is the velocity of the air relative to Earth. The vectors v$PA and only appears to do so as Earth, $vAE are related to $vPE using the following relative velocity equation: which is closer to the Sun, catches up and then passes the planet. v#$PE 5 v#$PA 1 #v$AE relative velocity velocity of an This equation applies whether the motion is in one, two, or three dimensions. For object relative to a specific frame of example, consider the one-dimensional situation in which the wind and the plane are both reference moving eastward. If the plane’s velocity relative to the air is 430 km/h [E], and the air’s 52 Chapter 1 NEL Section 1.5 velocity relative to the ground is 90 km/h [E], then the velocity of the plane relative to the ground is: DID YOU KNOW ? Wind Directions $vPE 5 $vPA 1 $vAE By convention, a west wind is a wind that blows from the west, so 5 430 km/h [E] 1 90 km/h [E] its velocity vector points east (e.g., $vPE 5 520 km/h [E] a west wind might be blowing at 45 km/h [E]). A southwest wind Thus, with a tail wind, the ground speed increases—a logical result. You can easily figure has the direction [45° N of E] or out that the plane’s ground speed in this example would be only 340 km/h [E] if the [45° E of N]. wind were a head wind (i.e., if $vAG = 90 km/h [W]). Before looking at relative velocities in two dimensions, make sure that you under- stand the pattern of the subscripts used in any relative velocity equation. As shown in Figure 3, the left side of the equation has a single relative velocity, while the right side has DID YOU KNOW ? the vector addition of two or more relative velocities. Note that the “outside” and the Navigation Terminology “inside” subscripts on the right side are in the same order as the subscripts on the left side. Air navigators have terms for some of the key concepts of relative velocity. Air speed is the speed of a nW PE = nW PA + nW AE nW CE = nW CW + nW WE plane relative to the air. Wind speed is the speed of the wind rel- ative to the ground. Ground speed is the speed of the plane relative nW LO = nWLM + nWMN + nWNO nW DG = nW DE + nW EF + nW FG to the ground. The heading is the direction in which the plane is aimed. The course, or track, is the Figure 3 path relative to Earth or the The pattern in relative velocity equations ground. Marine navigators use “heading,” “course,” and “track” in analogous ways. SAMPLE problem 1 An Olympic canoeist, capable of travelling at a speed of 4.5 m/s in still water, is crossing a river that is flowing with a velocity of 3.2 m/s [E]. The river is 2.2 3 102 m wide. (a) If the canoe is aimed northward, as in Figure 4, what is its velocity relative to the shore? (b) How long does the crossing take? (c) Where is the landing position of the canoe relative to its starting position? (d) If the canoe landed directly across from the starting position, at what angle would the canoe have been aimed? north shore N vWws E vWcw river u vWcs Figure 4 The situation Solution Using the subscripts C for the canoe, S for the shore, and W for the water, the known rela- tive velocities are: $vCW 5 4.5 m/s [N] $vWS 5 3.2 m/s [E] NEL Kinematics 53 LEARNING TIP (a) Since the unknown is v$CS, we use the relative velocity equation Alternative Symbols An alternative method of writing v$CS 5 v$CW 1 $vWS a relative velocity equation is to $vCS 5 4.5 m/s [N] 1 3.2 m/s [E] place the subscript for the observed object before the $v Applying the law of Pythagoras, we find: and the subscript for the frame of reference after the v$. Using v$CS 5 Ïw (4.5 mw /s)2 1w (3.2 mw /s)2 this method, the equation for v$CS 5 5.5 m/s our example of plane and air is $E 5 Pv$A 1 A$vE. Pv Trigonometry gives the angle v in Figure 4: 3.2 m/s v 5 tan21 }} 4.5 m/s v 5 35° The velocity of the canoe relative to the shore is 5.5 m/s [35° E of N]. (b) To determine the time taken to cross the river, we consider only the motion perpendi- cular to the river. Dd$ 5 2.2 3 102 m [N] v$CW 5 4.5 m/s [N] Dt 5 ? D$ d From $vCW 5 }}, we have: Dt D$d Dt 5 }} $vCW 2.2 3 102 m [N] 5 }} 4.5 m/s [N] Dt 5 49 s The crossing time is 49 s. (c) The current carries the canoe eastward (downstream) during the time it takes to cross the river. The downstream displacement