Ревинская О.Г., Кравченко Н.С. Изучение моделей физических процессов и явлений на компьютере
Подождите немного. Документ загружается.
ɆɨɞɌ-03 Ⱦɜɢɠɟɧɢɟ ɛɪɨɭɧɨɜɫɤɨɣ ɱɚɫɬɢɰɵ
171
10. ɋɪɟɞɧɢɣ ɤɜɚɞɪɚɬ ɫɦɟɳɟɧɢɹ ɩɪɹɦɨ ɩɪɨɩɨɪɰɢɨɧɚɥɟɧ ɜɪɟɦɟɧɢ ɧɚɛɥɸɞɟɧɢɹ.
Ʉɨɷɮɮɢɰɢɟɧɬ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ ɜ ɱɟɬɵɪɟ ɪɚɡɚ ɛɨɥɶɲɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɞɢɮɮɭɡɢɢ:
<r
2
> =4Dt. Ɉɬɫɸɞɚ ɥɟɝɤɨ ɩɨɥɭɱɢɬɶ ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭɡɢɢ D.
Ʉɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭɡɢɢ D ɜɵɪɚɠɚɟɬɫɹ ɱɟɪɟɡ ɩɨɫɬɨɹɧɧɭɸ Ȼɨɥɶɰɦɚɧɚ k:
d
kT
D
SK
3
. Ɂɧɚɹ ɬɟɦɩɟɪɚɬɭɪɭ T ɢ ɜɹɡɤɨɫɬɶ K ɠɢɞɤɨɫɬɢ, ɚ ɬɚɤɠɟ ɞɢɚɦɟɬɪ d ɱɚɫɬɢɰɵ,
ɩɨ ɤɨɷɮɮɢɰɢɟɧɬɭ ɞɢɮɮɭɡɢɢ ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ ɩɨɫɬɨɹɧɧɭɸ Ȼɨɥɶɰɦɚɧɚ.
11. ɉɨɜɬɨɪɢɬɟ ɷɤɫɩɟɪɢɦɟɧɬ ɞɥɹ ɞɥɢɬɟɥɶɧɨɫɬɢ ɷɤɫɩɟɪɢɦɟɧɬɚ 30 ɢ 45 ɫ, ɧɚɱɢ-
ɧɚɹ ɫ ɩɭɧɤɬɚ 5.
12. ȼɵɱɢɫɥɢɬɟ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɞɢɮɮɭɡɢɢ ɢ ɩɨɫɬɨɹɧɧɨɣ
Ȼɨɥɶɰɦɚɧɚ.
13. Ȼɪɨɭɧɨɜɫɤɚɹ ɱɚɫɬɢɰɚ ɢ ɠɢɞɤɨɫɬɶ ɧɚɯɨɞɹɬɫɹ ɜ
ɫɨɫɬɨɹɧɢɢ ɬɟɪɦɨɞɢɧɚɦɢɱɟ-
ɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ. ɉɨɷɬɨɦɭ ɫɪɟɞɧɸɸ ɫɤɨɪɨɫɬɶ ɱɚɫɬɢɰɵ ɦɚɫɫɨɣ m ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ,
ɡɧɚɹ ɬɟɦɩɟɪɚɬɭɪɭ ɫɪɟɞɵ:
m
kT2
2
! v . Ɇɚɫɫɭ ɫɮɟɪɢɱɟɫɤɨɣ ɱɚɫɬɢɰɵ ɦɨɠɧɨ ɨɩɪɟɞɟ-
ɥɢɬɶ, ɡɧɚɹ ɩɥɨɬɧɨɫɬɶ U ɢ ɞɢɚɦɟɬɪ d:
U
S
6
3
d
m .
Ʉɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭɡɢɢ D ɫɜɹɡɚɧ ɫɨ ɫɪɟɞɧɟɣ ɞɥɢɧɨɣ ɫɜɨɛɨɞɧɨɝɨ ɩɪɨɛɟɝɚ <O>
ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
!O! v
2
1
D
. Ɍɟɩɟɪɶ ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ ɫɪɟɞɧɸɸ ɞɥɢɧɭ
ɫɜɨɛɨɞɧɨɝɨ ɩɪɨɛɟɝɚ ɛɪɨɭɧɨɜɫɤɨɣ ɱɚɫɬɢɰɵ.
ɋɟɪɢɹ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ʋ 2. ɑɚɫɬɢɰɚ ɛɨɥɶɲɨɝɨ ɞɢɚɦɟɬɪɚ ɩɪɢ ɧɢɡɤɨɣ ɬɟɦ-
ɩɟɪɚɬɭɪɟ.
ȼ ɞɚɧɧɨɣ ɫɟɪɢɢ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɧɟɨɛɯɨɞɢɦɨ ɢɡɭɱɢɬɶ ɡɚɜɢɫɢɦɨɫɬɶ ɫɪɟɞɧɟɝɨ
ɫɦɟɳɟɧɢɹ ɱɚɫɬɢɰɵ ɨɬ ɜɪɟɦɟɧɢ ɧɚɛɥɸɞɟɧɢɹ ɞɥɹ ɱɚɫɬɢɰɵ ɛɨɥɶɲɨɝɨ ɞɢɚɦɟɬɪɚ ɩɪɢ
ɧɢɡɤɨɣ ɬɟɦɩɟɪɚɬɭɪɟ.
14. Ɍɟɦɩɟɪɚɬɭɪɭ ɫɪɟɞɵ ɦɨɠɧɨ ɢɡɦɟɧɹɬɶ ɜ ɩɪɟɞɟɥɚɯ ɨɬ ɬɨɱɤɢ ɡɚɬɜɟɪɞɟɜɚɧɢɹ
ɞɨ ɬɨɱɤɢ ɤɢɩɟɧɢɹ (ɭ ɤɚɠɞɨɣ ɠɢɞɤɨɫɬɢ ɫɜɨɣ ɞɢɚɩɚɡɨɧ ɬɟɦɩɟɪɚɬɭɪ, ɩɪɢ ɤɨɬɨɪɨɦ ɞɚɧ-
ɧɨɟ ɜɟɳɟɫɬɜɨ ɧɚɯɨɞɢɬɫɹ ɜ ɠɢɞɤɨɦ ɫɨɫɬɨɹɧɢɢ). ɋ ɩɨɦɨɳɶɸ ɫɱɟɬɱɢɤɚ «Ɍɟɦɩɟɪɚɬɭ-
ɪɚ» ɜɵɛɟɪɢɬɟ ɧɢɡɤɭɸ ɬɟɦɩɟɪɚɬɭɪɭ ɫɪɟɞɵ, ɧɨ ɨɬɥɢɱɚɸɳɭɸɫɹ ɨɬ ɦɢɧɢɦɚɥɶɧɨɣ (ɬɨɱɤɢ
ɡɚɬɜɟɪɞɟɜɚɧɢɹ). ɂɧɬɟɪɜɚɥ ɞɨɫɬɭɩɧɵɯ ɬɟɦɩɟɪɚɬɭɪ ɭɤɚɡɚɧ ɩɨɞ ɫɱɟɬɱɢɤɨɦ.
15. Ⱦɢɚɦɟɬɪ ɛɪɨɭɧɨɜɫɤɨɣ ɱɚɫɬɢɰɵ ɜ ɷɬɨɣ ɫɟɪɢɢ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɨɫɬɚɜɶɬɟ ɬɚ-
ɤɢɦ ɠɟ, ɤɚɤ ɜ ɩɪɟɞɵɞɭɳɟɣ ɫɟɪɢɢ.
