Chapter 2: Motion in Two Dimensions

Transcription

1 Chapter Motion in Two Dimensions Mini Inestigation Garbage Can Basketball, page 59 A. Answers may ary. Sample answer When launched from knee height, an underhand upward thrust is used such that the ball of paper traels in a parabolic arc. When launched from waist height, a downward arm motion is used to direct the ball of paper to moe in a cured trajectory downward into the trash can. When launched from shoulder height, an oerhand upward thrust is used to launch the paper ball in a parabolic arc that allows it to land in the trash can. Section.1 Motion in Two Dimensions A Scale Diagram Approach Tutorial 1 Practice, page Answers may ary. Sample answer I think a suitable scale would hae the ectors be about 5 cm long. Looking at the smaller displacement, if I diide 350 by 100, I get 3.5. Then if I diide 410 by 100, I get 4.1. Since 3.5 cm and 4.1 cm are both suitable lengths for my ectors I choose a scale of 1 cm 100 m.. Tutorial Practice, page (a) Gien d 1 7 cm [W]; d 46 cm [N] Required d T Analysis d T d 1 + d This figure shows the gien ectors, with the tip of d 1 joined to the tail of d. The resultant ector d T is drawn in black from the tail of d 1 to the tip of d. Using a compass, the direction of d T is [W 33 N]. d T measures 8.5 cm in length, so using the scale of 1 cm 10 cm, the actual magnitude of d T is 85 cm. Statement The sum of the two ectors is 85 cm [W 33 N]. (b) Gien d m [E 4 N]; d 94.8 m [S] Required d T Analysis d T d 1 + d Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions.1-1

2 This figure shows the gien ectors, with the tip of d 1 joined to the tail of d. The resultant ector d T is drawn in black from the tail of d 1 to the This figure shows the gien ectors, with the tip of d 1 joined to the tail of d. The resultant ector d T is drawn in black from the tail of d 1 to the tip of d. Using a compass, the direction of d T is [W 46 S]. d T measures 7.05 cm in length, so using the scale of 1 cm 10 m, the actual magnitude of d T is 70.5 m. Statement The sum of the two ectors is 70.5 m [W 46 S].. (a) Gien d m [W 35 S]; d 630 m [W 60 N] Required d T tip of d. Using a compass, the direction of d T is [W 3 N]. d T measures 7.4 cm in length, so using the scale of 1 cm 100 m, the actual magnitude of d T is 740 m. Statement The sum of the two ectors is 740 m [W 3 N]. (b) Gien d 740 m [W 3 N]; t 77 s Required a Analysis a d t a d t 740 m [W 3 N] 77 s a 9.6 m/s [W 3 N] Statement The cyclist s aerage elocity is 9.6 m/s [W 3 N]. Analysis d T d 1 + d Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions.1-

3 Section.1 Questions, page This figure shows the gien ectors, with the tip of d joined to the tail of d. The resultant ector d T is drawn in black from the tail of d 1 to the tip of d. Using a compass, the direction of d T is [E 58 S]. d T measures 6.7 cm in length, so using the scale of 1 cm 100 m, the actual magnitude of d T is 670 m. Statement The sum of the two ectors is 670 m [E 58 S]. 4. Gien d m [S]; d m [E] Required d T. Each ector can be written with the second direction first. The angle will then change to the complementary angle, so subtract the angle from 90. For example, [S 15 E] becomes [E 75 S]. 3. (a) The length of d 1 is 3.6 cm and the length of d is 5.7 cm. Using the scale of 1 cm 100 m, the actual ector of d actual ector of d is 570 m [S]. (b) Gien Figure 11 Required d T is 360 m [E] and the Analysis d T d 1 + d Analysis d T d 1 + d Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions.1-3

4 This figure shows the gien ectors, with the tip of d joined to the tail of d. The resultant ector d T is drawn in black from the tail of d 1 to the tip of d. Using a compass, the direction of d T is [S 31 E]. d T measures 7.0 cm in length, so using the scale of 1 cm 50 m, the actual magnitude of d T is 350 m. Statement The taxi s total displacement is 350 m [S 31 E]. 5. Gien d km [N]; d 4 km [E] Required d T Analysis d T d 1 + d This figure shows the gien ectors, with the tip of d 1 joined to the tail of d. The resultant ector d T is drawn in black from the tail of d to the tip of d. Using a compass, the direction of d T is [E 7 S]. d T measures 5.4 cm in length, so using the scale of 1 cm 5 m, the actual magnitude of d T is 7 m. Statement The horse s total displacement is 7 m [E 7 S]. 8. (a) Gien d 1 8 m [E 35 S]; d 45 m [S]; t 6.9 s This figure shows the gien ectors, with the tip of d joined to the tail of d. The resultant ector d T is drawn in black from the tail of d to the Required a Analysis a d T t tip of d. Using a compass, the direction of d T is [N 67 E]. d T measures 6.5 cm in length, so using the scale of 1 cm 4 km, the actual magnitude of d T is 6 km. Statement The total displacement of the two trips is 6 km [N 67 E]. 6. Yes, the answer would be the same. Whicheer order the ectors are placed, the final position, which is what determines the sum of the ectors, stays the same. 7. Gien d 1 15 m [N 3 E]; d 3 m [S 35 E] Required d T Analysis d T d 1 + d This figure shows the gien ectors, with the tip of d 1 joined to the tail of d. The resultant ector d T is drawn in black from the tail of d 1 to the tip of d. Using a compass, the direction of d T Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions.1-4

5 is [E 69 S]. d T measures 6.55 cm in length, so using the scale of 1 cm 10 m, the actual magnitude of d T is 65.5 m. a d T t 65.5 m [E 69 S] 6.9 s a 9.5 m/s [E 69 S] Statement The car s aerage elocity is 9.5 m/s [E 69 S]. (b) Gien d 1 8 m; d 45 m; t 6.9 s Required a Analysis a d T t a d 1 + d t a d + d 1 t 8 m+45 m 6.9 s a 11 m/s Statement The car s aerage speed is 11 m/s. 9. (a) Gien d km [N 30 E]; d 50.0 km [W] This figure shows the gien ectors, with the tip of d joined to the tail of d. The resultant ector d T is drawn in black from the tail of d to the tip of d. Using a compass, the direction of d T is [N]. d T measures 8.7 cm in length, so using the scale of 1 cm 10 km, the actual magnitude of d T is 87 km. Statement The aircraft s total displacement is 87 km [N]. (b) Gien d T 87 km [N]; t 10.0 min Required a Analysis a d T t a d T t 87 km [N] 10.0 min 60 min 1 h a km/h [N] Statement The aircraft s aerage elocity is km/h [N] or 50 km/h [N]. Required d T Analysis d T d 1 + d Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions.1-5