is $ 5 $vWSDt Dd 5 (3.2 m/s [E])(49 s) $ 5 1.6 3 102 m [E] Dd The landing position is 2.2 3 102 m [N] and 1.6 3 102 m [E] of the starting position. Using the law of Pythagoras and trigonometry, the resultant displacement is 2.7 3 102 m [36° E of N]. (d) The velocity of the canoe relative to the water, $vCW , which has a magnitude of 4.5 m/s, is the hypotenuse of the triangle in Figure 5. The resultant velocity $vCS must point directly north for the canoe to land directly north of the starting position. north shore N vWws E vWcw vWcs f Figure 5 The solution for part (d) 54 Chapter 1 NEL Section 1.5 The angle in the triangle is v#$ f 5 sin 21 }W} S #v$ CW 3.2 m/s 5 sin 21 }} 4.5 m/s f 5 45° The required heading for the canoe is [45° W of N]. SAMPLE problem 2 The air speed of a small plane is 215 km/h. The wind is blowing at 57 km/h from the west. Determine the velocity of the plane relative to the ground if the pilot keeps the plane aimed in the direction [34° E of N]. Solution vWAE We use the subscripts P for the plane, E for Earth or the ground, and A for the air. f $vPA 5 215 km/h [34° E of N] $vAE 5 57 km/h [E] vWPA N $vPE 5 ? vWPE u E v$PE 5 $vPA 1 $vAE 34° f = 90° + 34° This vector addition is shown in Figure 6. We will solve this problem by applying the f = 124° cosine and sine laws; however, we could also apply a vector scale diagram or components as described in Appendix A. Figure 6 Solving Sample Problem 2 using Using the cosine law: trigonometry v$PE2 5 v$PA2 1 v$AE2 2 2v$PAv$AE cos f 5 (215 km/h)2 1 (57 km/h)2 2 2(215 km/h)(57 km/h) cos 124° v$PE 5 251 km/h Using the sine law: sin v sin f }} 5 }} v$AE v$PE 57 km/h (sin 124°) sin v 5 }} 251 km/h v 5 11° LEARNING TIP The direction of v$PE is 34° 1 11° 5 45° E of N. Thus $vPE 5 251 km/h [45° E of N]. Subtracting Vectors When a relative velocity equation, such as $vPE 5 $vPA 1 $vAE, is Sometimes it is helpful to know that the velocity of object X relative to object Y has the rearranged to isolate either $vPA or v$AE , a vector subtraction same magnitude as the velocity of Y relative to X, but is opposite in direction: $vXY 5 2v$YX. must be performed. For example, Consider, for example, a jogger J running past a person P sitting on a park bench. $vPA 5 $vPE 2 $vAE is equivalent to If $vJP 5 2.5 m/s [E], then P is viewing J moving eastward at 2.5 m/s. To J, P appears to be $vPA 5 $vPE 1 (2v$AE). Appendix A moving at a velocity of 2.5 m/s [W]. Thus $vPJ 5 22.5 m/s [E] 5 2.5 m/s [W]. In the discusses vector arithmetic. next Sample Problem, we will use this relationship for performing a vector subtraction. NEL Kinematics 55 N SAMPLE problem 3 scale: 1.0 cm = 30 km/h A helicopter, flying where the average wind velocity is 38 km/h [25° N of E], needs to E achieve a velocity of 91 km/h [17° W of N] relative to the ground to arrive at the destina- tion on time, as shown in Figure 7. What is the necessary velocity relative to the air? vWHG = 91 km/h [17° W of N] Solution Using the subscripts H for the helicopter, G for the ground, and A for the air, we have the following relative velocities: v$HG 5 91 km/h [17° W of N] $vAG 5 38 km/h [25° N of E] vWAG = 38 km/h [25° N of E] $vHA 5 ? Figure 7 $vHG 5 $vHA 1 $vAG Situation for Sample Problem 3 We rearrange the equation to solve for the unknown: v$HA 5 v$HG 2 $vAG $vHA 5 v$HG 1 (2v$AG) where 2v$AG is 38 km/h [25° S of W] N Figure 8 shows this vector subtraction. By direct measurement on the scale diagram, we can see that the velocity of the helicopter relative to the air must be 94 km/h [41° W of N]. scale: 1.0 cm = 30 km/h The same result can be obtained using components or the laws of sines and cosines. 2vWAG E Practice vWHG Understanding Concepts 17° 1. Something is incorrect in each of the following equations. Rewrite each equation vWHA 24° to show the correction. (a) v$LE 5 $vLD 1 $vLE (b) $vAC 5 $vAB 2 $vBC Figure 8 (c) $vMN 5 $vNT 1 v$TM (Write down two correct equations.) Solution to Sample Problem 3 (d) $vLP 5 $vML 1 $vMN 1 $vNO 1 $vOP 2. A cruise ship is moving with a velocity of 2.8 m/s [fwd] relative to the water. A Answers group of tourists walks on the deck with a velocity of 1.1 m/s relative to the deck. 2. (a) 3.9 m/s [fwd] Determine their velocity relative to the water if they are walking toward (a) the bow, (b) the stern, and (c) the starboard. (The bow is the front of a ship, the (b) 1.7 m/s [fwd] stern is the rear, and the starboard is on the right side of the ship as you face (c) 3.0 m/s [21° right of fwd] the bow.) 3. 