Prawa Gaussa dla grawitacji ܨ→
ܩ݉ܯ
ܩܯ
ܧ→ = ݉ = ܽ = ݎଶ ∗ ݉ = ݎଶ
ܧ→ ∗ ݀ݏ→
Φ = ܧ→ ∗ ݀ݏ→ = ܧ ∗ ܵ ∗ cos ߙ
ܩܯ
ර ܧ→ ∗ ݀ݏ→ = ܧ ∗ 4ߨܴଶ = ݎଶ ∗ 4ߨܴଶ = 4ߨܩܯ
1. ׯ ܧ→ ∗ ݀ݏ→ = 4ߨܩܯ
2. ׯ ܧ→ ∗ ݀ݏ→ = 4ߨܩሺܯଵ + ܯଶሻ
3. ׯ ܧ→ ∗ ݀ݏ→ = 0
ܩܯ
݃ =
௭
ܴଶ
4ߨܩݎଷܯ
ݎߨܴ݃ଶ ∗ ݎଷ 4ߨ݃ݎଷ
ර ܧ→ ∗ ݀ݏ→ = 4ߨܩܯ
௭
௪ =
ܴଷ
=
ଷ
=
= ܧሺݎሻ ∗ 4ߨݎଶ
௭
ܴ௭
ܴ௭
݃ݎ
ܧሺݎሻ = ܴଷ ௭
4
ܯ
3 ߨݎଷ
௪ = 4
∗ ܯ௭
ଷ
3 ߨܴ௭
Praca i energia
ܹ = ܨ ∗ ݏ [ܬ]
ܹ = ܨ→ ∗ ݏ→ = ܨ ∗ ݏ ∗ ܿݏߙ
ௌ
ܹ = න ܨ ∗ ܨሺݏሻ ∗ ݀ݏ
ܹ
ܬ
ܲ = ݐ = [?] = ݏ = ܹ
ܨ ∗ ݏ ∗ cos ߙ
ܲ =
ݐ
= ܨ ∗ ݒ ∗ ܿݏߙ
݀ ܹሺݐሻ
ܲ = ݀ ݐ
௫బ
ݔଶ
න ݔ݀ݔ =
2
ܽݐଶ
ܹ = ܨ ∗ ݏ = ݉ ∗ ܽ ∗ 2
ܽଶݐଶ ݉ሺܽݐሻଶ ݉ݒଶ
ܨ = ݉ ∗ ܽ = ݉ ∗ 2 = 2 = 2
݉ →
→
ଵݒଵ + ܯଶݒଶ = 0
݉
݉
݉ ଶ
ݒ→
ଵݒଵ
ଵ
ଵݒଵ
ଶ = − ܯ ≅
∗ ሺ
ଶ
ܯଶ
2 ሻ
ܧ௭ ≅ 10ିଶସܧ
Energia potencjalna 1
݃ݎ
ܽ = ܴ ଶ
ܴ݃ଶ
݉ܽ =
ݎଶ ݉
݉ ∗ ܴଶ ∗ ݃
1
1 1
ܧ
ଶ
ଶ
ଶ
ଶ
ሺݎሻ = න
∗ න ݎିଶ݀ݎ = ݉ ∗ ܴ௭ ∗ ݃ ൬−
൬−
൰ =
ோ
ݎଶ
= ݉ ∗ ܴଶ ∗ ݃
ݎ൰ = ܴ݉݃௭
ݎ + ܴ
ோ
௭
1
1
= ܴ݉݃ଶ௭ ൬ܴ −
௭
ܴ௭ + ݎ൰ = ݉݃ሺݎ − ܴ௭ሻ = ݉݃ℎ
݉ݒଶ
ଶ
1 1
2 = ܴ݉݃௭ ൬− ݎ + ܴ ൰
௭
݉ݒଶ
݇݉
2 = ܴ݉݃௭ → ݒூூ = ඥ2ܴ݃௭ = 11 ݏ
1
݉݃ݎ
1 ܴ݉݃
݉݃
ܹ =
௭ + ݉݃ݎ
2 ൬݉݃ + ܴ ൰ ∗ ሺܴ௭ − ݎሻ =
൰ ሺܴ௭ − ݎሻ =
ሺܴ + ݎሻሺܴ௭ − ݎሻ =
௭
2 ൬
ܴ௭
2ܴ௭
݉݃
=
ଶ
2ܴ ሺܴ௭ − ݎଶሻ
௭
ܴ݉݃
ܹ
௭
→ =
2
ோ ݉݃ݎ
݉݃ ோ
݉݃
݉݃
ܹ = න
=
න ݎ ∗ ݀ݎ =
ோ =
ሺܴଶ௭ − ݎଶሻ
ܴ ∗ ݀ݎ
௭
ܴ௭
ܴ௭2 ሺݎଶሻ
2ܴ௭
Energia potencjalna 2
Praca jaką trzeba wykonać żeby przenieść ciało z nieskończoności do punktu odległego o r od środka innej planety