6 Section. Motion in Two Dimensions An Algebraic Approach Tutorial 1 Practice, page Gien d 1 7 m [W]; d 35 m [S] Required d T Analysis d T d 1 + d Let represent the angle d T makes with the x-axis. d T d 1 + d d T d 1 + d d T d 1 + d ( ) + ( 35 m) 7 m d T 44 m tan d d 1 tan 35 m 7 m tan 1.96 tan ( ) 5 Statement The sum of the two ectors is 44 m [W 5 S].. Gien d m [S]; d 17.8 m [E] Required d T Analysis d T d 1 + d Let represent the angle d T makes with the y-axis. d T d 1 + d d T d 1 + d d T d 1 + d ( ) + ( 17.8 m) 13. m d T. m tan d d 1 tan 17.8 m 13. m tan tan ( ) 53 The sum of the two ectors is. m [S 53 E] or [E 37 S]. Statement The sum of the two ectors is. m [E 37 S]. Tutorial Practice, page Gien d T 15 m [W 35 N] Required d x ; d y Analysis d T d x + d y Since the direction of d T is between west and north, the direction of d x is [W] and the direction of d y is [N]. sin d y d T d y d T sin ( )( sin35 ) 15 m m d y 8.6 m cos d x d T d x d T cos ( )( cos35 ) 15 m 1.9 m d x 1 m Statement The ector has a horizontal or x-component of 1 m [W] and a ertical or y-component of 8.6 m [N].. Gien d x 7. m [E]; d y 1.7 m [N] Required d T Analysis d T d 1 + d Let represent the angle d T makes with the x-axis. d T d x + d y d T d x + d y d T d x + d y ( ) + ( 1.7 m) 7. m d T 30.0 m tan d y d x tan 1.7 m 7. m tan tan ( ) 5 Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions.-1

7 Statement The sum of the two ectors is 30.0 m [E 5 N], which is the original ector from Sample Problem 1. Tutorial 3 Practice, page Gien d 1.78 cm [W]; d 6.5 cm [S 40 E] Required d T Analysis d T d 1 + d Determine the total x-component and y-component of d d 1x + d x.78 cm [W]+ 6.5 cm ( )( sin40 ) [E].78 cm [W] cm [E].78 cm [E] cm [E] cm [E] 1.4 cm [E] d 1y + d y 0 cm+ 6.5 cm cm [S] 4.79 cm [S] ( )( cos40 ) [S] Determine the magnitude of d T d T + d T + ( ) + ( cm) cm (two extra digits carried) d T 4.95 cm Let represent the angle d T makes with the x-axis. tan cm tan cm (two extra digits carried) tan tan ( ) 76 Statement The ant s total displacement is 4.95 cm [E 76 S].. Gien d 1.64 m [W 6 N]; d 3.1 m [S 1 E] Required d T Analysis d T d 1 + d Determine the total x-component and y-component of d T d 1x + d x.64 m ( )( cos6 ) [W]+( 3.1 m) ( sin1 ) [E].378 m [W] m [E].378 m [W] m [W] m [W] 1.71 m [W] d 1y + d y.64 m ( )( sin6 ) [N]+( 3.1 m) ( cos1 ) [S] m [N] m [S] m [S] m [S] m [S] 1.98 m [S] Determine the magnitude of d d T + d T + ( ) + ( m) (two extra digits carried) m d T.6 m Let represent the angle d T makes with the x-axis. tan tan m m (two extra digits carried) tan tan ( ) 49 Statement The total displacement of the paper airplane is.6 m [W 49 S]. Tutorial 4 Practice, page Answers may ary. Sample answer Imagine the rier current is flowing south and the canoe is pointed east. As long as the boat is pointed perpendicular to the current, the current has no effect on the time it takes to cross the rier. Think about the component ectors. Een though the canoe will be traelling in a direction between south and east, all its eastbound elocity is the same, no matter how fast the current is, or if there s any current at all. Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions.-

8 . (a) Gien d 0.0 m; 1.3 m/s Required t Analysis d t t d t d 0.0 m 1.3 m s s t 15 s Statement It will take the swimmer 15 s to cross the rier. (b) Gien x.7 m/s [W] Required d x Analysis x d x t d x x t d x x t.7 m s [W] s d x 4 m [W] ( ) (two extra digits carried) Statement The swimmer lands 4 m downstream from his intended location. Section. Questions, page (a) Gien d T 0 km [W 50 N] Required d x ; d y Analysis d T d x + d y Since the direction of d T is between west and north, the direction of d x is [W] and the direction of d y is [N]. sin d y d T d y d T sin ( )( sin50 ) 0 km d y 15 km cos d x d T d x d T cos ( )( cos50 ) 0 km d x 13 km Statement The ector has a ertical or y-component of 15 km [N] and a horizontal or x-component of 13 km [W]. (b) Gien d T 15 km [W 80 S] Required d x ; d y Analysis d T d x + d y Since the direction of d T is between west and south, the direction of d x is [W] and the direction of d y is [S]. sin d y d T d y d T sin ( )( sin80 ) 15 km km d y 15 km cos d x d T d x d T cos ( )( cos80 ) 15 km d x.6 km Statement The ector has a ertical or y-component of 15 km [S] and a horizontal or x-component of.6 km [W]. (c) Gien d T 40 km [N 65 E] Required d x ; d y Analysis d T d x + d y Since the direction of d T is between north and east, the direction of d x is [E] and the direction of d y is [N]. cos d y d T d y d T cos ( )( cos65 ) 40 km d y 17 km sin d x d T d x d T sin ( )( sin65 ) 40 km d x 36 km Statement The ector has a ertical or y-component of 17 km [N] and a horizontal or x-component of 36 km [E]. Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions.-3

9 . Gien d 5.1 km [E]; d 14 km [N] Required d Analysis d T d 1 + d Let represent the angle d T makes with the x-axis. d T d 1 + d d T d 1 + d d T d 1 + d ( ) + ( 14 km) 5.1 km d T 15 km tan d d 1 14 km tan 5.1 km tan.745 tan ( ) 70 Statement The sum of the two ectors is 15 km [E 70 N]. 3. Gien d 1 11 m [N 0 E]; d 9.0 m [E] Required d T Analysis d T d 1 + d Determine the total x-component and y-component of d T d 1x + d x 11 m ( )( sin0 ) [E]+ 9.0 m [E] 3.76 m [E]+ 9.0 m [E] 1.76 m [E] 13 m [E] d 1y + d y 11 m ( )( cos0 ) [N]+ 0 m m [N] 10 m [N] Determine the magnitude of d T d T + d T + ( ) + ( m) (two extra digits carried) 1.76 m d T 16 m Let represent the angle d T makes with the y-axis. tan tan 1.76 m m (two extra digits carried) tan 1.34 tan ( ) 51 Statement The total displacement of the football player is 16 m [N 51 E]. 4. Gien d m [S 5 W]; d m [N 30 E] Required d T Analysis d T d 1 + d Determine the total x-component and y-component of d T d 1x + d x 00.0 m ( )( sin5 ) [W]+( m) ( sin30 ) [E] m [W]+ 75 m [E] m [W] 75 m [W] m [W] 9.54 m [W] d 1y + d y 00.0 m ( )( cos5 ) [S]+( m) ( cos30 ) [N] m [S] m [N] m [S] m [S] m [S] m [S] Determine the magnitude of d d T + d T + ( ) + ( m) (one extra digit carried) m d T 5.3 m Let represent the angle d T makes with the y-axis. tan tan m m (one extra digit carried) Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions.-4