5.3 m/s [12° E of N] 3. The cruise ship in question 2 is travelling with a velocity of 2.8 m/s [N] off the 4. 7.2 3 102 km [30° S of W] coast of British Columbia, in a place where the ocean current has a velocity rela- from Winnipeg tive to the coast of 2.4 m/s [N]. Determine the velocity of the group of tourists in 2(c) relative to the coast. 4. A plane, travelling with a velocity relative to the air of 320 km/h [28° S of W], passes over Winnipeg. The wind velocity is 72 km/h [S]. Determine the displace- ment of the plane from Winnipeg 2.0 h later. Making Connections 5. Airline pilots are often able to use the jet stream to minimize flight times. Find out more about the importance of the jet stream in aviation. GO www.science.nelson.com 56 Chapter 1 NEL Section 1.5 Frames of Reference and SUMMARY Relative Velocity A frame of reference is a coordinate system relative to which motion can be observed. Relative velocity is the velocity of an object relative to a specific frame of refer- ence. (A typical relative velocity equation is v$PE 5 $vPA 1 $vAE , where P is the observed object and E is the observer or frame of reference.) Section 1.5 Questions Understanding Concepts (a) Assuming that the raindrops are falling straight down- 1. Two kayakers can move at the same speed in calm water. ward relative to Earth’s frame of reference, and that the One begins kayaking straight across a river, while the other speed of the train is 64 km/h, determine the vertical kayaks at an angle upstream in the same river to land speed of the drops. straight across from the starting position. Assume the (b) Describe sources of error in carrying out this type of speed of the kayakers is greater than the speed of the river estimation. current. Which kayaker reaches the far side first? Explain direction of train’s motion why. 2. A helicopter travels with an air speed of 55 m/s. The heli- copter heads in the direction [35° N of W]. What is its velocity relative to the ground if the wind velocity is (a) 21 m/s [E] and (b) 21 m/s [22° W of N]? 3. A swimmer who achieves a speed of 0.75 m/s in still water swims directly across a river 72 m wide. The swimmer lands on the far shore at a position 54 m downstream from the starting point. (a) Determine the speed of the river current. (b) Determine the swimmer’s velocity relative to the shore. (c) Determine the direction the swimmer would have to aim to land directly across from the starting position. 4. A pilot is required to fly directly from London, UK, to Rome, Italy in 3.5 h. The displacement is 1.4 3 103 km [43° E of S]. left hand right hand A wind is blowing with a velocity of 75 km/h [E]. Determine the required velocity of the plane relative to the air. Figure 9 Estimating the speed of falling raindrops Applying Inquiry Skills 5. A physics student on a train estimates the speed of falling Making Connections raindrops on the train car’s window. Figure 9 shows the 6. You have made a video recording of a weather report, student’s method of estimating the angle with which the showing a reporter standing in the wind and rain of a hurri- drops are moving along the window glass. cane. How could you analyze the video to estimate the wind speed? Assume that the wind is blowing horizontally, and that the vertical component of the velocity of the rain- drops is the same as the vertical component for the rain- drops in the previous question. NEL Kinematics 57 Chapter 1 SUMMARY D#v$ v#$f 2 v#$i Key Expectations a$av 5 }} 5 }} Dt Dt (1.2) analyze and predict, in quantitative terms, and explain $a 5 lim }} D#v$ (1.2) the linear motion of objects in the horizontal plane, Dt→0 Dt the vertical plane, and any inclined plane (for example, 1 $ 5 $v Dt 1 }}a$ (Dt)2 Dd i (1.2) a skier accelerating down a hillside) (1.1, 1.2, 1.3, 1.5) 2 analyze and predict, in quantitative terms, and explain $ 5 $v Dt 5 } (v$i 1 v$f ) Dd av } Dt 2 (1.2) the motion of a projectile in terms of