University Physics Volume 1

# Chapter 2

2.1

a. not equal because they are orthogonal; b. not equal because they have different magnitudes; c. not equal because they have different magnitudes and directions; d. not equal because they are antiparallel; e. equal.

2.2

16 m; $D→=−16mu^D→=−16mu^$

2.3

G = 28.2 cm, $θG=291°θG=291°$

2.4

$D→=(−5.0i^−3.0j^)cmD→=(−5.0i^−3.0j^)cm$; the fly moved 5.0 cm to the left and 3.0 cm down from its landing site.

2.5

5.83 cm, $211°211°$

2.6

$D → = ( −20 m ) i ^ D → = ( −20 m ) i ^$

2.7

35.2 m/s = 126.7 km/h

2.8

$G → = ( 10.25 i ^ − 26.22 j ^ ) cm G → = ( 10.25 i ^ − 26.22 j ^ ) cm$

2.9

D = 55.7 N; direction $65.7°65.7°$ north of east

2.10

$v^=0.8i^+0.6j^v^=0.8i^+0.6j^$, $36.87°36.87°$ north of east

2.11

$A→·B→=−57.3A→·B→=−57.3$, $F→·C→=27.8F→·C→=27.8$

2.13

$131.9 ° 131.9 °$

2.14

$W1=1.5JW1=1.5J$, $W2=0.3JW2=0.3J$

2.15

$A→×B→=−40.1k^A→×B→=−40.1k^$ or, equivalently, $|A→×B→|=40.1|A→×B→|=40.1$, and the direction is into the page; $C→×F→=+157.6k^C→×F→=+157.6k^$ or, equivalently, $|C→×F→|=157.6|C→×F→|=157.6$, and the direction is out of the page.

2.16

a. $−2k^−2k^$, b. 2, c. $153.4°153.4°$, d. $135°135°$

## Conceptual Questions

1.

scalar

3.

5.

parallel, sum of magnitudes, antiparallel, zero

7.

no, yes

9.

zero, yes

11.

no

13.

equal, equal, the same

15.

a unit vector of the x-axis

17.

They are equal.

19.

yes

21.

a. $C=A→·B→C=A→·B→$, b. $C→=A→×B→C→=A→×B→$ or $C→=A→−B→C→=A→−B→$, c. $C→=A→×B→C→=A→×B→$, d. $C→=AB→C→=AB→$, e. $C→+2A→=B→C→+2A→=B→$, f. $C→=A→×B→C→=A→×B→$, g. left side is a scalar and right side is a vector, h. $C→=2A→×B→C→=2A→×B→$, i. $C→=A→/BC→=A→/B$, j. $C→=A→/BC→=A→/B$

23.

They are orthogonal.

## Problems

25.

$h→=−49mu^h→=−49mu^$, 49 m

27.

30.8 m, $35.7°35.7°$ west of north (equivalently, $54.2°54.2°$ north of west or $125.7°125.7°$ from east)

29.

134 km, $80°80°$

31.

7.34 km, $63.5°63.5°$ south of east

33.

3.8 km east, 3.2 km north, 7.0 km

35.

14.3 km, $65°65°$

37.

a. $A→=+8.66i^+5.00j^A→=+8.66i^+5.00j^$, b. $B→=+3.01i^+3.99j^B→=+3.01i^+3.99j^$, c. $C→=+6.00i^−10.39j^C→=+6.00i^−10.39j^$, d. $D→=−15.97i^+12.04j^D→=−15.97i^+12.04j^$, f. $F→=−17.32i^−10.00j^F→=−17.32i^−10.00j^$

39.

a. 1.94 km, 7.24 km; b. proof

41.

3.8 km east, 3.2 km north, 2.0 km, $D→=(3.8i^+3.2j^)kmD→=(3.8i^+3.2j^)km$

43.

$P1(2.165m,1.250m)P1(2.165m,1.250m)$, $P2(−1.900m,3.290m)P2(−1.900m,3.290m)$, 4.55 m

45.