10 tan tan ( ) 11 Statement The total displacement of the boat is 5.3 m [S 11 W]. 5. Gien d 5 m [N 0 W]; d 35 m [S 15 E] Required d Analysis d T d 1 + d Determine the total x-component and y-component of d T d 1x + d x 5 m ( )( sin0 ) [W]+( 35 m) ( sin15 ) [E] m [W] m [E] m [E] m [E] m [E] 0.51 m [E] d 1y + d y 5 m ( )( cos0 ) [N]+( 35 m) ( cos15 ) [S] 3.49 m [N] m [S] 3.49 m [S] m [S] 10.3 m [S] m [S] Determine the magnitude of d T d T + d T + ( ) + ( 10.3 m) (two extra digits carried) m d T m Let represent the angle d T makes with the y-axis. tan tan m 10.3 m (one extra digit carried) tan tan ( ) 3 Statement The total displacement of the object is m [S 3 E]. 6. (a) Gien d km [W]; d 8.0 km [W 54 N] Analysis d T d 1 + d Determine the total x-component and y-component of d T d 1x + d x 4.3 km [W]+ 8.0 km ( )( cos54 ) [W] 4.3 km [W] km [W] 9.00 km [W] 9.0 km [W] d 1y + d y 0 km+ 8.0 km 6.47 km [N] 6.5 km [N] ( )( sin54 ) [N] Determine the magnitude of d T d T + d T + ( ) + ( 6.47 km) (two extra digits carried) 9.00 km d T 11 km Let represent the angle d T makes with the x-axis. tan 6.47 km tan 9.00 km (two extra digits carried) tan tan ( ) 36 Statement The total displacement gien by the two ectors is 11 km [W 36 N]. (b) Gien d 1 35 m [E 65 N]; d m [E 37 S] Required d T Analysis d T d 1 + d Determine the total x-component and y-component of d T d 1x + d x 35 m ( )( cos65 ) [E]+( m) ( cos37 ) [E] m [E] m [E] 3.36 m [E] 3 m [E] Required d T Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions.-5

11 d 1y + d y 35 m ( )( sin65 ) [N]+( m) ( sin37 ) [S] 31.7 m [N]+13.4 m [S] 31.7 m [N]13.4 m [N] m [N] 18 m [N] Determine the magnitude of d d T + d T + ( ) + ( m) (two extra digits carried) 3.36 m d T 37 m Let represent the angle d makes with the x-axis. tan tan m 3.36 m (one extra digit carried) tan tan ( ) 30 Statement The total displacement of the boat is 37 m [E 30 N]. 7. Gien d 5 m [N 30 W]; d 30.0 m [N 40 E]; d 3 35 m [S 5 W] Required d T Analysis d T d 1 + d + d 3 Determine the total x-component and y-component of d d 1x + d x + d 3x 5 m ( )( sin30 ) [W]+( 30.0 m) ( sin40 ) [E] ( )( sin5 ) [W] + 35 m 1.5 m [W]+19.8 m [E] m [W] 1.5 m [W]19.8 m [W] m [W] 8.01 m [W] 8.0 m [W] d 1y + d y + d 3y ( )( cos30 ) [N]+( 30.0 m) ( cos40 ) [N] ( )( cos5 ) [S] 5 m + 35 m 1.65 m [N]+.98 m [N] m [S] 1.65 m [N]+.98 m [N] 31.7 m [N] 1.91 m [N] 13 m [N] Determine the magnitude of d T d T + d T + ( ) + ( 1.91 m) (two extra digits carried) 8.01 m d T 15 m Let represent the angle d T makes with the y-axis. tan tan 8.01 m 1.91 m (two extra digits carried) tan tan ( ) 3 Statement The total displacement of the ectors is 15 m [N 3 W]. 8. (a) Gien d 5.1 km; 0.87 km/h Required t Analysis d t t d t d 5.1 km 0.87 km h 5.86 h t 5.9 h Statement It will take the swimmer 5.9 h to cross the rier. (b) Gien x.0 km/h [W] Required d x Analysis x d x t d x x t d x x t.0 km h [W] 5.86 h d x 1 km [W] ( ) (two extra digits carried) Statement The current has moed the swimmer 1 km downstream by the time she reaches the other side. Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions.-6

12 9. Time for the conductor to reach the other side Gien d 4.0 m; 1. m/s Required t Analysis d t t d t d 4.0 m 1. m s t 3.3 s Statement It will take the conductor 3.3 s to cross the railcar. The conductor s elocity relatie to the ground Gien m/s [N]; 1. m/s [E] Required T Analysis T 1 + Let represent the angle makes with the y-axis. T 1 + T 1 + T 1 + ( ) + ( 1. m/s) 4.0 m/s T 4. m/s tan 1 1. m/s tan 4.0 m/s 17 Statement The elocity of the conductor relatie to the ground is 4. m/s [N 17 E]. 10. Answers may ary. Sample answer (a) I prefer the algebraic method of adding ectors. It takes more time, but I think the answers are more accurate because there is no chance of making errors measuring. I prefer to use a diagram only to double check my answer by sketching the ectors. (b) If the naigator on a boat were working on a large map, it would probably be more useful to plot the ectors directly on the map. There would be no need for calculations and the naigator could use other map features to double check the resultant (like comparing distances and directions to landmarks on the map). Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions.-7

13 Section.3 Projectile Motion Tutorial 1 Practice, page (a) Gien d y 3 m; a y 9.8 m/s ; y 0 m/s Required t Analysis d y y t + 1 a y t d y 0+ 1 a y t t d y a y t d y a y t d y a y ( ) 3 m 9.8 m s.556 s t.6 s Statement The hockey puck is in flight for.6 s. (b) Gien t.6 s; a x 0 m/s ; x 8.6 m/s Required d x Analysis d x x t d x x t 8.6 m s.556 s d x m ( ) (two extra digits carried) Statement The range of the hockey puck is m.. Since the elocity of the puck does not affect its ertical motion, it would still take.6 s to hit the ground. The range of the puck would be half because it is traelling at half the elocity (horizontally). That means the range would be 11 m. Tutorial Practice, page Gien d y 17 m; a y 9.8 m/s ; i 7.3 m/s; 5 First determine the time of flight Required t Analysis d y y t + 1 a y t d y y t + 1 a y t i ( sin)t + 1 a y t 17 ( 7.3) ( sin5 )t + 1 ( 9.8 )t t 4.9t + 17 ( )( 17) ( ) t 3.085± ± t s or t.04 s The answer must be the positie alue. Statement The superhero is in flight for. s. Determine the range Required d x Analysis d x x t d x x t i cost 7.3 m s cos5 d x 15 m ( )(.04 s) (two extra digits carried) Statement The superhero traels 15 m horizontally before landing. Determine the final elocity Required f Analysis f fx + fy f fx + fy ( i cos)+ ( iy + a y t) i cos + i sin + a y t f 7.3 m s ( cos5 ) [right]+ 7.3 m s ( sin5 ) [up] m s [down] (.04 s ) (two extra digits carried) m/s [right] [up] m/s [down] m/s [right] [down] m/s [down] m/s [right]+18.5 m/s [down] Use the Pythagorean theorem f fx + fy f fx + fy f ( ) + ( 18.5 m/s) (two extra digits carried) m/s.0 10 m/s Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions.3-1