the horizontal and vertical components of its motion (1.4) vf2 5 vi2 1 2aDd (1.2) carry out experiments and/or simulations involving 1 Dd$ 5 $v Dt 2 }}a$ (Dt )2 (1.2) objects moving in two dimensions, and analyze and dis- f 2 play the data in an appropriate form (1.1, 1.2, 1.3, 1.4) Dvx vfx 2 vix aav,x 5 }} 5 }} (1.2) predict the motion of an object given its initial speed and Dt Dt Dvy vfy 2 viy direction of motion (e.g., terminal speed and projectile aav,y 5 }} 5 }} (1.2) motion) and test the predictions experimentally (1.3, 1.4) Dt Dt vfy 2 viy describe or construct technological devices that are ay 5 }} (1.3) Dt based on the concepts and principles related to projec- 1 2 tile motion (1.4) Dy 5 viy Dt 1 }} ay (Dt) (1.3) 2 (viy 1 vfy) Dy 5 }} Dt (1.3) 2 Key Terms vfy2 5 viy2 1 2ay Dy (1.3) kinematics average acceleration 1 2 Dy 5 vfy Dt 2 }} ay (Dt) (1.3) scalar quantity instantaneous 2 acceleration Dx instantaneous speed vix 5 }} (1.4) Dt average speed acceleration due to gravity v$PE 5 v$PA 1 $vAE (1.5) vector quantity free fall position terminal speed displacement projectile velocity projectile motion instantaneous velocity MAKE a summary horizontal range average velocity frame of reference Draw a large diagram showing the path of a ball undergoing tangent relative velocity projectile motion. Label several positions along the path acceleration (A, B, C, D, and E), and show as many details of the motion as you can. For example, indicate the magnitude and direc- tion (where possible) of the horizontal and vertical compo- Key Equations nents of the position, displacement, instantaneous velocity, and instantaneous acceleration at each position. Show what d vav 5 }} (1.1) happens to those quantities if you assume that air resistance Dt near the end of the path is no longer negligible. Finally, show $ 5 $ Dd d2 2 $ d1 (1.1) details related to frames of reference (for instance, one #$ frame of reference could be the playing field and another Dd v$av 5 }} (1.1) could be an athlete running parallel to the ball’s motion just Dt before catching the ball). In the diagrams and labels, include D#d$ v$ 5 lim }} (1.1) as many of the key expectations, key terms, and key equa- Dt→0 Dt tions from this chapter as you can. $ 5 Dd Dd $ 1 Dd $ 1… (1.1) 1 2 62 Chapter 1 NEL Chapter 1 SELF QUIZ Unit 1 Write numbers 1 to 11 in your notebook. Indicate beside 13. You drop a rubber stopper from your hand: the initial each number whether the corresponding statement is true position is your hand, and 1y is upward. Which (T) or false (F). If it is false, write a corrected version. graph in Figure 1 best represents the relationship? 1. You toss a ball vertically and step aside. The ball rises 14. You toss a ball directly upward: the initial position is and then falls down along the same path and hits the your hand, and 1y is downward. Which graph in ground. Since the ball reverses direction, it undergoes Figure 1 best represents the relationship? two-dimensional motion. 15. You release a cart from rest at the top of a ramp: the 2. The magnitude of the velocity of that same ball just initial position is at the top of the ramp, and 1y is up before landing is greater than its magnitude of initial the ramp. Which graph in Figure 1 best represents velocity upon leaving your hand. the relationship? 3. The acceleration of that ball at the top of the flight is 16. A car with an initial velocity of 25 m/s [E] experi- zero. ences an average acceleration of 2.5 m/s2 [W] for 4. The time for that ball to rise equals the time for it to 2.0 3 101 s. At the end of this interval, the velocity is fall. (a) 5.0 3 101 m/s [W] (d) 75 m/s [W] (b) 0.0 m/s (e) 75 m/s [E] 5. A jogger running four laps around a circular track at (c) 25 m/s [W] 4.5 m/s undergoes motion with constant velocity. 17. An acceleration has an eastward component of 6. The slope of the tangent to a curved line on a position- 2.5 m/s2 and a northward component of 6.2 m/s2. time graph gives the instantaneous velocity. The direction of the acceleration is 7. Megametres per hour per day is a possible unit of (a) [40° E of N] (d) [68° E of N] acceleration. (b) [50° E of N] (e) [68° N of E] 8. The magnitude of the acceleration due to gravity (c) [24° E of N] at Miami is greater than that at St. John’s, 18. You are a fullback running with an initial velocity of Newfoundland. 