1. c. $v=0$ requires both $v_x=0$ and $v_y=0$. Neither x nor y can be changing.

2. d. The parallel component of $\vec{a}$ is opposite $\vec{v}$ and will cause the particle to slow down. The perpendicular component of $\vec{a}$ will cause the particle to change direction downward.

3. d. A projectile’s acceleration $\vec{a}=-gĵ$ does not depend on its mass. The second marble has the same initial velocity and the same acceleration, so it follows the same trajectory and lands at the same position.

4. f. The plane’s velocity relative to the helicopter is $\vec{v}{}_{\text{PH}}=\vec{v}{}_{\text{PH}}=\vec{v}{}_{\text{PG}}+\vec{v}{}_{\text{GH}}=\vec{v}{}_{\text{PG}}-\vec{v}{}_{\text{HG}}$, where G is the ground. The vector addition shows that $\vec{v}{}_{\text{PH}}$ is to the right and down with a magnitude greater than the 100 m/s of $\vec{v}{}_{\text{PG}}$.

5. b. An initial cw rotation causes the particle’s angular position to become increasingly negative. The speed drops to half after reversing direction, so the slope becomes positive and is half as steep as the initial slope. Turning through the same angle returns the particle to $\theta =0\text{°}$.

6. $a{}_B>a{}_E>a{}_A=a{}_C>a{}_D$. Centripetal acceleration is $v^2/r$. Doubling r decreases $a_r$ by a factor of 2. Doubling v increases $a_r$ by a factor of 4. Reversing direction doesn’t change $a_r$.

7. c. $\omega$ is negative because the rotation is cw. Because $\omega$ is negative but becoming less negative, the change $\text{Δ}\omega$ is positive. So $\alpha$ is positive.

3. C

5.

1. Slowing

2. Curving downward

7. $2.2\text{m}\text{/}{\text{s}}^2$

9.

1. 0 m, 2 m/s; 0 m, 8.3 m/s

2. $90\text{°}$ south of the $+x\text{-}\text{axis}$, $14\text{°}$ north of the $+x\text{-}\text{axis}$

11. 19.6 m

13.

1. ${\overrightarrow{v}}_0=(2.0î+4.0\widehat{j})\text{ m/s,}{\overrightarrow{v}}_2=(2.0î+0.0\widehat{j})\text{ m/s,}{\overrightarrow{v}}_3=(2.0î+2.0\widehat{j})\text{ m/s}$

2. $-2.0\text{m}\text{/}{\text{s}}^2$

3. $63\text{°}$ above the $+x\text{-}\text{axis}$

15.

1. 0.064 s

2. 780 m/s

17.

1. 276 m farther

2. 12.75 s longer

19.

1. 50 m east (downstream)

2. 2.2 m/s

21.

1. $42\text{°}$ west of north

2. 45 s

23. 8.8 rev

25.

27. $8\text{°/}\text{h}$

29. 22 cm/s

31. 43 m

33.

1. $3.0\times {10}^4\text{m}\text{/}\text{s}$

2. $2.0\times {10}^{-7}\text{rad}\text{/}\text{s}$

3. $6.0\times {10}^{-3}\text{m}\text{/}{\text{s}}^2$

35. $v=5.7\text{m/s, }{a}_r=108\text{m}\text{/}{\text{s}}^2$

37.

1. 30 rad/s

2. 20 rad/s

39. 98 rpm

41.

1. 53 rpm

2. 12.3 rev

43. 38 rev

45. $(\frac{1}{2}b{t}^2+v_{0x})î+(e^{-ct}+v_{0y})ĵ$

47. $15\text{°}\text{,}75\text{°}$

49.

1. 12 m/s

2. 0.90 m

51. $13\text{°}$

53. 7.6 m

55.

1. 13 m/s

2. $48\text{°}$

57. $470\text{m}\text{/}{\text{s}}^2$

59.

2. $3.9\times {10}^2\text{m}$

3. $7.4\text{°}$ below the horizontal

61. 69 m/s, $21\text{°}$

63.

1. 32 rad/s

2. $1.7\times {10}^2\text{rad}\text{/}{\text{s}}^2$

3. 28 rev

65.

1. $1.75\times {10}^4\text{m}\text{/}{\text{s}}^2$

2. $4.4\times {10}^3\text{m}\text{/}{\text{s}}^2$

67.

1. $v=\sqrt{2\alpha \text{Δ}\theta R}$

2. $a=2\alpha \text{Δ}\theta R$

69.

1. 12 m/s

2. 36 rev

71.

1. 1.0 km/s

2. $1.4\times {10}^2\text{rad}\text{/}\text{s}$

3. 40,000 rev

73.

1. 6.3 s

2. 84 rad

75. $2.3\times {10}^2{\text{rad/s}}^2$

77. 550 rpm

79.

2. 30 m west

81. 3.8 m

83. $\theta ={\displaystyle \frac{\pi }{2}}-{\displaystyle \frac{\varphi }{2}}$

85. $10\text{°}$