14 Let represent the angle f makes with the x-axis. tan fy fx 18.5 m s m s (two extra digits carried).799 tan ( ) 70 Statement The superhero s final elocity is.0 10 m/s [right 70 down].. The ball thrown at an angle aboe the horizontal will take longer to reach the ground. It has an initial elocity with a ertical component, so it will take more time for it to reach the ground due to graity. The difference in the times to reach the ground depends on the initial height and the initial elocity of the first ball. Section.3 Questions, page The horizontal and ertical motions of a projectile take the same amount of time.. Gien d x 0.0 m; a y 9.8 m/s ; x 10.0 m/s; y 0 m/s Determine the time of flight first Required t Analysis x d x t t d x x t d x x 0.0 m 10.0 m s t.00 s Statement It takes the tennis ball.00 s to reach the ground. Determine the ertical displacement Required d y Analysis d y y t + 1 a y t d y y t + 1 a y t m.00 s s d y.0 10 m ( ) Statement The water tower is.0 10 m high. 3. (a) To gie a projectile the greatest time of flight, launch it at 90 from the ground because this angle maximizes the ertical component of the elocity. At 90 from the ground, all the elocity is straight up instead of some of the elocity going into the horizontal component. (b) To gie a projectile the greatest time of flight, launch it at 45. If the angle is less than 45, the flight will be too short to trael any farther. If the angle is greater than 45, the horizontal component of the elocity is too short to trael any farther. 4. (a) The ball experiences no horizontal acceleration. The ball accelerates 9.8 m/s down due to graity. (b) The ball experiences no horizontal acceleration. The ball accelerates 9.8 m/s down due to graity. (c) The ball experiences no horizontal acceleration. The ball accelerates 9.8 m/s down due to graity. 5. The arrow strikes the ground before reaching the target. It would take the arrow more than 1 s to trael to 60 m to the target when traelling at 55 m/s. But in 1 s, the arrow would fall more than 1.5 m due to graity. For her next shot, the archer should increase the initial elocity, aim higher, or a combination of the two. 6. (a) Gien i 6 m/s; 60 ; a y 9.8 m/s ; fy 0 m/s Required t Analysis fy iy + a y t t fy iy a y t fy iy a y ( )( sin60 ) 9.8 m/s 0 6 m/s.5 m s 9.8 m s.98 s t.3 s Statement It takes the acrobat.3 s to reach his maximum height. (b) Gien i 6 m/s; 60 ; a y 9.8 m/s ; fy 0 m/s Determine the maximum height Required d y Analysis d y fy + iy t Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions.3-

15 d y fy + iy t 0.5 m s 5.88 m d y 6 m (.98 s) (two extra digits carried) Statement The maximum height of the acrobat is 6 m. Determine the time to reach half the maximum height (13 m) the second time Required t Analysis d y iy t + 1 a y t d y iy t + 1 a y t 13 (.5)t + 1 ( 9.8 )t 0.5t 4.9t 13 ( )( 13) ( ) t.5± ± t 0.68 s or t 3.9 s The first time is on his way up, so the correct time is 3.9 s. Statement It takes the acrobat 3.9 s to reach a point halfway back down to the ground. 7. Gien i 0 m/s; 45 ; a y 9.8 m/s ; fy 0 m/s Determine the time of flight Required t Analysis d y iy t + 1 a y t d y iy t + 1 a y t 0 i ( sin45 )t + 1 a y t 0 i ( sin45 )+ 1 a t, t 0 y ( ) ( sin45 ) 1 a t sin45 y i t i a y 0 m s ( sin45 ) 9.8 m s.886 s t.9 s Statement The time of flight of the golf ball is.9 s. Determine the horizontal distance Required d x Analysis d x ix t d x ix t i ( cos45 )t 0 m s sin45 d x 41 m ( )(.9 s) Statement The golfer was 41 m from the hole when he hit the ball. Determine the maximum height Required d y Analysis fy iy + a y d y d y fy iy a y d y fy iy a y ( ) ( ) ( )( sin45 ) 0 sin45 i 9.8 m/s 0 0 m/s 19.6 m/s 00 m s 19.6 m s d y m Statement The golf ball reached a maximum height of m. 8. Gien d y 1 m; a y 9.8 m/s ; i 4.5 m/s; 5 First determine the time of flight Required t Analysis d y y t + 1 a y t Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions.3-3

16 d y y t + 1 a y t i ( sin)t + 1 a y t 1 ( 4.5) ( sin5 )t + 1 ( 9.8 )t t 4.9t + 1 ( )( 1) ( ) t 1.90± ± t 1.38 s or t s The answer must be the positie alue. Statement The beanbag is in flight for 1.8 s. Determine the range Required d x Analysis d x x t d x x t i cost 4.5 m s cos5 d x 7. m ( )( s) (two extra digits carried) Statement The student s friend must stand 7. m from the building to catch the beanbag. Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions.3-4