7.2 m/s [N]. You swerve to avoid a tackle, and after 9. The quadratic formula must be used to solve 2.0 s are moving at 7.2 m/s [W]. Your average acceler- problems involving the quadratic equation ation over the time interval is vf2 5 vi2 1 2aDd. (a) 0 m/s2 10. A model rocket launched in a vacuum chamber at an (b) 5.1 m/s2 [45° N of W] angle of 45° above the horizontal, undergoes projec- (c) 1.0 3 101 m/s2 [45° N of W] tile motion. (d) 3.6 m/s2 [S] 11. If #v$AB 5 8.5 m/s [E], then #v$BA 5 –8.5 m/s [W]. (e) 5.1 m/s2 [45° W of S] 19. A tennis ball is thrown into the air with an initial Write numbers 12 to 19 in your notebook. Beside each velocity that has a horizontal component of 5.5 m/s number, write the letter corresponding to the best choice. and a vertical component of 3.7 m/s [up]. If air 12. You toss a ball vertically upward from your hand: the resistance is negligible, the speed of the ball at the top initial position is your hand, and 1y is upward. Of of the trajectory is the position-time graphs shown in Figure 1, which (a) zero (c) 5.5 m/s (e) 9.2 m/s best represents the relationship? (b) 3.7 m/s (d) 6.6 m/s (a) y (b) (c) y (d) (e) y y y 0 t 0 t t t t 0 0 0 Figure 1 Graphs of vertical position as a function of time for questions 12–15 NEL An interactive version of the quiz is available online. Kinematics 63 GO www.science.nelson.com Chapter 1 REVIEW Understanding Concepts N 1. (a) In Canada, the speed limit on many highways is A= 5.1 km 100 km/h. Convert this measurement to metres per second to three significant digits. 38° (b) The fastest recorded speed of any animal is the E dive of the peregrine falcon, which can reach 97 m/s. How fast is the dive in kilometres per hour? (c) Suggest a convenient way of converting kilome- 19° tres per hour to metres per second, and metres B= 6.8 km per second to kilometres per hour. S Figure 1 2. For each dimensional operation listed, state the type of quantity that results (speed, length, etc.). 10. Assume that a displacement vector can be drawn from a person’s nose to his or her toes. For a town of 2000 1 2 1 2 L L (a) L 3 T 21 (b) }}3 3 T (c) }}2 3 T 3 T people, estimate the resultant displacement vector of T T the sum of all the nose-to-toes vectors at (a) 5 P.M. and 3. A student reads a test question that asks for a dis- (b) 5 A.M. Explain your reasoning. tance, gives a time interval of 3.2 s and a constant 11. In the Canadian Grand Prix auto race, the drivers acceleration of magnitude 5.4 m/s2. Uncertain which travel a total distance of 304.29 km in 69 laps around equation applies, the student tries dimensional the track. If the fastest lap time is 84.118 s, what is the analysis and selects the equation $d 5 $a(Dt)2. average speed for this lap? (a) Is the equation dimensionally correct? (b) Describe the limitation of relying on dimensional 12. According to a drivers’ handbook, your safest separa- analysis to remember equations. tion distance from the car ahead at a given speed is the distance you would travel in 2.0 s at that speed. 4. For motion with constant velocity, compare: What is the recommended separation distance (a) instantaneous speed with average speed (a) in metres and (b) in car lengths, if your speed is (b) instantaneous velocity with average velocity 115 km/h? (c) instantaneous speed with average velocity 13. An eagle flies at 24 m/s for 1.2 3 103 m, then glides at 5. How can a velocity-time graph be used to determine 18 m/s for 1.2 3 103 m. Determine (a) displacement and (b) acceleration? (a) the time interval for this motion 6. Can a component of a vector have a magnitude (b) the eagle’s average speed during this motion greater than the vector’s magnitude? Explain. 