16. ɉɨɜɬɨɪɢɬɟ ɷɤɫɩɟɪɢɦɟɧɬɵ ɫ ɩɭɧɤɬɚ 4 ɩɨ ɩɭɧɤɬ 13.
ɋɟɪɢɹ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ʋ 3. ɑɚɫɬɢɰɚ ɦɚɥɨɝɨ ɞɢɚɦɟɬɪɚ ɩɪɢ ɧɢɡɤɨɣ ɬɟɦɩɟ-
ɪɚɬɭɪɟ.
ȼ ɞɚɧɧɨɣ ɫɟɪɢɢ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɧɟɨɛɯɨɞɢɦɨ ɢɡɭɱɢɬɶ ɡɚɜɢɫɢɦɨɫɬɶ ɫɪɟɞɧɟɝɨ
ɫɦɟɳɟɧɢɹ ɱɚɫɬɢɰɵ ɨɬ ɜɪɟɦɟɧɢ ɧɚɛɥɸɞɟɧɢɹ ɞɥɹ ɱɚɫɬɢɰɵ ɦɚɥɨɝɨ ɞɢɚɦɟɬɪɚ ɩɪɢ ɧɢɡ-
ɤɨɣ ɬɟɦɩɟɪɚɬɭɪɟ.
17. Ɍɟɦɩɟɪɚɬɭɪɭ ɫɪɟɞɵ ɜ ɷɬɨɣ ɫɟɪɢɢ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɨɫɬɚɜɶɬɟ ɬɚɤɨɣ ɠɟ,
ɤɚɤ ɜ
ɩɪɟɞɵɞɭɳɟɣ ɫɟɪɢɢ (ɫɦ. ɩɭɧɤɬ 14).
18. Ⱦɢɚɦɟɬɪ ɛɪɨɭɧɨɜɫɤɨɣ ɱɚɫɬɢɰɵ ɦɨɠɧɨ ɢɡɦɟɧɹɬɶ ɜ ɩɪɟɞɟɥɚɯ ɨɬ 0,5 ɞɨ
2,5 ɦɤɦ. ɋ ɩɨɦɨɳɶɸ ɛɟɝɭɧɤɚ ɩɪɨɤɪɭɬɤɢ «Ⱦɢɚɦɟɬɪ» ɜɵɛɟɪɢɬɟ ɱɚɫɬɢɰɭ ɦɚɥɨɝɨ ɞɢɚ-
Ɏɢɡɢɤɚ ɠɢɞɤɨɫɬɟɣ ɢ ɝɚɡɨɜ
172
ɦɟɬɪɚ (ɦɟɧɶɲɟ 1,5 ɦɤɦ). Ɍɨɱɧɨɟ ɡɧɚɱɟɧɢɟ ɜɵɛɪɚɧɧɨɝɨ ɞɢɚɦɟɬɪɚ ɭɤɚɡɵɜɚɟɬɫɹ ɧɚɞ ɛɟ-
ɝɭɧɤɨɦ ɩɪɨɤɪɭɬɤɢ.
19. ɉɨɜɬɨɪɢɬɟ ɷɤɫɩɟɪɢɦɟɧɬɵ ɫ ɩɭɧɤɬɚ 4 ɩɨ ɩɭɧɤɬ 13.
20. ɇɚ ɨɞɧɨɦ ɝɪɚɮɢɤɟ ɩɨɫɬɪɨɣɬɟ ɬɪɢ ɡɚɜɢɫɢɦɨɫɬɢ ɫɪɟɞɧɟɝɨ ɤɜɚɞɪɚɬɚ ɫɦɟɳɟ-
ɧɢɹ ɱɚɫɬɢɰɵ ɨɬ ɜɪɟɦɟɧɢ ɧɚɛɥɸɞɟɧɢɹ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɬɪɟɦ ɩɪɨɜɟɞɟɧɧɵɦ ɫɟɪɢ-
ɹɦ ɷɤɫɩɟɪɢɦɟɧɬɨɜ. Ɍɚɧɝɟɧɫ ɭɝɥɚ ɧɚɤɥɨɧɚ ɤɚɠɞɨɝɨ ɝɪɚɮɢɤɚ ɜ ɱɟɬɵɪɟ ɪɚɡɚ ɛɨɥɶɲɟ
ɤɨɷɮɮɢɰɢɟɧɬɚ ɞɢɮɮɭɡɢɢ. ȼɵɱɢɫɥɢɬɟ ɬɚɧɝɟɧɫ ɭɝɥɚ ɧɚɤɥɨɧɚ ɢ ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭ-
ɡɢɢ ɩɨ ɤɚɠɞɨɦɭ ɝɪɚɮɢɤɭ.
21. ȼɵɱɢɫɥɢɬɟ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɩɨɫɬɨɹɧɧɨɣ Ȼɨɥɶɰɦɚɧɚ, ɢɫɩɨɥɶɡɭɹ ɡɧɚɱɟɧɢɹ
ɩɨɫɬɨɹɧɧɨɣ, ɩɨɥɭɱɟɧɧɵɟ ɜɨ ɜɫɟɯ ɫɟɪɢɹɯ ɷɤɫɩɟɪɢɦɟɧɬɨɜ.
22. ɋɞɟɥɚɣɬɟ ɜɵɜɨɞɵ.
ɋɪɚɜɧɢɬɟ ɪɟɡɭɥɶɬɚɬɵ ɩɟɪɜɨɣ ɢ ɜɬɨɪɨɣ ɫɟɪɢɢ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɢ ɫɞɟɥɚɣɬɟ ɜɵ-
ɜɨɞ: ɤɚɤ ɡɚɜɢɫɹɬ ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭɡɢɢ ɢ ɞɥɢɧɚ ɫɜɨɛɨɞɧɨɝɨ ɩɪɨɛɟɝɚ ɨɬ ɬɟɦɩɟɪɚɬɭ-
ɪɵ ɠɢɞɤɨɫɬɢ, ɜ
ɤɨɬɨɪɨɣ ɧɚɯɨɞɢɬɫɹ ɱɚɫɬɢɰɚ.
ɋɪɚɜɧɢɬɟ ɪɟɡɭɥɶɬɚɬɵ ɜɬɨɪɨɣ ɢ ɬɪɟɬɶɟɣ ɫɟɪɢɢ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɢ ɫɞɟɥɚɣɬɟ ɜɵ-
ɜɨɞ: ɤɚɤ ɡɚɜɢɫɢɬ ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭɡɢɢ ɢ ɞɥɢɧɚ ɫɜɨɛɨɞɧɨɝɨ ɩɪɨɛɟɝɚ ɨɬ ɞɢɚɦɟɬɪɚ
ɛɪɨɭɧɨɜɫɤɨɣ ɱɚɫɬɢɰɵ.
ɋɪɚɜɧɢɬɟ ɡɧɚɱɟɧɢɹ ɞɥɢɧɵ ɫɜɨɛɨɞɧɨɝɨ ɩɪɨɛɟɝɚ ɛɪɨɭɧɨɜɫɤɨɣ ɱɚɫɬɢɰɵ ɫ ɪɚɡɦɟ-
ɪɚɦɢ ɱɚɫɬɢɰɵ, ɚ ɬɚɤɠɟ ɫ ɬɚɛɥɢɱɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɞɥɢɧɵ ɫɜɨɛɨɞɧɨɝɨ ɩɪɨɛɟɝɚ ɦɨɥɟ-
ɤɭɥ ɠɢɞɤɨɫɬɟɣ ɢ ɝɚɡɨɜ. Ɉɛɴɹɫɧɢɬɟ ɪɚɡɧɢɰɭ.
