College Physics 2e

# Chapter 28

## Problems & Exercises

1.

(a) 1.0328

(b) 1.15

3.

$5 . 96 × 10 − 8 s 5 . 96 × 10 − 8 s$

5.

$0.800 c 0.800 c$

7.

$0 . 140 c 0 . 140 c$

9.

(a) $0.745c0.745c$

(b) $0.99995c0.99995c$ (to five digits to show effect)

11.

(a) 0.996

(b) $γγ$ cannot be less than 1.

(c) Assumption that time is longer in moving ship is unreasonable.

12.

48.6 m

14.

(a) 1.387 km = 1.39 km

(b) 0.433 km

(c) $L = L 0 γ = 1.387 × 103m 3.20 = 433.4 m = 0.433 km L = L 0 γ = 1.387 × 103m 3.20 = 433.4 m = 0.433 km$

Thus, the distances in parts (a) and (b) are related when $γ=3.20γ=3.20$.

16.

(a) 4.303 y (to four digits to show any effect)

(b) 0.1434 y

(c) $Δt = γΔt 0 ⇒ γ = Δt Δt 0 = 4 . 303 y 0 . 1434 y = 30 . 0 Δt = γΔt 0 ⇒ γ = Δt Δt 0 = 4 . 303 y 0 . 1434 y = 30 . 0$

Thus, the two times are related when $γ= 30 . 00 γ= 30 . 00$.

18.

(a) 0.250

(b) $γγ$ must be ≥1

(c) The Earth-bound observer must measure a shorter length, so it is unreasonable to assume a longer length.

20.

(a) $0.909c0.909c$

(b) $0.400c0.400c$

22.

$0 . 198 c 0 . 198 c$

24.

a) $658 nm658 nm$

b) red

c) $v/c=9.92×10−5v/c=9.92×10−5$ (negligible)

26.

$0 . 991 c 0 . 991 c$

28.

$− 0 . 696 c − 0 . 696 c$

30.

$0 . 01324 c 0 . 01324 c$

32.

$u′=cu′=c$, so

$u= v+u′ 1 + ( vu′/ c 2 ) = v+c 1 + ( vc / c 2 ) = v+c 1 + ( v / c ) = c ( v+c ) c+v = c u= v+u′ 1 + ( vu′/ c 2 ) = v+c 1 + ( vc / c 2 ) = v+c 1 + ( v / c ) = c ( v+c ) c+v = c$

34.

a) $0.99947c0.99947c$

b) $1.2064×1011 y1.2064×1011 y$

c) $1.2058×1011 y1.2058×1011 y$ (all to sufficient digits to show effects)

35.

$4 . 09 × 10 –19 kg ⋅ m/s 4 . 09 × 10 –19 kg ⋅ m/s$

37.

(a) $3.000000015 ×1013 kg⋅m/s3.000000015 ×1013 kg⋅m/s$.

(b) Ratio of relativistic to classical momenta equals 1.000000005 (extra digits to show small effects)

39.

$2.9957 × 10 8 m/s 2.9957 × 10 8 m/s$

41.

(a) $1.121×10–8 m/s1.121×10–8 m/s$

(b) The small speed tells us that the mass of a proton is substantially smaller than that of even a tiny amount of macroscopic matter!

43.

$8.20 × 10 − 14 J 8.20 × 10 − 14 J$

0.512 MeV

45.

$2 . 3 × 10 − 30 kg 2 . 3 × 10 − 30 kg$

47.

(a) $1 . 11 × 10 27 kg 1 . 11 × 10 27 kg$

(b) $5 . 56 × 10 − 5 5 . 56 × 10 − 5$

49.

$7 . 1 × 10 − 3 kg 7 . 1 × 10 − 3 kg$

$7 . 1 × 10 − 3 7 . 1 × 10 − 3$

The ratio is greater for hydrogen.

51.

208

$0.999988 c 0.999988 c$

53.

$6.92 × 10 5 J 6.92 × 10 5 J$

1.54

55.

(a) $0 . 914 c 0 . 914 c$

(b) The rest mass energy of an electron is 0.511 MeV, so the kinetic energy is approximately 150% of the rest mass energy. The electron should be traveling close to the speed of light.

57.

90.0 MeV

59.

(a) $E 2 = p 2 c 2 + m 2 c 4 = γ 2 m 2 c 4 , so that p 2 c 2 = γ 2 − 1 m 2 c 4 , and therefore pc 2 mc 2 2 = γ 2 − 1 E 2 = p 2 c 2 + m 2 c 4 = γ 2 m 2 c 4 , so that p 2 c 2 = γ 2 − 1 m 2 c 4 , and therefore pc 2 mc 2 2 = γ 2 − 1$

(b) yes

61.

$1 . 07 × 10 3 1 . 07 × 10 3$

63.

$6 . 56 × 10 − 8 kg 6 . 56 × 10 − 8 kg$

$4.37 × 10 − 10 4.37 × 10 − 10$

65.

$0.314 c0.314 c$

$0.99995 c 0.99995 c$

67.

(a) 1.00 kg

(b) This much mass would be measurable, but probably not observable just by looking because it is 0.01% of the total mass.

69.

(a) $6 . 3 × 10 11 kg/s 6 . 3 × 10 11 kg/s$

(b) $4 . 5 × 10 10 y 4 . 5 × 10 10 y$

(c) $4 . 44 × 10 9 kg 4 . 44 × 10 9 kg$

(d) 0.32%

73.

(a)

(b)

(c) No.

(d) Yes.

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