14. Describe the motion represented by each graph 7. (a) Can the sum of two vectors of the same magni- shown in Figure 2. tude be a zero vector? (a) (b) (b) Can the sum of two vectors of unequal magni- tudes be a zero vector? W d Wv (c) Can the sum of three vectors, all of unequal mag- nitudes, be a zero factor? In each case, give an example if “yes,” an explanation t t 0 0 if “no.” Figure 2 8. A golfer drives a golf ball 214 m [E] off the tee, then hits it 96 m [28° N of E], and finally putts the ball 15. A firefighter slides 4.5 m down a pole and runs 6.8 m 12 m [25° S of E]. Determine the displacement from to a fire truck in 5.0 s. Determine the firefighter’s the tee needed to get a hole-in-one using (a) a vector (a) average speed and (b) average velocity. scale diagram and (b) components. Compare your 16. Over a total time of 6.4 s, a field hockey player runs answers. 16 m [35° S of W], then 22 m [15° S of E]. Determine 9. Determine the vector that must be added to the sum the player’s (a) resultant displacement and of $ A1$ B in Figure 1 to give a resultant displacement (b) average velocity. of (a) 0 and (b) 4.0 km [W]. 64 Chapter 1 NEL Unit 1 17. The cheetah, perhaps the fastest land animal, can (b) How long does the motion take if the magnitude maintain speeds as high as 100 km/h over short time of the initial velocity is 8.0 m/s in the direction of intervals. The path of a cheetah chasing its prey at top the acceleration? speed is illustrated in Figure 3. State the cheetah’s 24. An airplane, travelling initially at 240 m/s [28° S of instantaneous velocity, including the approximate E], takes 35 s to change its velocity to 220 m/s [28° E direction, at positions D, E, and F. of S]. What is the average acceleration over this time interval? N 25. A race car driver wants to attain a velocity of 54 m/s F [N] at the end of a curved stretch of track, experi- E encing an average acceleration of 0.15 m/s2 [S] for motion 95 s. What is the final velocity? 26. A camera is set up to take photographs of a ball undergoing vertical motion. The camera is 5.2 m D E above the ball launcher, a device that can launch the Figure 3 ball with an initial velocity 17 m/s [up]. Assuming that the ball goes straight up and then straight down past the camera, at what times after the launch will 18. A car travelling initially at 42 km/h on the entrance the ball pass the camera? ramp of an expressway accelerates uniformly to 105 km/h in 26 s. 27. Figure 4 shows a velocity-time graph for a squirrel (a) How far, in kilometres, does the car travel over walking along the top of a fence. this time interval? (a) Draw the corresponding acceleration-time graph (b) Determine the magnitude of the average acceler- of the motion. ation in kilometres per hour per second. (b) Draw the corresponding position-time graph from 0.0 s to 1.0 s. (Be careful: for the first 0.50 s, 19. In a thrill ride at an amusement park, the cars start this graph is not a straight line.) from rest and accelerate rapidly, covering the first 15 m [fwd] in 1.2 s. 4.0 (a) Calculate the average acceleration of the cars. vW (m/s [E]) (b) Determine the velocity of the cars at 1.2 s. 2.0 (c) Express the magnitude of the acceleration in terms of g$. 0 t (s) 20. Determine the constant acceleration needed for a 1.0 2.0 3.0 bullet to reach a muzzle velocity of 4.0 3 102 m/s –2.0 [fwd], provided friction is zero and the muzzle is Figure 4 0.80 m long. 21. A rocket begins its third stage of launch at a velocity 28. Venus, with an orbit of radius 1.08 3 1011 m, takes of 2.28 3 102 m/s [fwd]. It undergoes a constant 1.94 3 107 s to complete one orbit around the Sun. acceleration of 6.25 3 101 m/s2, while travelling (a) What is the average speed in metres per second, 1.86 km, all in the same direction. What is the rocket’s and kilometres per hour? velocity at the end of this motion? (b) Determine the magnitude of the average velocity 22. In its final trip upstream to its spawning territory, a after it has completed half a revolution around salmon jumps to the top of a waterfall 1.9 m high. the Sun. What is the minimum vertical velocity needed by the (c) Determine the magnitude of the average acceler- salmon to reach the top of the waterfall? ation after it has completed a quarter revolution 23. A bus travels 2.0 3 102 m with a constant accelera- around the Sun. tion of magnitude 1.6 m/s2. (a) How long does the motion take if the magnitude of the initial velocity is 0.0 m/s? NEL Kinematics 65 29. (a) What are the horizontal and vertical components Applying Inquiry Skills of the acceleration of a projectile? 