8.60 m, $A(25m,0.647π)A(25m,0.647π)$, $B(32m,0.75π)B(32m,0.75π)$

47.

a. $A→+B→=−4i^−6j^A→+B→=−4i^−6j^$, $|A→+B→|=7.211,θ=236°|A→+B→|=7.211,θ=236°$; b. $A→−B→=–2i^+2j^A→−B→=–2i^+2j^$, $|A→−B→|=22,θ=135°|A→−B→|=22,θ=135°$

49.

a. $C→=(5.0i^−1.0j^−3.0k^)m,C=5.92mC→=(5.0i^−1.0j^−3.0k^)m,C=5.92m$;
b. $D→=(4.0i^−11.0j^+15.0k^)m,D=19.03mD→=(4.0i^−11.0j^+15.0k^)m,D=19.03m$

51.

$D→=(3.3i^−6.6j^)kmD→=(3.3i^−6.6j^)km$, $i^i^$ is to the east, 7.34 km, $−63.5°−63.5°$

53.

a. $R→=−1.35i^−22.04j^R→=−1.35i^−22.04j^$, b. $R→=−17.98i^+0.89j^R→=−17.98i^+0.89j^$

55.

$D→=(200i^+300j^)ydD→=(200i^+300j^)yd$, D = 360.5 yd, $56.3°56.3°$ north of east; The numerical answers would stay the same but the physical unit would be meters. The physical meaning and distances would be about the same because 1 yd is comparable with 1 m.

57.

$R → = ( −3 i ^ − 16 j ^ ) m R → = ( −3 i ^ − 16 j ^ ) m$

59.

$E→=EE^E→=EE^$, $Ex=+178.9V/mEx=+178.9V/m$, $Ey=−357.8V/mEy=−357.8V/m$, $Ez=0.0V/mEz=0.0V/m$, $θE=−tan−1(2)θE=−tan−1(2)$

61.

a. $–34.290 R→B=(–12.278i^+7.089j^+2.500k^)km–34.290R→B=(–12.278i^+7.089j^+2.500k^)km$, $R→D=(−34.290i^+3.000k^)kmR→D=(−34.290i^+3.000k^)km$; b. $|R→B−R→D|=23.131km|R→B−R→D|=23.131km$

63.

a. 0, b. 0, c. –0.866, d. –17.32

65.

$θ i = 64.12 ° , θ j = 150.79 ° , θ k = 77.39 ° θ i = 64.12 ° , θ j = 150.79 ° , θ k = 77.39 °$

67.

a. $−120k^−120k^$, b. $0k^0k^$, c. $–94k^–94k^$, d. $–240k^–240k^$, e. $4.0k^4.0k^$, f. $−3.0k^−3.0k^$, g. $15k^15k^$, h. 0

69.

a. 0, b. 0, c. $+–20,000k^+–20,000k^$

71.

a. 18.4 km and 26.2 km, b. 31.5 km and 5.56 km

73.

a. $(r,π−φ)(r,π−φ)$, b. $(2r,φ+2π)(2r,φ+2π)$, c. $(3r,−φ)(3r,−φ)$

75.

$d PM = 6.2 nmi = 11.4 km , d NP = 7.2 nmi = 13.3 km d PM = 6.2 nmi = 11.4 km , d NP = 7.2 nmi = 13.3 km$

77.

proof

79.

a. 10.00 m, b. $5πm5πm$, c. 0

81.

22.2 km/h, $35.8°35.8°$ south of west

83.

270 m, $4.2°4.2°$ north of west

85.

$B→=−4.0i^+3.0j^B→=−4.0i^+3.0j^$ or $B→=4.0i^−3.0j^B→=4.0i^−3.0j^$

87.

proof

## Challenge Problems

89.

$G H = 19 N / 17 ≈ 4.6 N G H = 19 N / 17 ≈ 4.6 N$

91.

proof