23. ɋɞɟɥɚɣɬɟ ɢ ɨɛɨɫɧɭɣɬɟ ɩɪɨɝɧɨɡ: ɤɚɤ ɛɭɞɟɬ ɞɜɢɝɚɬɶɫɹ ɱɚɫɬɢɰɚ ɦɚɥɨɝɨ
ɞɢɚɦɟɬɪɚ ɩɪɢ ɜɵɫɨɤɨɣ ɬɟɦɩɟɪɚɬɭɪɟ, ɤɚɤɢɦɢ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɩɨɥɭɱɟɧɧɵɦ ɜ ɪɚɛɨ-
ɬɟ ɡɧɚɱɟɧɢɹɦ ɛɭɞɭɬ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɞɥɢɧɚ ɫɜɨɛɨɞɧɨɝɨ ɩɪɨɛɟɝɚ ɢ ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭ-
ɡɢɢ. ɇɚ ɝɪɚɮɢɤɟ, ɩɨɫɬɪɨɟɧɧɨɦ ɜ ɩɭɧɤɬɟ 20, ɩɭɧɤɬɢɪɨɦ ɢɡɨɛɪɚɡɢɬɟ ɩɪɟɞɩɨɥɨɠɢɬɟɥɶ-
ɧɵɣ ɯɨɞ ɡɚɜɢɫɢɦɨɫɬɢ ɫɪɟɞɧɟɝɨ ɫɦɟɳɟɧɢɹ ɨɬ ɜɪɟɦɟɧɢ ɧɚɛɥɸɞɟɧɢɹ ɞɥɹ ɷɬɨɝɨ ɫɥɭɱɚɹ.
Ɍɚɛɥɢɰɚ
ɉɪɨɞɨɥɠɢɬɟɥɶɧɨɫɬɶ ɷɤɫɩɟɪɢɦɟɧɬɚ, ɫ
ʋ
X-ɤɨɨɪɞɢɧɚɬɚ ɱɚɫ-
ɬɢɰɵ, ɦɤɦ
Y-ɤɨɨɪɞɢɧɚɬɚ ɱɚɫ-
ɬɢɰɵ, ɦɤɦ
Ʉɜɚɞɪɚɬ ɫɦɟɳɟɧɢɹ
ɱɚɫɬɢɰɵ r
2
, ɦɤɦ
2
1
2
…
15
ɋɪɟɞɧɢɣ ɤɜɚɞɪɚɬ ɫɦɟɳɟɧɢɹ <r
2
>, ɦɤɦ
2
Ʉɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭɡɢɢ D, ɦɤɦ
2
/ɫ
ɉɨɫɬɨɹɧɧɚɹ Ȼɨɥɶɰɦɚɧɚ, Ⱦɠ/Ʉ
ɆɨɞɌ-03 Ⱦɜɢɠɟɧɢɟ ɛɪɨɭɧɨɜɫɤɨɣ ɱɚɫɬɢɰɵ
173
4. Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
1. Ʉɚɤɚɹ ɱɚɫɬɢɰɚ ɧɚɡɵɜɚɟɬɫɹ ɛɪɨɭɧɨɜɫɤɨɣ?
2. Ʉɚɤ ɫɪɟɞɧɢɣ ɤɜɚɞɪɚɬ ɨɬɤɥɨɧɟɧɢɹ ɛɪɨɭɧɨɜɫɤɨɣ ɱɚɫɬɢɰɵ ɨɬ ɩɨɥɨɠɟɧɢɹ
ɪɚɜɧɨɜɟɫɢɹ ɡɚɜɢɫɢɬ ɨɬ ɜɪɟɦɟɧɢ? Ⱦɨɤɚɡɚɬɶ.
3. Ʉɚɤɭɸ ɮɢɡɢɱɟɫɤɭɸ ɜɟɥɢɱɢɧɭ ɧɚɡɵɜɚɸɬ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɞɢɮɮɭɡɢɢ.
4. Ʉɚɤ ɫɪɟɞɧɢɣ ɤɜɚɞɪɚɬ ɨɬɤɥɨɧɟɧɢɹ ɛɪɨɭɧɨɜɫɤɨɣ ɱɚɫɬɢɰɵ ɫɜɹɡɚɧ ɫ ɤɨ-
ɷɮɮɢɰɢɟɧɬɨɦ ɞɢɮɮɭɡɢɢ?
5. Ʉɚɤ ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭɡɢɢ ɫɜɹɡɚɧ ɫ ɜɹɡɤɨɫɬɶɸ ɫɪɟɞɵ?
6. Ʉɚɤ ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭɡɢɢ ɫɜɹɡɚɧ
ɫɨ ɫɪɟɞɧɟɣ ɞɥɢɧɨɣ ɫɜɨɛɨɞɧɨɝɨ
ɩɪɨɛɟɝɚ ɱɚɫɬɢɰɵ?
174
ɅȺȻɈɊȺɌɈɊɇȺə ɊȺȻɈɌȺ ɆɨɞɌ-04
ɉɈ ɂɁɍɑȿɇɂɘ ɆɈȾȿɅȿɃ ɎɂɁɂɑȿɋɄɂɏ
ɉɊɈɐȿɋɋɈȼ ɂ əȼɅȿɇɂɃ ɇȺ ɄɈɆɉɖɘɌȿɊȿ
Ɋɚɫɩɪɟɞɟɥɟɧɢɟ Ɇɚɤɫɜɟɥɥɚ
ɐɟɥɶ ɪɚɛɨɬɵ: ɢɡɭɱɟɧɢɟ ɨɫɨɛɟɧɧɨɫɬɟɣ ɫɬɚɬɢɫɬɢɱɟɫɤɨɝɨ ɪɚɫɩɪɟɞɟ-
ɥɟɧɢɹ ɱɚɫɬɢɰ ɩɨ ɚɛɫɨɥɸɬɧɵɦ ɡɧɚɱɟɧɢɹɦ ɫɤɨɪɨɫɬɢ. ɂɫɫɥɟɞɨɜɚɧɢɟ ɡɚɜɢ-
ɫɢɦɨɫɬɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɨɬ ɦɚɫɫɵ ɱɚɫɬɢɰ ɝɚɡɚ, ɟɝɨ ɬɟɦɩɟ-
ɪɚɬɭɪɵ ɢ ɧɚɩɪɚɜɥɟɧɢɹ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ.
