1. e. The acceleration decreases inversely with the square of the distance. At height $R_{\text{e}}$, the distance from the center of the earth is $2R_{\text{e}}$.

2. c. Newton’s third law requires $F_{1\text{on}2}=F_{2\text{on}1}$.

3. b. $g_{\text{surface}}=GM/R^2$. Because of the square, a radius twice as large balances a mass four times as large.

4. In absolute value, $U{}_{\text{E}}>U{}_{\text{A}}=U{}_{\text{B}}=U{}_{\text{D}}>U{}_{\text{C}}$. $\left|U_{\text{G}}\right|$ is proportional to $m_1{m}_2\text{/}r$.

5. a. $T^2$ is proportional to $r^3$, or $T$ is proportional to $r^{3/2}$. $4^{3/2}=8$.

1. ${\displaystyle \frac{F_{\text{s on m}}}{F_{\text{e on m}}}}=2.18$

3.

1. $6.7\times {10}^{-9}\text{N}$

2. $6.8\times {10}^{-9}\text{N}$

5. $2.3\times {10}^{-7}\text{N}$

7. 12 cm

9.

1. $274{\text{m/s}}^2$

2. $5.90\times {10}^{-3}{\text{m/s}}^2$

11. $39.3{\text{m/s}}^2$

13.

1. $3.0\times {10}^{24}\text{kg}$

2. $0.89{\text{m/s}}^2$

15. $59.5\text{km}\text{/}\text{s}$

17. 10 km/s

19. $2.4\times {10}^{30}\text{kg}$

21. 14.7 km

23.

1. 11 km/s

2. 8.9 km/s

25. $M_{\text{p}}=1.5\times {10}^{25}\text{ kg, }{M}_{\text{s}}=5.2\times {10}^{30}\text{ kg}$

27. $2.9\times {10}^9\text{m}$

29. 4.2 h

31.

1. $9.9R_{\text{s}}$

2. $2.4\times {10}^4\text{m}\text{/}\text{s}$

33. $8.67\times {10}^7\text{m}$

35.

1. $1.3\times {10}^{-6}\text{N,}83\text{°}$ cw from the $+y\text{-axis}$

2. $2.3\times {10}^{27}\text{N,}7\text{.}5\text{°}$ ccw from the $-y\text{-axis}$

37. $-1.48\times {10}^{-7}\text{J}$

39. 46 kg and 104 kg

41. $v=\sqrt{{\displaystyle \frac{2GMh}{R(R+h)}}}$

43. $4.1\times {10}^{24}\text{kg}$

45. 4.7 km

47. Asteroid 1: $1.032(GM\text{/}R{)}^{1\text{/}2}$, asteroid 2: $0.516(GM\text{/}R{)}^{1\text{/}2}$

49. 0.414R

51. $1.2\times {10}^{11}\text{J}$

53. $5.0\times {10}^6\text{m}$

55.

1. 0.0066c

2. $4.5\times {10}^6{M}_{\text{s}}$

57. $T=[{\displaystyle \frac{4{\pi }^2{r}^3}{G}}{\displaystyle \frac{1}{M+m/4}}{]}^{1/2}$

59.

1. $2.36\times {10}^{20}\text{m}$

2. 24

3. $1.9\times {10}^{41}\text{kg}$

4. $9.4\times {10}^{10}$

61. 4.49 km/s

63.

3. $6.21\times {10}^7\text{m}$

65.

3. 1680 m/s

66.$g_{\mathrm{eff}}=4.8\text{m}\text{/}{\text{s}}^2$

67. Each has a speed of $3.0\times {10}^4\text{m}\text{/}\text{s.}$

69.

1. $5.8\times {10}^{22}\text{kg}$

2. $1.3\times {10}^6\text{m}$