37. A baseball player wants to measure the initial speed of (b) How would your answer change if both compo- a ball when the ball has its maximum horizontal nents of the motion experience air resistance? range. 30. A child throws a snowball with a horizontal velocity (a) Describe how this could be done using only a of 18 m/s directly toward a tree, from a distance of metre stick or a measuring tape. 9.0 m and a height above the ground of 1.5 m. (b) Describe sources of random and systematic error (a) After what time interval does the snowball hit the in this activity. tree? 38. You obtain the following data from an experiment (b) At what height above the ground will the snow- involving motion on an essentially frictionless air ball hit the tree? table inclined at an angle to the horizontal: (c) Determine the snowball’s velocity as it strikes the tree. length of air table side 62.0 cm 31. Determine the initial velocity of a projectile that is vertical distance from lab bench to 9.9 cm launched horizontally, and falls 1.5 m while moving the elevated end of the air table 16 m horizontally. vertical distance from lab bench 4.3 cm 32. You are standing in a train moving at constant to the lower end of the air table velocity relative to Earth’s frame of reference. You drop a ball to the floor. What is the path of the ball (a) Determine the angle of incline of the air table. (a) in your frame of reference and (b) from the frame (b) Determine the magnitude of the acceleration of of reference of a person standing stationary beside an air puck parallel to the incline of the table. the train? (Hint: Use g$ and the value you found for the 33. A plane is travelling with an air speed of 285 km/s angle of incline.) [45° S of E]. A wind is blowing at 75 km/h [22° E (c) What are the possible sources of random and sys- of N] relative to the ground. Determine the velocity tematic error in this experiment? of the plane relative to the ground. 39. Figure 5 shows a demonstration of projectile motion 34. A swimmer who can swim at a speed of 0.80 m/s in that usually warrants applause from the audience. still water heads directly across a river 86 m wide. The At the instant a dart is launched at a high velocity, a swimmer lands at a position on the far bank 54 m target (often a cardboard monkey) drops from a downstream from the starting point. Determine uspended position downrange from the launching (a) the speed of the current device. Show that if the dart is aimed directly at the (b) the velocity of the swimmer relative to the shore target, it will always strike the falling target. (Use a (c) the direction of departure that would have taken specific set of numbers.) the swimmer directly across the river 35. The displacement from London, UK, to Rome is 1.4 3 103 km [43° E of S]. A wind is blowing with a target velocity of 75 km/h [E]. The pilot wants to fly directly from London to Rome in 3.5 h. What velocity relative direction aimed path of target to the air must the pilot maintain? 36. A football is placed on a line 25 m from the goal post. dart launcher path of projectile The placement kicker kicks the ball directly toward the post, giving the ball an initial velocity of 21.0 m/s [47° above the horizontal]. The horizontal bar of the goal post is 3.0 m above the field. How far above or below the bar will the ball travel? Figure 5 In this “monkey-hunter” demonstration, launching the dart causes the target to drop. 66 Chapter 1 NEL Unit 1 Making Connections (a) (b) 40. An impatient motorist drives along a city bypass at an average speed of 125 km/h. The speed limit is 100 km/h. (a) If the bypass is 17 km long, how many minutes does the driver save by breaking the speed limit? (b) The driver consumes about 20% more fuel at the higher speed than at the legal limit. Can you sug- (c) (d) gest a reason? 41. An electromagnetic signal, travelling at the speed of light (3.0 3 108 m/s), travels from a ground station on Earth to a satellite 4.8 3 107 m away. The satellite receives the signal and, after a delay of 0.55 s, sends a return signal to Earth. Figure 6 (a) What is the time interval between the transmis- sion from the ground station and the reception of Extension the return signal at the station? 