1. Ɍɟɨɪɟɬɢɱɟɫɤɨɟ ɫɨɞɟɪɠɚɧɢɟ
1.1. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ Ɇɚɤɫɜɟɥɥɚ ɩɨ ɫɤɨɪɨɫɬɹɦ
ȼ ɩɪɨɰɟɫɫɟ ɫɬɚɧɨɜɥɟɧɢɹ ɦɨɥɟɤɭɥɹɪɧɨ-ɤɢɧɟɬɢɱɟɫɤɨɣ ɬɟɨɪɢɢ ɝɚɡɨɜ
ɧɟɤɨɬɨɪɨɟ ɜɪɟɦɹ ɫɱɢɬɚɥɨɫɶ, ɱɬɨ ɜɫɟ ɦɨɥɟɤɭɥɵ ɝɚɡɚ ɜ ɭɫɥɨɜɢɹɯ ɬɟɪɦɨɞɢ-
ɧɚɦɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɞɜɢɠɟɬɫɹ ɫ ɨɞɢɧɚɤɨɜɵɦɢ ɫɤɨɪɨɫɬɹɦɢ. Ɉɞɧɚɤɨ
ɞɜɢɠɟɧɢɟ ɩɪɨɢɫɯɨɞɢɬ ɜ ɪɚɡɧɵɯ ɧɚɩɪɚɜɥɟɧɢɹɯ, ɬɨɝɞɚ ɫɬɨɥɤɧɨɜɟɧɢɹ ɦɨ-
ɥɟɤɭɥ ɩɪɨɢɫɯɨɞɹɬ ɩɨɞ ɪɚɡɥɢɱɧɵɦɢ ɭɝɥɚɦɢ. ɗɬɨ ɞɨɥɠɧɨ ɩɪɢɜɨɞɢɬɶ ɤ ɢɡ-
ɦɟɧɟɧɢɸ ɫɤɨɪɨɫɬɢ ɫɬɚɥɤɢɜɚɸɳɢɯɫɹ ɦɨɥɟɤɭɥ. ɋɬɨɥɤɧɨɜɟɧɢɹ ɦɨɥɟɤɭɥ ɝɚ-
ɡɚ ɦɟɠɞɭ
ɫɨɛɨɣ ɧɨɫɹɬ ɫɥɭɱɚɣɧɵɣ ɯɚɪɚɤɬɟɪ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɯ ɫɤɨɪɨɫɬɢ
ɞɨɥɠɧɵ ɹɜɥɹɬɶɫɹ ɫɥɭɱɚɣɧɵɦɢ ɮɢɡɢɱɟɫɤɢɦɢ ɜɟɥɢɱɢɧɚɦɢ. Ɉɛɥɚɫɬɶ ɜɨɡ-
ɦɨɠɧɵɯ ɡɧɚɱɟɧɢɣ ɷɬɨɣ ɮɢɡɢɱɟɫɤɨɣ ɜɟɥɢɱɢɧɵ ɧɢɱɟɦ ɧɟ ɨɝɪɚɧɢɱɟɧɚ, ɩɨ-
ɷɬɨɦɭ ɫɤɨɪɨɫɬɢ ɦɨɥɟɤɭɥ ɜ ɝɚɡɟ ɜ ɭɫɥɨɜɢɹɯ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɨɝɨ ɪɚɜɧɨ-
ɜɟɫɢɹ – ɧɟɩɪɟɪɵɜɧɚɹ ɫɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢɧɚ, ɞɥɹ ɨɩɢɫɚɧɢɹ ɤɨɬɨɪɨɣ ɧɟɨɛ-
ɯɨɞɢɦɨ ɩɨɫɬɪɨɢɬɶ ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ ɮɭɧɤɰɢɸ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
)(v
G
f (ɫɦ.
ɉɪɢɥɨɠɟɧɢɟ 6.1). Ɏɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɦɨɥɟɤɭɥ ɝɚɡɚ ɩɨ ɫɤɨɪɨɫɬɹɦ
ɛɵɥɚ ɜɵɜɟɞɟɧɚ Ⱦɠ. Ɇɚɤɫɜɟɥɥɨɦ ɜ 1859 ɝ. ɢ ɧɨɫɢɬ ɟɝɨ ɢɦɹ.
Ƚɚɡ ɧɚɯɨɞɢɬɫɹ ɜ ɫɨɫɬɨɹɧɢɢ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ, ɜ ɨɬ-
ɫɭɬɫɬɜɢɢ ɜɧɟɲɧɢɯ ɩɨɥɟɣ, ɞɟɣɫɬɜɢɟ ɤɨɬɨɪɵɯ ɦɨɝɥɨ ɛɵ ɢɡɦɟɧɢɬɶ ɫɤɨɪɨ-
ɫɬɢ ɯɚɨɬɢɱɟɫɤɢ ɞɜɢɠɭɳɢɯɫɹ ɦɨɥɟɤɭɥ. ɑɚɫɬɢɰɵ ɝɚɡɚ ɞɜɢɠɭɬɫɹ ɪɚɜɧɨ-
ɦɟɪɧɨ ɢ ɩɪɹɦɨɥɢɧɟɣɧɨ, ɧɚɩɪɚɜɥɟɧɢɟ ɞɜɢɠɟɧɢɹ ɫɨɜɩɚɞɚɟɬ ɫ
ɧɚɩɪɚɜɥɟɧɢ-
ɟɦ ɫɤɨɪɨɫɬɢ.
ɉɨɫɤɨɥɶɤɭ ɫɤɨɪɨɫɬɶ
v
G
ɦɨɥɟɤɭɥ (ɱɚɫɬɢɰ) ɝɚɡɚ – ɧɟɩɪɟɪɵɜɧɚɹ ɜɟɤ-
ɬɨɪɧɚɹ ɫɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢɧɚ, ɬɨ ɜɟɪɨɹɬɧɨɫɬɶ dP ɨɛɧɚɪɭɠɢɬɶ ɱɚɫɬɢɰɭ ɫɨ
ɫɤɨɪɨɫɬɶɸ, ɩɪɢɧɢɦɚɸɳɟɣ ɡɧɚɱɟɧɢɟ ɢɡ ɢɧɬɟɪɜɚɥɚ ɨɬ
v
G
ɞɨ vv
GG
d , ɩɪɨ-
ɩɨɪɰɢɨɧɚɥɶɧɚ ɨɛɴɟɦɭ dV ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɫɤɨɪɨɫɬɟɣ, ɜ ɩɪɟɞɟɥɚɯ ɤɨɬɨɪɨɝɨ
ɦɨɠɟɬ ɛɵɬɶ ɢɡɨɛɪɚɠɟɧ ɜɟɤɬɨɪ ɫɤɨɪɨɫɬɢ (ɫɦ. ɉɪɢɥɨɠɟɧɢɟ 6.1.2)
ɆɨɞɌ-04 Ɋɚɫɩɪɟɞɟɥɟɧɢɟ Ɇɚɤɫɜɟɥɥɚ
175
dVfdP )(v
G
.
Ⱦɥɢɧɚ ɢɧɬɟɪɜɚɥɚ ɫɤɨɪɨɫɬɟɣ
v
G
d
ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɚ, ɩɨɷɬɨɦɭ ɫɤɨɪɨ-
ɫɬɢ ɦɨɥɟɤɭɥ, ɩɨɩɚɞɚɸɳɢɟ ɜ ɢɧɬɟɪɜɚɥ [
v
G
, vv
G
G
d ], ɢ ɩɨ ɜɟɥɢɱɢɧɟ ɢ ɩɨ
ɧɚɩɪɚɜɥɟɧɢɸ ɨɬɥɢɱɚɸɬɫɹ ɧɟɡɧɚɱɢɬɟɥɶɧɨ. ȼ ɷɬɨɦ ɫɦɵɫɥɟ ɦɨɠɧɨ ɝɨɜɨ-
ɪɢɬɶ, ɱɬɨ
)(v
G
f – ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɦɨɥɟɤɭɥ ɝɚɡɚ ɩɨ ɫɤɨɪɨɫɬɹɦ ɜ
ɡɚɞɚɧɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ.