17 Section.4 Physics Journal Section.4 Questions, page Answers may ary. Sample answer Albert Einstein considered Galileo to be the father of modern science because Galileo was the first to explore science through experiments instead of purely logical means.. Galileo chose to use a ramp to perform his acceleration experiment because objects in freefall increase their speed far too quickly to measure accurately. 3. Answers may ary. Sample answer could include one of the following Galileo discoered four of Jupiter s satellites, using a telescope that he deised. He was also the first to detect sunspots. He refuted Aristotle s theory that the moon was a smooth sphere. With the help of his telescope, Galileo showed that the Moon s surface had alleys and pot marks. 4. Answers may ary. Sample answers could include three of any of the following Until the mid nineteenth century, it was widely belieed that there were only nine planets in our solar system. But due to discoeries of other bodies in our solar system that hae the traits of a planet but were larger than Pluto. Pluto was reclassified as a dwarf planet in 006. Other newly discoered bodies were also classified as dwarf planets. So officially today, there are only eight planets in our solar system. Before the Human Genome Project started in 1984, it was widely belieed that humans shared 98.5 % of their genes with monkeys, that percentage is now 96. Howeer, it was discoered, that humans share a great percentage of the genes with other species that are ery unlike humans. In fact, 61 % of human genes are a match to genes of a household fly. Prior to the twenty-first century, it was belieed that the ice caps of the polar region did not change much in size or mass oer time. Howeer, recent studies hae shown that these masses are actually decreasing at a dramatic rate much faster than in preious years. With global warming, new phenomena are emerging such as the shrinking polar ice cap, more iolent weather patterns, and odd weather patterns for arious regions around the globe. 5. Answers may ary. Sample answer In the sixteenth and seenteenth centuries, Galileo s experiments and published works were often criticized by fellow scholars who followed Aristotle or by the Church. In 161, he published Discourse on Floating Objects, which refuted Aristotle s iews on elocity of objects and their mass, through experiments. Galileo s work was challenged in four print articles. In 1613, he published Letters on Sunspots. With the aid of his telescope, he showed that the Moon s surface was not smooth and hypothesized that the Earth was not stationary. This was in opposition to widely held iews by the Church and he was called a heretic. In 1614, he was denounced by some members of the Catholic church when he suggested that the Church s teachings and science be separate subjects. In 164, he wrote Dialogue on the Tides, which discussed Ptolemy s and Copernicus s theories on the physics of tides. The Church censors changed the title to Dialogue on the Two Chief World Systems and allowed it to be printed in 163. Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions.4-1

18 Chapter Self-Quiz, page (b). (a) 3. (c) 4. (b) 5. (a) 6. (c) 7. (d) 8. (b) 9. (b) 10. True 11. True 1. False. To find the direction of the ector 30 m [N W], point north and then turn west. 13. False. The horizontal component of a ector using the cardinal directions is the component of that ector that points east or west. 14. False. The magnitude of a ector with components 4.0 m [W] and 7.0 m [S] is 8.1 m. 15. False. The direction of the resultant ector with components 5. m/s [S] and 8.5 m/s [E] is [E 31 S]. 16. False. The ector 57.0 m [S E] has an x-component of 1.4 m [E]. Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions -1

19 Chapter Reiew, pages Knowledge 1. (b). (d) 3. (b) 4. (a) 5. (b) 6. (c) 7. (c) 8. (a) 9. (a) 10. False. A diagram with a scale of 1 cm 10 cm means that 1 cm on the diagram represents 10 cm in real life. 11. True 1. False. To add two ectors on a diagram, join them tip to tail. 13. False. The resultant ector is the ector that results from adding the gien ectors. 14. False. When gien the x- and y-component ectors, the Pythagorean theorem should be used to determine the magnitude of the displacement ector. 15. True 16. False. The x-component of the ector 8.0 m [S 45 W] is 5.7 m [W]. 17. True 18. True 19. False. When two objects are dropped from the same height at the same time, both objects will land at the same time when there is no air resistance. 0. (a)(ii) (b)(i) (c)(i) (d)() (e)(iii) Understanding 1. Answers may ary. Sample answers (a) A map or a building blueprints would hae a scale that is smaller than the real-world measurement because you would want to see the whole area at a reasonable size. (a) A diagram of a cell, atoms, or a microchip would hae a scale that is larger than the realworld measurement because the original objects are too small to see in detail.. For each ector, the magnitude stays the same, but the cardinal directions are replaced by their opposites. (a) d 17 m [W 63 S] d opposite 17 m [E 63 N] (b) d 79 cm [E 56 N] d opposite 79 cm [W 56 S] (c) d 44 km [S 7 E] d opposite 44 km [N 7 W] 3. Diagram size Real-world size 3.4 cm 170 m 0.75 cm 37.5 m 85.0 mm 45 m 5.0 cm 150 m Row 1 Multiply by 50 and change the units to metres. 3.4 cm m Row Diide by 50 and change the units to cenitmetres m cm Row 3 Multiply by 50 and change the units to metres cm m Row 4 Diide by 50 and change the units to cenitmetres. 150 m cm 4. (a) (b) Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions -

20 (c) This figure shows the gien ectors, with the tip of d 1 joined to the tail of d. The resultant ector d T is drawn in black from the tail of d 1 to the tip of d. Using a compass, the direction of d T is [W 54 N]. d T measures 5. cm in length, so using the scale of 1 cm 50 m, the actual magnitude of d T is 1300 m. Statement The net displacement of the drier is 1300 m [W 54 N]. 8. Gien d 1.0 m [N]; d 0.80 m [N 45 W] Required d T 5. (a) Since 10 is 50 times.4, an appropriate scale is 1 cm 50 m. (b) Since 360 is 150 times.4, an appropriate scale is 1 cm 150 km. (c) Since 100 is 500 times.4, an appropriate scale is 1 cm 500 m. 6. For each ector, determine the complementary angle, then reerse the order of the directions. (a) d 566 m [W 18 N] d 566 m [N 7 W] (b) d 37 cm [E 68 S] d 37 cm [S E] (c) d 7150 km [S 38 W] d 7150 km [W 5 S] 7. Gien d m [W]; d 1050 m [N] Required d T Analysis d T d 1 + d Analysis d T d 1 + d This figure shows the gien ectors, with the tip of d 1 joined to the tail of d. The resultant ector d T is drawn in black from the tail of d to the tip of d. Using a compass, the direction of d T is [N 1 W]. d T measures 5. cm in length, so using the scale of 1 cm 0.5 m, the actual magnitude of d T is.6 m. Statement The net displacement of the cue ball is.6 m [N 1 W]. 9. d x d y d T Use the Pythagorean theorem to determine each missing magnitude d x + d y d T. Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions -3

21 Row 1 d T d x + d y d T d x + d y ( ) + ( 4.0) 3.0 d T 5.0 Row d T d x + d y d y d T d x ( ) ( 8.0) 10.0 d y 6.0 Row 3 d T d x + d y d x d T d y ( ) ( 5.00) 7.81 d x 6.00 Row 4 d T d x + d y d y d T d x ( ) ( 4.00) 8.06 d y d x d y 3.0 [E] 4.0 [N] E 53 N 5.00 [W] 7.00 [N] W 54.5 N 8.0 [E] 1.1 [S] E 14.4 S 351 [W] 456 [N] W 5.4 N Use the tangent function tan d y d x. Row 1 Find the missing angle. tan d y d x tan tan (two extra digits carried) tan 1 (1.333) 53 Row Find the missing angle. tan d y d x tan tan 1.40 tan 1 (1.40) 54.5 Row 3 Find the missing component ector. tan d y d x tan 14.4 d y 8.0 (0.57)(8.0) d y 1.1 d y Row 4 Find the missing component ector. tan d y d x tan d x d x d x (a) Gien d T 5 m [W 7 S] Required d x ; d y Analysis d T d x + d y Since the direction of d T is between west and south, the direction of d x is [W] and the direction of d y is [S]. sin d y d T d y d T sin ( )( sin7 ) 5 m d y 49 m cos d x d T d x d T cos ( )( cos7 ) 5 m d x 16 m Statement The ector has a horizontal or x-component of 16 m [W] and a ertical or y-component of 49 m [S]. Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions -4

