College Physics for AP® Courses

# Chapter 28

1.

(a) 1.0328

(b) 1.15

3.

$5 . 96 × 10 − 8 s 5 . 96 × 10 − 8 s size 12{5 "." "96" times "10" rSup { size 8{ - 8} } " s"} {}$

5.

$0.800c0.800c$

7.

$0 . 140 c 0 . 140 c size 12{0 "." "140"c} {}$

9.

(a) $0.745c0.745c size 12{0 "." "745"c} {}$

(b) $0.99995c0.99995c size 12{0 "." "99995"c} {}$ (to five digits to show effect)

11.

(a) 0.996

(b) $γγ size 12{γ} {}$ 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) $Δt = γΔt 0 ⇒ γ = Δt Δt 0 = 4 . 303 y 0 . 1434 y = 30 . 0 Δt = γΔt 0 ⇒ γ = Δt Δt 0 = 4 . 303 y 0 . 1434 y = 30 . 0 size 12{"Δt"="γΔt" rSub { size 8{0} } drarrow γ= { {"Δt"} over {"Δt" rSub { size 8{0} } } } = { {4 "." "303 y"} over {0 "." "1434 y"} } = {underline {"30" "." 0}} } {}$

Thus, the two times are related when $γ= 30 . 00 γ= 30 . 00 size 12{ ital "γ=""30" "." "00"} {}$.

18.

(a) 0.250

(b) $γγ size 12{γ} {}$ must be ≥1

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

20.

(a) $0.909c0.909c size 12{0 "." "909"c} {}$

(b) $0.400c0.400c size 12{0 "." "400"c} {}$

22.

$0 . 198 c 0 . 198 c size 12{0 "." "198"c} {}$

24.

a) $658 nm658 nm size 12{"658""nm"} {}$

b) red

c) $v/c=9.92×10−5v/c=9.92×10−5 size 12{v/ ital "c="9 "." "92" times "10" rSup { size 8{ - 5} } } {}$ (negligible)

26.

$0 . 991 c 0 . 991 c size 12{0 "." "991"c} {}$

28.

$−0.696c−0.696c size 12{0 "." "696"c} {}$

30.

$0 . 01324 c 0 . 01324 c size 12{0 "." "01324" c} {}$

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 size 12{alignl { stack { ital "u=" { { ital "v+u'"} over {1+ $$ital "vu""'/"c rSup { size 8{2} }$$ } } = { { ital "v+c"} over {1+ $$ital "vc"/c rSup { size 8{2} }$$ } } = { { ital "v+c"} over {1+ $$v/c$$ } } {} # { {c $$ital "v+c"$$ } over { ital "c+v"} } = {underline {c}} {} } } } {}

34.

a) $0.99947c0.99947c$

b) $1.2064×1011 y1.2064×1011 y size 12{1 "." "2064" times "10" rSup { size 8{"11"} } " y"} {}$

c) $1.2058×1011 y1.2058×1011 y size 12{1 "." "2058" times "10" rSup { size 8{"11"} } " y"} {}$ (all to sufficient digits to show effects)

35.

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

37.

(a) $3.000000015 ×1013 kg⋅m/s3.000000015 ×1013 kg⋅m/s size 12{ {underline {3 "." "000000015 " times "10" rSup { size 8{"13"} } " kg" cdot "m/s"}} } {}$.

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

39.

$2.9957 × 10 8 m/s 2.9957 × 10 8 m/s size 12{ {underline {2 "." "988" times "10" rSup { size 8{8} } " m/s"}} } {}$

41.

(a) $1.121×10–8 m/s1.121×10–8 m/s size 12{1 "." "121" times "10" rSup { size 8{"-8"} } " m/s"} {}$

(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.

$2 . 3 × 10 − 30 kg 2 . 3 × 10 − 30 kg size 12{2 "." 3 times "10" rSup { size 8{ - "30"} } "kg"} {}$

47.

(a) $1 . 11 × 10 27 kg 1 . 11 × 10 27 kg size 12{1 "." "11" times "10" rSup { size 8{ - "27"} } "kg"} {}$

(b) $5 . 56 × 10 − 5 5 . 56 × 10 − 5 size 12{5 "." "56" times "10" rSup { size 8{ - 5} } } {}$

49.

$7 . 1 × 10 − 3 kg 7 . 1 × 10 − 3 kg$

$7 . 1 × 10 − 3 7 . 1 × 10 − 3 size 12{7 "." 1 times "10" rSup { size 8{ - 3} } } {}$

The ratio is greater for hydrogen.

51.

208

$0.999988c0.999988c$

53.

$6.92 × 10 5 J 6.92 × 10 5 J size 12{6 "." "92" times "10" rSup { size 8{5} } J} {}$

1.54

55.

(a) $0 . 914 c 0 . 914 c size 12{0 "." "914"c} {}$

(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.

$1 . 07 × 10 3 1 . 07 × 10 3 size 12{1 "." "07" times "10" rSup { size 8{3} } } {}$

63.

$6 . 56 × 10 − 8 kg 6 . 56 × 10 − 8 kg size 12{6 "." "56" times "10" rSup { size 8{ - 8} } "kg"} {}$

$4.37 × 10 − 10 4.37 × 10 − 10 size 12{4 "." "20" times "10" rSup { size 8{ - "12"} } } {}$

65.

$0.314 c0.314 c size 12{c} {}$

$0.99995c0.99995c size 12{c} {}$

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%

1.

(a)

3.

The relativistic Doppler effect takes into account the special relativity concept of time dilation and also does not require a medium of propagation to be used as a point of reference (light does not require a medium for propagation).

5.

Relativistic kinetic energy is given as $KE rel =(γ−1)m c 2 KE rel =(γ−1)m c 2$

where $γ= 1 1− v 2 c 2 γ= 1 1− v 2 c 2$

Classical kinetic energy is given as $KE class = 1 2 m v 2 KE class = 1 2 m v 2$

At low velocities $v=0 v=0$, a binomial expansion and subsequent approximation of $γ γ$ gives:

$γ=1+ 1 v 2 2 c 2 γ=1+ 1 v 2 2 c 2$ or $γ−1= 1 v 2 2 c 2 γ−1= 1 v 2 2 c 2$

Substituting $γ−1 γ−1$ in the expression for $KE rel KE rel$ gives

$KE rel =[ 1 v 2 2 c 2 ]m c 2 = 1 2 m v 2 = KE class KE rel =[ 1 v 2 2 c 2 ]m c 2 = 1 2 m v 2 = KE class$

Hence, relativistic kinetic energy becomes classical kinetic energy when $v≪c v≪c$.