ȼ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɜɟɤɬɨɪ ɫɤɨɪɨɫɬɢ
v
G
ɨɞɧɨɡɧɚɱɧɨ
ɨɩɪɟɞɟɥɹɟɬɫɹ ɱɟɪɟɡ ɟɝɨ ɩɪɨɟɤɰɢɢ
x
v ,
y
v
,
z
v . ɉɪɢ ɫɬɨɥɤɧɨɜɟɧɢɹɯ ɦɨɥɟ-
ɤɭɥ ɝɚɡɚ ɦɟɠɞɭ ɫɨɛɨɣ ɤɚɠɞɚɹ ɩɪɨɟɤɰɢɹ ɫɤɨɪɨɫɬɢ ɢɡɦɟɧɹɟɬɫɹ ɧɟɡɚɜɢɫɢ-
ɦɨ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɡɚɤɨɧɨɦ ɫɨɯɪɚɧɟɧɢɹ ɢɦɩɭɥɶɫɚ. ɉɨɷɬɨɦɭ ɩɪɨɟɤɰɢɢ
ɫɤɨɪɨɫɬɢ
x
v ,
y
v ,
z
v ɧɚ ɩɪɨɢɡɜɨɥɶɧɨ ɜɵɛɪɚɧɧɭɸ ɞɟɤɚɪɬɨɜɭ ɫɢɫɬɟɦɭ ɤɨ-
ɨɪɞɢɧɚɬ ɹɜɥɹɸɬɫɹ ɧɟɡɚɜɢɫɢɦɵɦɢ ɫɥɭɱɚɣɧɵɦɢ ɜɟɥɢɱɢɧɚɦɢ. ȼɟɪɨɹɬɧɨɫɬɶ
x
dP ɬɨɝɨ, ɱɬɨ x-ɤɨɦɩɨɧɟɧɬɚ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰɵ ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɟ ɢɡ
ɢɧɬɟɪɜɚɥɚ ɨɬ
x
v ɞɨ
xx
dvv , ɪɚɜɧɚ
xxx
ddP vv )(I .
Ⱥɧɚɥɨɝɢɱɧɨ,
yyy
ddP vv )(I ɢ
zzz
ddP vv )(I
– ɜɟɪɨɹɬɧɨɫɬɢ ɬɨɝɨ,
ɱɬɨ y- ɢ z-ɤɨɦɩɨɧɟɧɬɵ ɫɤɨɪɨɫɬɢ ɩɪɢɧɢɦɚɸɬ ɡɧɚɱɟɧɢɹ ɢɡ ɢɧɬɟɪɜɚɥɨɜ [
y
v ,
yy
dvv ] ɢ [
z
v
,
zz
dvv
], ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ȼ ɭɫɥɨɜɢɹɯ ɬɟɪɦɨɞɢɧɚɦɢɱɟ-
ɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɜɫɟ ɧɚɩɪɚɜɥɟɧɢɹ ɞɜɢɠɟɧɢɹ ɦɨɥɟɤɭɥ ɜ ɝɚɡɟ ɪɚɜɧɨɩɪɚɜ-
ɧɵ. ɉɨɷɬɨɦɭ ɬɪɢ ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɢ ɪɚɜɧɨɩɪɚɜɧɵ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɩɟɪɟ-
ɨɛɨɡɧɚɱɟɧɢɹ ɢ ɨɩɢɫɵɜɚɸɬɫɹ ɨɞɢɧɚɤɨɜɵɦɢ ɮɭɧɤɰɢɹɦɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ I.
Ɍɨɝɞɚ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɜɫɟ ɬɪɢ ɧɟɡɚɜɢɫɢɦɵɟ ɤɨɦɩɨɧɟɧɬɵ ɫɤɨɪɨɫɬɢ
ɩɪɢɧɢɦɚɸɬ ɡɧɚɱɟɧɢɹ ɜ ɭɤɚɡɚɧɧɵɯ ɢɧɬɟɪɜɚɥɚɯ ɨɞɧɨɜɪɟɦɟɧɧɨ, ɜɵɱɢɫɥɹ-
ɟɬɫɹ ɤɚɤ ɩɪɨɢɡɜɟɞɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɟɣ
zyx
dPdPdPdP ,
dVf )(v
G
=
zyxzyx
ddd vvvvvv )()()( III .
Ɉɛɴɟɦ dV ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɫɤɨɪɨɫɬɟɣ, ɤɨɬɨɪɵɣ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɢɧ-
ɬɟɪɜɚɥɭ [
v
G
, vv
GG
d ], ɜ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɪɚɜɟɧ
zyx
ddddV vvv . ɉɨɷɬɨɦɭ ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
)(v
G
f
ɦɨɥɟɤɭɥ ɝɚɡɚ
ɩɨ ɫɤɨɪɨɫɬɹɦ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɩɪɨɢɡɜɟɞɟɧɢɟ ɮɭɧɤɰɢɣ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
ɦɨɥɟɤɭɥ ɝɚɡɚ ɩɨ ɞɟɤɚɪɬɨɜɵɦ ɤɨɦɩɨɧɟɧɬɚɦ ɫɤɨɪɨɫɬɢ
)(v
G
f
= )()()(
zyx
vvv III .
Ɏɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ, ɤɚɤ ɢ ɜɟɪɨɹɬɧɨɫɬɢ, ɩɨ-
ɥɨɠɢɬɟɥɶɧɵɟ, ɩɨɷɬɨɦɭ ɦɨɠɧɨ ɩɪɨɥɨɝɚɪɢɮɦɢɪɨɜɚɬɶ ɩɨɥɭɱɟɧɧɨɟ ɜɵɪɚ-
ɠɟɧɢɟ
Ɏɢɡɢɤɚ ɠɢɞɤɨɫɬɟɣ ɢ ɝɚɡɨɜ
176
)(ln v
G
f = )(ln)(ln)(ln
zyx
vvv III .
Ⱥɪɝɭɦɟɧɬɵ ɮɭɧɤɰɢɣ ɫɩɪɚɜɚ ɢ ɫɥɟɜɚ ɜ ɷɬɨɦ ɭɪɚɜɧɟɧɢɢ ɫɜɹɡɚɧɵ ɫɨ-
ɨɬɧɨɲɟɧɢɟɦ
2222
zyx
vvvv
. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɷɬɢ ɞɜɚ ɭɪɚɜɧɟɧɢɹ ɫɥɟ-
ɞɭɟɬ ɪɟɲɚɬɶ ɫɨɜɦɟɫɬɧɨ:
°
¯
°
®
III
.
),(ln)(ln)(ln)(ln
2222
zyx
zyx
f
vvvv
vvvv
G
ɉɟɪɟɦɟɧɧɵɟ
x
v ,
y
v
,
z
v – ɧɟɡɚɜɢɫɢɦɵɟ. ɉɪɨɞɢɮɮɟɪɟɧɰɢɪɭɟɦ ɨɛɚ
ɭɪɚɜɧɟɧɢɹ ɩɨ ɨɞɧɨɣ ɢɡ ɧɢɯ, ɧɚɩɪɢɦɟɪ, ɩɨ
x
v
)(
)(
)(
)(
x
x
x
f
f
v
v
v
v
v
v
I
I
c
w
w
c
G
G
,
x
x
v
v
v
v
w
w
.
Ɍɨɝɞɚ
)(
)(
1
)(
)(1
x
x
x
f
f
v
v
vv
v
v I
I
c
c
G
G
. Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ
)(
)(
1
)(
)(1
y
y
y
f
f
v
v
vv
v
v I
I
c
c
G
G
ɢ
)(
)(
1
)(
)(1
z
z
z
f
f
v
v
vv
v
v I
I
c
c
G
G
.