ܩܯ
ܽ = ݎଶ
ஶ ܩܯ݉
ஶ
ܹ→ஶ න
= ܩܯ݉ න ݎିଶ݀ݎ
ݎଶ ݀ݎ
1 ஶ ܩܯ݉
ܧூூ = ܹஶ→ = ܩܯ݉ ൬− ݎ൰ =
ݎ
Podstawowe wzory na całki ݎାଵ
න ݎ݀ݎ = ݊ + 1
1
න ݎିଶ݀ݎ = − ݎ
Zderzenia ciał
Δ→ = ݉ݒ→
→
ଶ − ݉ݒଵ = ܨ→ ∗ ݐ
ܨ ∗ ݐ = ܫ ሺę݀ ݏ݅łݕሻ
Δ→ = ܨ→ ∗ ݐ
Zderzenie sprężyste
݉ݒ→ + 0 = ݉ݒ→
→
ଵ + ݉ݒଶ
݉ݒ→ଶ ݉ݒ→ଶ
→ ଶ
ଵ
݉ݒଶ
2 = 2
+ 2
ݒ→ = ݒ→
→
ଵ + ݒଶ
ݒ→మ = ݒ→ଶ
→మ
ଵ + ݒଶ
ሺݒ→
→
→ଶ
→ଶ
ଵ + ݒଶ ሻଶ = ݒଵ + ݒଶ
ݒ→ଶ
→ଶ
→ →
ଵ + ݒଶ + 2ݒଵ ݒଶ → ߙ = 90°
݉ଵݒଵ + ݉ଶݒଶ = ݉ଵݑଵ + ݉ଶݑଶ
݉ ଶ
ଶ
ଶ
ଶ
ଵݒଵ
݉ଶݒଶ ݉ଵݑଵ ݉ଶݑଶ
2 + 2 = 2 + 2
2݉
ݑ
ଵݒଵ + ሺ݉ଶ − ݉ଵሻݒଶ
ଵ =
݉
ଵ + ݉ଶ
2݉
ݑ
ଶݒଶ + ሺ݉ଵ − ݉ଶሻݒଵ
ଶ =
݉
ଵ + ݉ଶ
Zderzenia plastyczne
݉ଵݒଵ + ݉ଶݒଶ = ሺ݉ଵ + ݉ଶሻ ∗ ܸ
ܸ = భ௩భାమ௩మ
భାమ
݉ ଶ
ଶ
ଵݒଵ
݉ଶݒଶ ሺ݉ଵ + ݉ଶሻܸଶ
2 + 2 >
2
݉ ଶ + ݉ ଶ ሺ݉
ܹ = ଵݒଵ
ଶݒଶ
ଵ + ݉ଶሻܸଶ
2
−
2
ܸ = ݒ + ܯ ∗ 0
݉ + ܯ
൫ܯ + ݉൯ܸଶ
2
= ൫ܯ + ݉൯݃ℎ
݉ଶ
ଶ
∗ ݒ
൫ܯ + ݉൯ ∗ ඥ2݃ܪ
2൫ܯ + ݉
݉
൯ଶ = ݃ℎ → ݒ
Energia
ସ
1g słoniny –
ଶ
1g węglowodanów –
ଷ∗ଵల
Benzyna
– 1g 30݇ܬ
1݈݇ܿܽ = 4180ܬ
200ܹ ∗ 3600 ∗ 24
20000
= 864݃
Siły oporu ośrodka
݈ = ݒ ∗ ݐ
݉ = ܵ ∗ ݈ ∗ ρ
݉ݒଶ ܵ ∗ ݈ ∗ ߩ ∗ ݒଶ ܵ ∗ ݒ ∗ ݐ ∗ ߩ ∗ ݒଶ ܵ ∗ ݐ ∗ ߩ ∗ ݒଷ
ܧ = 2 =
2
=
2
=
2
ܧ ܵߩݒଷ
ݐ = 2
1
ܨ ∗ ݒ = 2ߩ ∗ ܵ ∗ ݒଷ
1
ܨ = 2ߩ ∗ ܵ ∗ ݒଶ
1
ܨ = 2ߩܵݒଶܥ௫ሺ1− 0.2ሻ
ݒ = ඥ2݃ܪ
1
ܨ
ଶ
= 2 ߩ௪ ∗ ܵ ∗ ݒ
1
ܨ = 2ߩܵ ∗ 2݃ܪ
ܪ = ߩ ∗ ܵ ∗ 2݃ = ߩ ∗ ܵ
100 ∗ 100 ∗ 2
1000
1
ܪ = 1000∗ 20 ∗ 0.05 = 1000∗ 0.05 = 0.05 = 20݉