22 (b) Gien d T 38 km [E 14 N] Required d x ; d y Analysis d T d x + d y Since the direction of d T is between east and north, the direction of d x is [E] and the direction of d y is [N]. sin d y d T d y d T sin ( )( sin14 ) 38 km d y 9. km cos d x d T d x d T cos ( )( cos14 ) 38 km d x 37 km Statement The ector has a horizontal or x-component of 37 km [E] and a ertical or y-component of 9. km [N]. (c) Gien d T 9 m [S 8 W] Required d x ; d y Analysis d T d x + d y Since the direction of d T is between south and west, the direction of d x is [W] and the direction of d y is [S]. cos d y d T d y d T cos ( )( cos8 ) 9 m d y 13 m sin d x d T d x d T sin ( )( sin8 ) 9 m d x 91 m Statement The ector has a horizontal or x-component of 91 m [W] and a ertical or y-component of 13 m [S]. 3. (a) Gien d x 5.0 m [W]; d y.9 m [S] Required d T Analysis d T d x + d y Let represent the angle d T makes with the x-axis. d T d x + d y d T d x + d y d T d x + d y ( ) + (.9 m) 5.0 m d T 5.8 m tan d y d x tan.9 m 5.0 m 30 Statement The sum of the two ectors is 5.8 m [W 30 S]. (b) Gien d x 18 m [E]; d y 5. m [N] Required d T Analysis d T d x + d y Let represent the angle d T makes with the x-axis. d T d x + d y d T d x + d y d T d x + d y ( ) + ( 5. m) 18 m d T 19 m tan d y d x tan 5. m 18 m 16 Statement The sum of the two ectors is 19 m [E 16 N]. (c) Gien d x 64 km [W]; d y 31 km [N] Required d T Analysis d T d x + d y Let represent the angle d T makes with the x-axis. d T d x + d y d T d x + d y d T d x + d y ( ) + ( 31 km) 64 km d T 71 km Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions -5

23 tan d y d x 31 km tan 64 km 6 Statement The sum of the two ectors is 71 km [W 6 N]. 33. Gien d 1.0 m [left]; d 5.0 m [up] Required d T Analysis d T d 1 + d d T d 1 + d d T d 1 + d d T d 1 + d ( ) + ( 5.0 m).0 m d T 5.4 m Statement The magnitude of the resultant ector is 5.4 m. 34. Gien d.0 m [left]; d 6.0 m [up] Required d T Analysis d T d 1 + d d T d 1 + d d T d 1 + d d T d 1 + d ( ) + ( 6.0 m).0 m d T 6.3 m Statement The magnitude of the resultant ector is 6.3 m. 35. Gien d m [down]; d 4.0 m [right] Required d T Analysis d T d 1 + d d T d 1 + d d T d 1 + d d T d 1 + d ( ) + ( 4.0 m) 7.0 m d T 8.1 m Statement The magnitude of the resultant ector is 8.1 m. 36. Gien d 4 m [W 1 S]; d 33 m [E 5 S] Required d T Analysis d T d 1 + d Determine the total x-component and y-component of d T d 1x + d x 4 m ( )( cos1 ) [W]+( 33 m) ( cos5 ) [E] 3.48 m [W]+ 0.3 m [E] 3.48 m [W] 0.3 m [W] 3.16 m [W] 3. m [W] d 1y + d y 4 m ( )( sin1 ) [S]+( 33 m) ( sin5 ) [S] 4.99 m [S] m [S] m [S] 31 m [S] Determine the magnitude of d T d T + d T + ( ) + ( m) (two extra digits carried) 3.16 m d T 31 m Let represent the angle d T makes with the x-axis. tan tan m (two extra digits carried) 3.16 m 84 Statement The dog s displacement is 31 m [W 84 S]. 37. (a) Gien d 36 m;.0 m/s Required t Analysis d t t d t d 36 m.0 m s t 18 s Statement It will take the student 18 s to cross the rier. (b) Gien 6. m/s [W];!.0 m/s [N] Required T Analysis T 1 + Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions -6

24 Let represent the angle T makes with the x-axis. T 1 + T 1 + T 1 + ( ) + (.0 m/s) 6. m/s T 6.5 m/s tan 1.0 m/s tan 6. m/s 18 Statement The resulting elocity of the boat is 6.5 m/s [W 18 N]. (c) Gien x 6. m/s [W] Required d x Analysis x d x t d x x t d x x t 6. m s [W] 18 s m [W] d x m [W] ( ) Statement The student lands m or 110 m downstream from her destination. 38. The beanbag thrown at an angle will hit the ground first. Both beanbags hae the same initial elocity but different ertical components of the elocity. Since both accelerate down at the same rate (graity), the beanbag thrown at angle will hae a shorter time of flight because its initial ertical elocity was less. 39. Gien i 15 m/s; 50 Required ix ; iy Analysis i ix + iy iy sin i iy i sin ( )( sin50 ) 15 m/s iy 11 m/s cos ix i ix i cos ( )( cos50 ) 15 m/s ix 9.6 m/s Statement The initial elocity has a horizontal or x-component of 9.6 m/s and a ertical or y-component of 11 m/s. 40. (a) Gien d y 1.3 m; a y 9.8 m/s ; y 0 m/s Required t Analysis d y y t + 1 a y t d y 0+ 1 a y t t d y a y t d y a y t d y a y ( ) 1.3 m 9.8 m s s t 0.5 s Statement The time of flight of the beanbag should be 0.5 s. (b) Gien t s; x 4. m/s Required d x Analysis d x x t d x x t 4. m s s d x. m ( ) (two extra digits carried) Statement The range of the beanbag should be. m. 41. (a) Gien a y 9.8 m/s ; t 3.78 s; 45.0 Required i Analysis i i" sin Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions -7

25 Determine the y-component fy iy +a y t iy fy a y t ( ) 3.78 s m/s 18.5 m/s iy 19 m/s Use the sine function sin i# i i i# sin 18.5 m/s (two extra digits carried) sin45.0 i 6. m/s Statement The cannon is fired with an initial elocity of 6. m/s. (b) Gien iy 18.5 m/s; t 3.78 s Required d y Analysis $x d x t d x $x t Since the initial angle is 45.0, the horizontal and ertical components of the elocity hae the same magnitude. d x ix t d x iy t 18.5 m s 3.78 s d x 70.0 m ( ) (two extra digits carried) Statement The range of the cannon is 70.0 m. This figure shows the gien ectors, with the tip of d ( joined to the tail of d, then the tip of d ) joined to the tail of d *. The resultant ector d T is drawn in black from the tail of d + to the tip of d,. Using a compass, the direction of d T is [N 30 E]. d T measures 5.6 cm in length, so using the scale of 1 cm 0 m, the actual magnitude of d T is 11 m. Statement The total displacement for his trip is 11 m [N 30 E]. 43. Gien d - 0 m [E 40 N]; d 360 m [N 30 W]; t s Determine the displacement of the boat Required d T Analysis d T d 1 + d Analysis and Application 4. (a) The trip would be represented by three ectors a 50 m [N] ector, a 50 m [E] ector, and another 50 m [N] ector. A possible scale would be 1 cm 0 m, so each ector is.5 cm long. (b) Gien d % 50 m [N]; d & 50 m [E]; d ' 50 m [N] Required d T Analysis d T d 1 + d + d 3 This figure shows the gien ectors, with the tip of d. joined to the tail of d. The resultant ector d T is drawn in black from the tail of d 1 to the tip of d. Using a compass, the direction of d T Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions -8