Ɂɧɚɱɢɬ,
)(
)(
1
)(
)(
1
)(
)(
1
z
z
zy
y
yx
x
x
v
v
vv
v
vv
v
v I
I
c
I
I
c
I
I
c
.
Ʉɚɠɞɚɹ ɞɪɨɛɶ ɫɨɞɟɪɠɢɬ ɮɭɧɤɰɢɢ ɬɨɥɶɤɨ ɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ. ɉɟ-
ɪɟɦɟɧɧɵɟ
x
v ,
y
v
,
z
v ɹɜɥɹɸɬɫɹ ɧɟɡɚɜɢɫɢɦɵɦɢ. Ɏɭɧɤɰɢɢ ɨɬ ɪɚɡɧɵɯ ɧɟɡɚ-
ɜɢɫɢɦɵɯ ɩɟɪɟɦɟɧɧɵɯ ɫɨɜɩɚɞɚɸɬ ɩɪɢ ɥɸɛɨɦ ɡɧɚɱɟɧɢɢ ɩɟɪɟɦɟɧɧɵɯ
ɬɨɥɶɤɨ, ɟɫɥɢ ɷɬɨ ɪɚɜɧɵɟ ɤɨɧɫɬɚɧɬɵ. Ɉɛɨɡɧɚɱɢɦ ɷɬɭ ɤɨɧɫɬɚɧɬɭ ɱɟɪɟɡ –2a,
ɬɨɝɞɚ
a
x
x
x
2
)(
)(
1
I
I
c
v
v
v
,
a
y
y
y
2
)(
)(
1
I
I
c
v
v
v
, a
z
z
z
2
)(
)(
1
I
I
c
v
v
v
.
ɉɨɥɭɱɢɥɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɧɚ ɮɭɧɤɰɢɸ ɪɚɫɩɪɟɞɟɥɟ-
ɧɢɹ ɦɨɥɟɤɭɥ ɩɨ ɨɞɧɨɣ ɢɡ ɞɟɤɚɪɬɨɜɵɯ ɤɨɦɩɨɧɟɧɬ ɫɤɨɪɨɫɬɢ. Ɋɟɲɚɹ ɟɝɨ,
ɩɨɥɭɱɢɦ
xx
x
x
da
d
vv
v
v
2
)(
)(
I
I
2
)(
ln
x
x
a
A
v
v
I
ɢɥɢ )exp()(
2
xx
aA vv I .
A – ɧɨɪɦɢɪɨɜɨɱɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ. Ɉɧ ɞɨɥɠɟɧ ɨɛɟɫɩɟɱɢɜɚɬɶ ɧɨɪɦɢɪɨɜɤɭ
ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ I ɧɚ ɟɞɢɧɢɰɭ (ɫɦ. ɉɪɢɥɨɠɟɧɢɟ 6.1.1). ɉɪɨɟɤɰɢɹ
ɫɤɨɪɨɫɬɢ ɦɨɠɟɬ ɩɪɢɧɢɦɚɬɶ ɤɚɤ ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɬɚɤ ɢ ɨɬɪɢɰɚɬɟɥɶɧɵɟ
ɡɧɚɱɟɧɢɹ,
x
v (f, f), ɩɨɷɬɨɦɭ
ɆɨɞɌ-04 Ɋɚɫɩɪɟɞɟɥɟɧɢɟ Ɇɚɤɫɜɟɥɥɚ
177
1)exp()(
2
I
³³
f
f
f
f
xxxx
daAd vvvv .
ɂɧɬɟɝɪɚɥ
a
da
xx
S
³
f
f
vv )exp(
2
(ɫɦ. ɉɪɢɥɨɠɟɧɢɟ 6.2), ɡɧɚɱɢɬ ɧɨɪɦɢ-
ɪɨɜɨɱɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ
S
a
A
.
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɧɫɬɚɧɬɵ a ɢɫɩɨɥɶɡɭɟɦ ɬɟɨɪɟɦɭ ɨ ɪɚɫɩɪɟɞɟɥɟ-
ɧɢɢ ɷɧɟɪɝɢɢ ɩɨ ɫɬɟɩɟɧɹɦ ɫɜɨɛɨɞɵ, ɤɨɬɨɪɚɹ ɝɨɜɨɪɢɬ, ɱɬɨ ɧɚ ɨɞɧɭ ɫɬɟɩɟɧɶ
ɫɜɨɛɨɞɵ (ɜ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɷɬɨ, ɧɚɩɪɢɦɟɪ, ɧɚɩɪɚɜɥɟɧɢɟ
OX) ɩɪɢɯɨɞɢɬɫɹ ɷɧɟɪɝɢɹ, ɪɚɜɧɚɹ
kT
2
1
. ȼ ɨɬɫɭɬɫɬɜɢɢ ɜɧɟɲɧɢɯ ɩɨɥɟɣ
ɱɚɫɬɢɰɵ ɝɚɡɚ ɦɚɫɫɨɣ m ɨɛɥɚɞɚɸɬ ɬɨɥɶɤɨ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɟɣ
2
2
vm
.
Ɍɚɤ ɤɚɤ ɱɚɫɬɢɰɵ ɞɜɢɠɭɬɫɹ ɯɚɨɬɢɱɟɫɤɢ, ɬɨ ɬɟɩɥɨɜɚɹ ɷɧɟɪɝɢɹ
kT
2
1
ɫɨɨɬ-
ɜɟɬɫɬɜɭɟɬ ɫɪɟɞɧɟɣ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ
!H
x
ɜ ɧɚɩɪɚɜɥɟɧɢɢ OX. Ⱦɥɹ
ɜɵɱɢɫɥɟɧɢɹ ɫɪɟɞɧɟɣ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ
!H
x
ɧɟɨɛɯɨɞɢɦɨ (ɫɦ.
ɉɪɢɥɨɠɟɧɢɟ 6.1.1) ɜɵɩɨɥɧɢɬɶ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɜɟɥɢɱɢɧɵ
2
2
x
mv
ɫɨɜɦɟɫɬ-
ɧɨ ɫ ɮɭɧɤɰɢɟɣ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
)(
x
vI :
³³
f
f
f
f
S
I ! H
xxxxx
xx
x
da
am
d
mm
vvvvv
vv
)exp(
2
)(
22
22
22
.
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɬɟɨɪɟɦɨɣ ɨ ɪɚɜɧɨɪɚɫɩɪɟɞɟɥɟɧɢɢ ɷɧɟɪɝɢɢ ɩɨ ɫɬɟɩɟɧɹɦ
ɫɜɨɛɨɞɵ ɞɨɥɠɧɨ ɩɨɥɭɱɢɬɶɫɹ, ɱɬɨ
!H
x
=
kT
2
1
.
ɂɧɬɟɝɪɚɥ
3
22
4
)exp(
a
da
xxx
S
³
f
f
vvv (ɫɦ. ɉɪɢɥɨɠɟɧɢɟ 6.2), ɡɧɚɱɢɬ
kT
a
am
2
1
4
2
3
S
S
kT
m
a
2
.
Ʉɚɤ ɢ ɩɪɟɞɩɨɥɚɝɚɥɨɫɶ ɜɵɲɟ, ɤɨɧɫɬɚɧɬɚ a ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɧɚɩɪɚɜɥɟ-
ɧɢɹ ɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɟɦɩɟɪɚɬɭɪɨɣ ɢ ɦɚɫɫɨɣ ɦɨɥɟɤɭɥ ɝɚɡɚ.