46. A helicopter flies directly toward a vertical cliff. When (b) Relate your answer in (a) to the time delays you the helicopter is 0.70 km from the cliff face, it trans- observe when live interviews are conducted via mits a sonar signal. It receives the reflected signal 3.4 s satellite communication on television. later. If the signal propagates at 3.5 3 102 m/s, what is 42. Kinematics in two dimensions can be extended to the speed of the helicopter? kinematics in three dimensions. What motion factors 47. A truck with one headlight is travelling at a constant would you expect to analyze in developing a com- speed of 18 m/s when it passes a stopped police car. puter model of the three-dimensional motion of The cruiser sets off in pursuit at a constant accelera- asteroids to predict how close they will come to Earth? tion of magnitude 2.2 m/s2. 43. A patient with a detached retina is warned by an eye (a) How far does the cruiser travel before catching specialist that any braking acceleration of magnitude the truck? greater than 2g$ risks pulling the retina entirely away (b) How long does the pursuit last? (Hint: Consider from the sclera. Help the patient decide whether to drawing a graph.) play a vigorous racket sport such as tennis. Use esti- 48. A car with an initial velocity of 8.0 3 101 km/h [E] mated values of running speeds and stopping times. accelerates at the constant rate of 5.0 (km/h)/s, 44. Your employer, a medical research facility specializing reaching a final velocity of 1.0 3 102 km/h [45° S in nanotechnology, asks you to develop a microscopic of E]. Determine (a) the direction of the acceleration motion sensor for injection into the human blood- and (b) the time interval. stream. Velocity readings from the sensor are to be 49. Derive an equation for the horizontal range of a pro- used to detect the start of blockages in arteries, capil- jectile with a landing point at a different altitude from laries, and veins. its launch point. Write the equation in terms of the (a) What physics principles and equations would you initial velocity, the acceleration due to gravity, the need to consider in brainstorming the design? launch angle, and the vertical component of the (b) Describe one possible design of the device. How, displacement. in your design, are data obtained from the 50. A sunbather, drifting downstream on a raft, dives off device? the raft just as it passes under a bridge and swims 45. Figure 6 shows four different patterns of fireworks against the current for 15 min. She then turns and explosions. What conditions of the velocity of the swims downstream, making the same total effort and fireworks device at the instant of the explosion could overtaking the raft when it is 1.0 km downstream account for each shape? from the bridge. What is the speed of the current in the river? Sir Isaac Newton Contest Question NEL Kinematics 67 chapter 2 In this chapter, you will be able to define and describe Dynamics To climb a vertical formation like the one shown in Figure 1, a rock climber must exert forces against the rock walls. The walls push back with reaction forces, and it is the upward com- concepts and units from the ponents of those forces that help the climber move upward. In this chapter, you will learn analysis of forces how to analyze the forces and components of forces for stationary and moving objects. distinguish between Forces cause changes in velocity. Thus, what you learned in Chapter 1 will be explored accelerating and further to help you understand why objects speed up, slow down, or change directions. nonaccelerating frames of In other words, you will explore the nature of the forces that cause acceleration. reference Chapter 2 concludes with a look at motion from different frames of reference. You will determine the net force find that Newton’s laws of motion apply in some frames of reference, but not in others. acting on an object and its resulting acceleration by analyzing experimental data using vectors and their REFLECT on your learning components, graphs, and 1. A dog-sled team is pulling a loaded toboggan trigonometry B up a snow-covered hill as illustrated in #$, B #$, C #$, A analyze and predict, in Figure 2. Each of th