26 is [N 1.4 W]. d / measures 4.5 cm in length, so using the scale of 1 cm 100 m, the actual magnitude of d T is 450 m. Statement The boat s displacement is 450 m [N 1.4 W]. Determine the aerage elocity of the boat Required a Analysis a d t a d t 450 m [N 1.4 W] s 0.45 m/s [N 1.4 W] a.0 10 m/s [N 1.4 W] Statement The boat s aerage elocity is.0 10 m/s [N 1.4 W]. 44. Answers may ary. Sample answer (a) Since I would draw c from the tail of a to the tip of b (when they were tip to tail), I would draw a and c touching tails, and the displacement from the tip of a to the tip of c would represent b. (b) As in part (a), I would draw what I knew of the ector addition to determine the magnitude and direction of the missing ector. This time, I would put the tips of b and c together, and a would be the ector from the tail of c to the tail of b. (c) It does not matter which component ector because ectors can be added in either order and you would always get the same resultant ector. (d) Gien 0.1 cm [W]; c 4.3 cm [W 45 N] Required d T Analysis d T d 1 + d 45. Gien d 4 5 blocks [N]; d 5 blocks [E] Required d T Analysis d T d 1 + d d T d 1 + d 5 blocks+5 blocks 50 m ( 10 blocks) 1 block d T 500 m Statement The total distance she traels is 500 m or.5 km. 46. (a) The most direct route is 1.5 blocks [E], 9 blocks [S], and another 1.5 blocks [E]. (b) Gien 40.0 km/h; d blocks [E]; d 5 9 blocks [S]; d blocks [E] Required t Analysis d t t d t d 1.5 blocks+9 blocks+ 1.5 blocks 40.0 km/h 1 blocks 50 m 40.0 km/h 1 block 3000 m 1 km 40.0 km 1000 m h 60 min h 1 h t 4.5 min Statement It will take the student 4.5 min to get to the market. 47. Gien d 7 3 blocks [N]; d 8 5 blocks [W] Required d T This figure shows the gien ectors, with the tail of a joined to the tail of c. The missing ector b is drawn in black from the tip of a to the tip of c. Using a compass, the direction of b is [W 74 N]. b measures 3.4 cm in length. Statement Vector b is 3.4 m [W 74 N]. Analysis 9 T Let represent the angle d T makes with the x-axis. 3 blocks [N] 750 m [N] 5 blocks [W] 150 m [W] Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions -9

27 d T d 1 + d d T d 1 + d d T d 1 + d ( ) + ( 150 m) 750 m d T 1460 m tan d d 1 tan 150 m 750 m 59 Statement The student s net displacement is 1460 m [N 59 W]. 48. Gien d 1 5 blocks [W]; d 11 blocks [S]; d blocks [W] Required d T Analysis d T d x + d y Let represent the angle d T makes with the x-axis. 5 blocks [W] 150 m [W] 11 blocks [S] 750 m [S] blocks [W] 500 m [W] d x 150 m [W]+ 500 m [W] 1750 m [W] d y 750 m [S] d T d x + d y d T d x + d y d T d x + d y ( ) + ( 750 m) 1750 m d T 360 m tan d y d x tan 750 m 1750 m 57.5 Statement The student s net displacement is 360 m [W 57.5 S]. 49. (a) Gien 5 km/h; d ; 5 blocks [W]; < 5 blocks [S] Required t Analysis d t t d t d 5 blocks+5 blocks 5 km/h 10 blocks 50 m 5 km/h 1 block 500 m 1 km 5 km 1000 m h 60 min 0.1 h 1 h t 6.0 min Statement It takes her 6.0 min to drie home from the school. (b) Gien d 1 5 blocks [W]; d 5 blocks [S]; t 6.0 min Determine the total displacement Required d T Analysis > T > 1 + > Let represent the angle? T makes with the x-axis. 5 blocks 150 m d T d 1 + d d T d 1 + d d T d 1 + d ( ) + ( 150 m) 150 m d T 1770 m tan d d 1 tan 150 m 150 m 45 Statement The net displacement from school to home is 1770 m [W 45 S]. Determine the aerage elocity Required a Analysis a d t a d t 1770 m [W 45 S] 6.0 min a 18 km/h 60 min 1 h 1 km 1000 m Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions -10

28 Statement Her aerage elocity from school to home is 18 km/h [W 45 S]. 50. Gien 5. km/h; 5 blocks [W]; d A blocks [S]; d B 5 blocks [E] Required C Analysis d t t d t d 5 blocks+ blocks+5 blocks 5. km/h 1 blocks 50 m 5. km/h 1 block 3000 m 1 km 60 min 5. km 1000 m 1 h h t 7.14 min Statement It took the student 7.14 min to get from school to the library. 51. Gien d x 750 m; d T 1100 m Determine how far north the boat traelled Required d x Analysis d T d x + d y d y d T d x d y d T d x d y d T d x ( ) + ( 750 m) DDEE m d y 800 m Statement The boat traelled 800 m north. Determine the direction the boat traelled Required Analysis sin d x d T Let represent the angle d T makes with the y-axis. sin d x d T 750 m 1100 m 43 Statement The boat traelled in the direction [N 43 E] 5. Gien d x 13 m; [N 3 W] Required d T Analysis sin d x d T d T d x sin d T d x sin 13 m sin3 d T 5 m Statement He needs to kick the ball at least 5 m in order to make it through the centre of the posts. 53. Gien F G 11 m [N]; t s; d H 6 m [W 4 N]; t 1. s Required a Analysis a d t Determine the total x-component and y-component of d T d 1x + d x 0+ 6 m ( )( cos4 ) [W] 19.3 m [W] 19 m [W] d 1y + d y 11 m [N]+ 6 m ( )( sin4 ) [N] 11 m [N] m [N] 8.40 m [N] 8 m [N] Determine the magnitude of d T d T + d T + ( ) + ( 8.40 m) (two extra digits carried) 19.3 m d T 34 m Let represent the angle d T makes with the y-axis. tan tan 19.3 m 8.40 m 34 (one extra digit carried) Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions -11