Ɏɢɡɢɤɚ ɠɢɞɤɨɫɬɟɣ ɢ ɝɚɡɨɜ
178
Ɍɨɝɞɚ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɱɚɫɬɢɰ ɝɚɡɚ ɩɨ ɞɟɤɚɪɬɨɜɵɦ ɤɨɦɩɨ-
ɧɟɧɬɚɦ ɫɤɨɪɨɫɬɢ ɢɦɟɸɬ ɜɢɞ
¸
¸
¹
·
¨
¨
©
§
S
I
kT
m
kT
m
x
x
2
exp
2
)(
2
v
v
,
¸
¸
¹
·
¨
¨
©
§
S
I
kT
m
kT
m
y
y
2
exp
2
)(
2
v
v
,
¸
¸
¹
·
¨
¨
©
§
S
I
kT
m
kT
m
z
z
2
exp
2
)(
2
v
v
.
Ⱥ ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɱɚɫɬɢɰ ɝɚɡɚ ɩɨ ɫɤɨɪɨɫɬɹɦ
)(v
G
f
= )()()(
zyx
vvv III ɩɪɢɦɟɬ ɜɢɞ
¸
¸
¹
·
¨
¨
©
§
¸
¹
·
¨
©
§
S
kT
m
kT
m
f
2
exp
2
)(
2
2
3
v
v
G
.
ɗɬɚ ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɨɩɢɫɵɜɚɟɬ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɫɤɨ-
ɪɨɫɬɶ ɱɚɫɬɢɰ ɝɚɡɚ ɛɭɞɟɬ ɧɚɯɨɞɢɬɶɫɹ ɜ ɷɥɟɦɟɧɬɚɪɧɨɦ ɨɛɴɟɦɟ
zyx
ddddV vvv
ɜɛɥɢɡɢ ɫɤɨɪɨɫɬɢ v
G
ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɫɤɨɪɨɫɬɟɣ.
1.2. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ Ɇɚɤɫɜɟɥɥɚ ɩɨ ɚɛɫɨɥɸɬɧɵɦ ɡɧɚɱɟɧɢɹɦ
ɫɤɨɪɨɫɬɢ (ɩɨ ɦɨɞɭɥɸ ɫɤɨɪɨɫɬɢ)
ɏɨɬɹ ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
)(v
G
f
ɜɵɜɨɞɢɥɚɫɶ ɞɥɹ ɫɤɨɪɨɫɬɟɣ ɧɟ-
ɡɧɚɱɢɬɟɥɶɧɨ ɨɬɥɢɱɚɸɳɢɯɫɹ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ («ɮɢɤɫɢɪɨɜɚɧɧɨɝɨ ɧɚɩɪɚɜ-
ɥɟɧɢɹ ɫɤɨɪɨɫɬɢ»), ɟɟ ɹɜɧɵɣ ɜɢɞ
¸
¸
¹
·
¨
¨
©
§
¸
¹
·
¨
©
§
S
kT
m
kT
m
f
2
exp
2
)(
2
2
3
v
v
G
ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɧɚɩɪɚɜɥɟɧɢɹ, ɚ ɡɚ-
ɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɚɛɫɨɥɸɬɧɨɝɨ ɡɧɚɱɟɧɢɹ (ɦɨɞɭɥɹ) ɫɤɨɪɨɫɬɢ
vv
G
. ɋɥɟ-
ɞɨɜɚɬɟɥɶɧɨ, ɜɟɪɨɹɬɧɨɫɬɶ ɨɛɧɚɪɭɠɢɬɶ ɱɚɫɬɢɰɭ, ɞɜɢɠɭɳɭɸɫɹ ɫɨ ɫɤɨɪɨ-
ɫɬɶɸ ɜ ɨɛɴɟɦɟ dV, ɜ ɨɞɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɛɭɞɟɬ ɬɚɤɨɣ ɠɟ, ɤɚɤ ɢ ɜ ɞɪɭɝɨɦ
ɧɚɩɪɚɜɥɟɧɢɢ.
Ɍɨɝɞɚ ɢɦɟɟɬ ɫɦɵɫɥ ɜɵɱɢɫɥɢɬɶ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɱɚɫɬɢɰɚ ɢɦɟ-
ɟɬ ɫɤɨɪɨɫɬɶ, ɚɛɫɨɥɸɬɧɨɟ ɡɧɚɱɟɧɢɟ ɤɨɬɨɪɨɣ ɥɟɠɢɬ ɜ ɢɧɬɟɪɜɚɥɟ ɨɬ
v ɞɨ
vv d . ȼɫɟ ɧɚɩɪɚɜɥɟɧɢɹ ɞɜɢɠɟɧɢɹ ɦɨɥɟɤɭɥ ɧɟɡɚɜɢɫɢɦɵ, ɩɨɷɬɨɦɭ ɢɫɤɨ-
ɦɚɹ ɜɟɪɨɹɬɧɨɫɬɶ ɜɵɱɢɫɥɹɟɬɫɹ ɤɚɤ ɫɭɦɦɚ ɜɟɪɨɹɬɧɨɫɬɟɣ, ɚ ɞɥɹ ɧɟɩɪɟɪɵɜ-
ɧɨɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ – ɤɚɤ ɢɧɬɟɝɪɚɥ ɩɨ ɜɫɟɦ ɜɨɡɦɨɠɧɵɦ ɧɚɩɪɚɜɥɟ-
ɧɢɹɦ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ.
ɆɨɞɌ-04 Ɋɚɫɩɪɟɞɟɥɟɧɢɟ Ɇɚɤɫɜɟɥɥɚ
179
Ɍɨ ɟɫɬɶ ɜɟɪɨɹɬɧɨɫɬɶ dP ɨɛɧɚɪɭɠɢɬɶ ɱɚɫɬɢɰɭ, ɞɜɢɠɭɳɭɸɫɹ ɫɨ ɫɤɨ-
ɪɨɫɬɶɸ ɜ ɨɛɴɟɦɟ dV ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɫɤɨɪɨɫɬɟɣ ɜ «ɮɢɤɫɢɪɨɜɚɧɧɨɦ ɧɚ-
ɩɪɚɜɥɟɧɢɢ»,
dVfdP )(v
G
ɢɥɢ
dV
kT
m
kT
m
dP
¸
¸
¹
·
¨
¨
©
§
¸
¹
·
¨
©
§
S
2
exp
2
2
2
3
v
ɧɟɨɛɯɨɞɢɦɨ ɩɪɨɢɧɬɟɝɪɢɪɨɜɚɬɶ ɩɨ ɜɫɟɦ ɜɨɡɦɨɠɧɵɦ ɧɚɩɪɚɜɥɟɧɢɹɦ ɞɜɢ-
ɠɟɧɢɹ. ȼ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ, ɜ ɤɨɬɨɪɨɣ ɨɛɴɟɦ ɜ ɩɪɨɫɬɪɚɧɫɬ-
ɜɟ ɫɤɨɪɨɫɬɟɣ ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɜɢɞɟ
zyx
ddddV vvv , ɜɵɩɨɥɧɹɬɶ ɢɧɬɟɝɪɢ-
ɪɨɜɚɧɢɟ ɩɨ ɧɚɩɪɚɜɥɟɧɢɹɦ ɧɟɭɞɨɛɧɨ. ȼ ɫɮɟɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɬɨɬ ɠɟ ɷɥɟ-
ɦɟɧɬɚɪɧɵɣ ɨɛɴɟɦ dV ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɜɢɞɟ
MTT ddddV sin
2
vv , ɝɞɟ ɭɝ-
ɥɵ T ɢ M ɡɚɞɚɸɬ ɧɚɩɪɚɜɥɟɧɢɟ ɢ ɦɨɝɭɬ ɢɡɦɟɧɹɬɶɫɹ ɜ ɩɪɟɞɟɥɚɯ: 0 d T d S,
0 d M d 2S. Ɍɨɝɞɚ
MTT
¸
¸
¹
·
¨
¨
©
§
¸
¹
·
¨
©
§
S
ddd
kT
m
kT
m
dP sin
2
exp
2
2
2
2
3
vv
v
.
ɉɪɨɢɧɬɟɝɪɢɪɨɜɚɜ ɩɨ ɭɝɥɚɦ ɜ ɡɚɞɚɧɧɵɯ ɩɪɟɞɟɥɚɯ, ɩɨɥɭɱɢɦ ɢɫɤɨ-
ɦɭɸ ɜɟɪɨɹɬɧɨɫɬɶ
v
dP :
³³
SS
TTM
0
2
0
sin ddPddP
v
,
³³
SS
TTM
¸
¸
¹
·
¨
¨
©
§
¸
¹
·
¨
©
§
S
0
2
0
2
2
2
3
sin
2
exp
2
ddd
kT
m
kT
m
dP vv
v
v
,
vv
v
v
d
kT
m
kT
m
dP
2
2
2
3
2
exp
2
4
¸
¸
¹
·
¨
¨
©
§
¸
¹
·
¨
©
§
S
S
.
ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ ɜɟɪɨɹɬɧɨɫɬɶ
v
dP
ɬɨɝɨ, ɱɬɨ
ɚɛɫɨɥɸɬɧɚɹ ɫɤɨɪɨɫɬɶ ɱɚɫɬɢɰ ɥɟɠɢɬ ɜ ɢɧɬɟɪɜɚɥɟ ɨɬ
v ɞɨ vv d , ɬɚɤɠɟ
ɞɨɥɠɧɚ ɜɵɱɢɫɥɹɬɶɫɹ ɱɟɪɟɡ ɧɟɤɨɬɨɪɭɸ ɮɭɧɤɰɢɸ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
)(v
F
ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
vv
v
dFdP )(
.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
)(v
F
ɦɨɥɟɤɭɥ ɝɚɡɚ ɩɨ
ɚɛɫɨɥɸɬɧɵɦ ɡɧɚɱɟɧɢɹɦ ɫɤɨɪɨɫɬɢ ɢɦɟɟɬ ɜɢɞ
¸
¸
¹
·
¨
¨
©
§
¸
¹
·
¨
©
§
S
S
kT
m
kT
m
F
2
exp
2
4)(
2
2
2
3
v
vv
.
Ɏɢɡɢɤɚ ɠɢɞɤɨɫɬɟɣ ɢ ɝɚɡɨɜ
180
ɗɬɚ ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɨɩɢɫɵɜɚɟɬ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɬɨ-
ɝɨ, ɱɬɨ ɚɛɫɨɥɸɬɧɚɹ ɫɤɨɪɨɫɬɶ ɱɚɫɬɢɰɵ ɥɟɠɢɬ ɜ ɢɧɬɟɪɜɚɥɟ [
v , vv d ] ɩɪɢ
ɜɫɟɯ ɜɨɡɦɨɠɧɵɯ ɧɚɩɪɚɜɥɟɧɢɹɯ ɞɜɢɠɟɧɢɹ.
Ɏɭɧɤɰɢɹ
)(v
F
ɹɜɥɹɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨɣ ɢ ɩɨɥɨɠɢɬɟɥɶɧɨ ɨɩɪɟɞɟɥɟɧ-
ɧɨɣ ɧɚ ɢɧɬɟɪɜɚɥɟ [0, f).
ɉɪɨɚɧɚɥɢɡɢɪɨɜɚɜ ɡɚɜɢɫɢɦɨɫɬɶ
ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
)(v
F
ɨɬ ɚɛ-
ɫɨɥɸɬɧɨɝɨ ɡɧɚɱɟɧɢɹ ɫɤɨɪɨɫɬɢ, ɦɨɠɧɨ
ɡɚɦɟɬɢɬɶ, ɱɬɨ ɫ ɪɨɫɬɨɦ
v
ɩɥɨɬɧɨɫɬɶ
ɜɟɪɨɹɬɧɨɫɬɢ ɫ ɨɞɧɨɣ ɫɬɨɪɨɧɵ ɪɚɫɬɟɬ
ɡɚ ɫɱɟɬ
2
v , ɚ ɫ ɞɪɭɝɨɣ – ɭɛɵɜɚɟɬ ɡɚ
ɫɱɟɬ ɷɤɫɩɨɧɟɧɬɵ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
)(v
F
ɞɨɥɠɧɚ
ɢɦɟɬɶ ɦɚɤɫɢɦɭɦ (ɪɢɫ. 1). Ɉɩɪɟɞɟɥɢɦ,
ɩɪɢ ɤɚɤɨɦ ɡɧɚɱɟɧɢɢ ɫɤɨɪɨɫɬɢ ɮɭɧɤ-
ɰɢɹ
)(v
F
ɢɦɟɟɬ ɦɚɤɫɢɦɭɦ. Ⱦɥɹ ɷɬɨɝɨ
ɜɵɱɢɫɥɢɦ ɩɪɨɢɡɜɨɞɧɭɸ ɨɬ ɮɭɧɤɰɢɢ
)(v
F
ɢ ɩɪɢɪɚɜɧɹɟɦ ɟɟ ɤ ɧɭɥɸ:
0
2
2
2
exp
2
exp2
2
4
)(
2
2
2
2
3
¸
¸
¹
·
¨
¨
©
§
¸
¹
·
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¹
·
¨
©
§
S
S
kT
m
kT
m
kT
m
kT
m
d
dF v
v
v
v
v
v
v
,
0
2
1
2
exp
2
8
2
2
2
3
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¹
·
¨
©
§
S
S
kT
m
kT
m
kT
m v
v
v
.
ɉɪɨɢɡɜɨɞɧɚɹ ɨɛɪɚɳɚɟɬɫɹ ɜ
ɧɨɥɶ ɩɪɢ
0 v , f v ɢ
m
kT2
v
. ɂɡ ɮɨɪɦɭɥɵ ɞɥɹ
)(v
F
ɜɢɞɧɨ, ɱɬɨ ɩɥɨɬɧɨɫɬɶ
ɜɟɪɨɹɬɧɨɫɬɢ ɹɜɥɹɟɬɫɹ ɩɨɥɨ-
ɠɢɬɟɥɶɧɨ ɨɩɪɟɞɟɥɟɧɧɨɣ
ɮɭɧɤɰɢɟɣ. ɉɪɢ
0 v
ɢ
f v
ɨɧɚ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ, ɫɥɟɞɨ-
ɜɚɬɟɥɶɧɨ, ɩɪɢ
m
kT2
v
ɢɦɟ-
ɟɬ ɦɚɤɫɢɦɭɦ.
v
v
2
exp(–av
2
)
F(v)
Ɋɢɫ. 1
F(v)
vv
ɜ
v
ɤɜ
<v>
Ɋɢɫ. 2