29 Determine the aerage elocity a d t d T t 1 +t m [N 34 W] 0.55 s+ 1. s m [N 34 W] 1.75 s a 0 m/s [N 34 W] Statement The puck s aerage elocity is 0 m/s [N 34 W]. 54. Gien x 5. m/s; d x 35 m; d y 5 m Required y Analysis y I y t Determine the time to cross the rier x d x t t d x x 35 m 5. m s s t 6.7 s Determine the speed of the rier current y d y t 5 m s y 3.7 m/s (two extra digits carried) Statement The speed of the current is 3.7 m/s. 55. (a) The student is on the south bank of an eastwest rier, so the resulting direction of the elocity of the boat must be north. (b) The student should point the boat north, then turn it west into the current at an angle that will result in the boat traelling directly across the rier. (c) Gien d y 50 m; x 1.1 m/s; T 3.8 m/s Required Analysis sin J T sin y T 1.1 m s 3.8 m s 17 Statement The student should point the boat [N 17 W]. (d) Gien d y 50 m; T 3.8 m/s Required t Analysis d t t d t d 50 m cos m s t 14 s Statement It takes the student 14 s to cross the rier. 56. Gien d y 1. m; a y 9.8 m/s ; x 1.5 m/s; y 0 m/s Determine the time of flight Required t Analysis d y y t + 1 a y t d y 0+ 1 a y t t d y a y t d y a y t d y a y ( ) 1. m 9.8 m s s t 0.49 s Statement The hockey puck is in flight for 0.49 s. Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions -1

30 Determine the range Required d x Analysis d x x t d x x t 1.5 m s s d x 0.74 m ( ) (two extra digits carried) Statement The range of the hockey puck is 0.74 m. (b) Gien d y 1. m; a y 9.8 m/s ; x 1.5 m/s; y 0 m/s Required f Analysis K Kx + Ky fy a y t ( )( s) (two extra digits carried) 9.8 m/s [down] fy 4.85 m/s [down] Use the Pythagorean theorem f fx + fl f fx + fl f ( ) + ( 4.85 m/s) (one extra digit carried) 1.5 m/s 5.1 m/s Let represent the angle f makes with the x-axis. tan fy fx 4.85 m s 1.5 m (one extra digit carried) s 73 Statement The puck s final elocity is 5.1 m/s [73 aboe horizontal]. 57. (a) Gien a y 9.8 m/s ; i 16.5 m/s; 35 Required t Analysis d y y t + 1 a y t d y y t + 1 a y t i ( sin)t + 1 a y t 0 ( m/s)( 4.9 m/s )M (M 0) M m s 4.9 m s s M 1.9 s Statement The soccer ball s time of flight is 1.9 s. (b) Gien i 16.5 m/s; 35 Required d x Analysis d x x t d x x t i cost 16.5 m s cos35 d x 6 m ( )( s) (two extra digits carried) Statement The soccer ball s range is 6 m. (c) Gien a y 9.8 m/s ; i 16.5 m/s; 35 ; fy 0 m/s Required d y Analysis fy iy + a y d y d y fy iy a y d y fy iy a y ( ) ( ) ( )( sin35 ) 0 sin35 i 9.8 m/s m/s 19.6 m/s m s 19.6 m s N y 4.6 m Statement The soccer ball reached a maximum height of 4.6 m. 58. (a) Gien a y 9.8 m/s ; t. s; d x 17 m; d y 5. m Required i Analysis O Ox + Oy 0 ( 16.5 m/s) ( sin35 )t + 1 ( 9.8 m/s)t 0 ( m/s)t ( 4.9 m/s )t Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions -13

31 Determine the x-component ix d x t 17 m. s 7.77 m/s ix 7.7 m/s Determine the y-component fy iy + P y d y iy fy P y d y iy fy P y d y ( )( 5. m) m/s m /s m/s iy 10 m/s Use the Pythagorean theorem f fx + fy f fx + fy f ( ) + ( m/s) (two extra digits carried) 7.77 m/s 13 m/s Statement The soccer ball is kicked with an initial elocity of 13 m/s. (b) Gien a y 9.8 m/s ; t. s; d x 17 m; d y 5. m Required Analysis tan iy ix tan iy ix m s 7.77 m (two extra digits carried) s 53 Statement The soccer ball is kicked at an angle of 53. Ealuation 59. Answers may ary. Sample answer Scale drawings require accuracy in scale, drawing, and measurement in order to get the correct answer. It can also be time consuming. It is less effectie than calculating a precise answer. 60. Answers may ary. Sample answer Time is a scalar because it always has the same direction forward in time. The analogy in the question relates time to distance and ordering magnitudes only. 61. (a) The ertical distance is being manipulated to obsere differences in the horizontal displacement (range) and time of flight. Initial elocity and angle of launch are being controlled, while acceleration due to graity is constant. (b) Student B s data will be the most alid because she is repeating her launches ten times to aoid any errors due to measurement or malfunction. (c) Answers may ary. Sample answer Errors can happen due to bad measuring or problems keeping the initial elocity and angle controlled. Haing more than one person measuring time and distance can preent errors, as well as making sure the launch mechanism is working consistently. Reflect on Your Learning 6. (a) When drawing two ectors that are added together, draw the first ector then draw the second ector, keeping its size and direction, starting at the tip of the first. The resultant ector starts at the tail of the first ector and ends at the tip of the second. (b) Answers may ary. Sample answer To subtract two ectors, you could use the rules for ector addition in reerse. Rearrange your ector subtraction to look like an addition problem and apply the steps of addition to find the missing ector. First, draw the resultant ector. You know that this connects the tail of one ector to the tip of the other. Then draw the other known ector starting its tail at the same point as the tail of the resultant ector. The missing ector is the ector connecting the tip of the second ector to the tip of the resultant ector. (c) Answers may ary. Sample answer I prefer the algebraic method of ector addition. Sometimes it seems like a lot more work since ectors must be first broken down into components, then added, then the resultant ector must be determined, but this method is more accurate. As long as you are careful with your algebra there are fewer errors and mistakes than using scale diagrams. 63. Students should discuss any gaps in their understanding of motion problems and how they could learn more about soling motion problems. Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions -14

32 Research 64. Students answers should be a few paragraphs that discuss the origin and deelopment of the compass rose. 65. Students papers should discuss the Cartesian coordinate system and its deelopment. Descartes should be mentioned as well as a reference to at least one other type of coordinate system such as a three-dimensional rectilinear, polar, or spherical coordinate system. 66. Students reports should discuss the uses of accelerometers in nature. The examples gien include the study of shark mating behaiour, the study of gliding behaiour of lemurs, and turning laptops into earthquake sensors. 67. Students should describe both radar and laser deices. The physics behind each technology should be briefly outlined in one or two paragraphs. Copyright 011 Nelson Education Ltd. Chapter Motion in Two Dimensions -15