2.3 Energiya va Impuls — Masalalar

Jami: 25 ta | Yechim bilan: ✅

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Asosiy Masalalar (1-10)

Masala 1 ⭐⭐

5 kg jism 10 m/s tezlik bilan harakatlanmoqda. Kinetik energiyasini toping.

Yechim

$KE = \frac{1}{2}mv^2 = \frac{1}{2}(5)(100) = 250$ J

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Masala 2 ⭐⭐

2 kg jism 20 m balandlikdan tushirildi. Yerga yetgandagi tezligi? (Energiya saqlanishi)

Yechim

$mgh = \frac{1}{2}mv^2$

$v = \sqrt{2gh} = \sqrt{2 \times 10 \times 20} = \sqrt{400} = 20$ m/s

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Masala 3 ⭐⭐

100 N kuch jismni 5 m ga siljitdi (kuch va harakat bir yo'nalishda). Bajarilgan ish?

Yechim

$W = Fd\cos\theta = 100 \times 5 \times \cos 0° = 500$ J

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Masala 4 ⭐⭐

Motor 2000 W quvvat bilan ishlaydi. 1 daqiqada qancha ish bajaradi?

Yechim

$W = P \times t = 2000 \times 60 = 120000$ J = 120 kJ

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Masala 5 ⭐⭐

Prujina (k = 400 N/m) 0.2 m ga siqildi. Potensial energiyasi?

Yechim

$PE = \frac{1}{2}kx^2 = \frac{1}{2}(400)(0.04) = 8$ J

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Masala 6 ⭐⭐

3 kg jism 8 m/s tezlik bilan harakatlanmoqda. Impulsini toping.

Yechim

$p = mv = 3 \times 8 = 24$ kg·m/s

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Masala 7 ⭐⭐

10 N kuch 5 s davomida jismga ta'sir qildi. Impuls (impulse) qancha?

Yechim

$J = F \times t = 10 \times 5 = 50$ N·s = 50 kg·m/s

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Masala 8 ⭐⭐

1500 kg avtomobil 20 m/s dan 30 m/s ga tezlashdi. Kinetik energiya o'zgarishi?

Yechim

$\Delta KE = \frac{1}{2}m(v_2^2 - v_1^2) = \frac{1}{2}(1500)(900 - 400)$ $= 750 \times 500 = 375000$ J = 375 kJ

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Masala 9 ⭐⭐

50 N kuch 60° burchak ostida 4 m ga siljitdi. Ish?

Yechim

$W = Fd\cos\theta = 50 \times 4 \times \cos 60° = 200 \times 0.5 = 100$ J

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Masala 10 ⭐⭐

2 kg jism 5 m/s, 3 kg jism -2 m/s tezlikda. Sistemaning umumiy impulsi?

Yechim

$p_{total} = m_1v_1 + m_2v_2 = 2(5) + 3(-2) = 10 - 6 = 4$ kg·m/s

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O'rtacha Masalalar (11-18)

Masala 11 ⭐⭐⭐

2 kg jism 10 m balandlikdan prujinaga (k = 5000 N/m) tushdi. Prujina maksimal qancha siqiladi?

Yechim

Energiya saqlanishi: $mgh = \frac{1}{2}kx^2$

Lekin jism prujinani siqganda ham balandlik yo'qotadi: $mg(h + x) = \frac{1}{2}kx^2$

$2 \times 10 \times (10 + x) = \frac{1}{2}(5000)x^2$

$200 + 20x = 2500x^2$

$2500x^2 - 20x - 200 = 0$

$x = \frac{20 + \sqrt{400 + 2000000}}{5000} = \frac{20 + 1414}{5000} = 0.287$ m

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Masala 12 ⭐⭐⭐

Elastik to'qnashuv: 4 kg jism 6 m/s, 2 kg jism tinch. To'qnashuvdan keyingi tezliklar?

Yechim

$v_1' = \frac{(m_1-m_2)v_1}{m_1+m_2} = \frac{(4-2)(6)}{6} = \frac{12}{6} = 2$ m/s

$v_2' = \frac{2m_1 v_1}{m_1+m_2} = \frac{2(4)(6)}{6} = 8$ m/s

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Masala 13 ⭐⭐⭐

5 kg jism 30° qiya tekislikdan 10 m sirpanib tushdi. Ishqalanish $\mu = 0.2$. Oxirgi tezlik?

Yechim

Balandlik o'zgarishi: $h = 10\sin 30° = 5$ m

Ishqalanish ishi: $W_f = -\mu mg\cos 30° \times d = -0.2(5)(10)(0.866)(10) = -86.6$ J

Energiya: $mgh + W_f = \frac{1}{2}mv^2$

$5(10)(5) - 86.6 = \frac{1}{2}(5)v^2$

$250 - 86.6 = 2.5v^2$

$v = \sqrt{65.36} = 8.08$ m/s

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Masala 14 ⭐⭐⭐

Noelastik to'qnashuv: 1000 kg avtomobil 20 m/s va 1500 kg avtomobil 10 m/s (qarama-qarshi). Yakuniy tezlik va yo'qotilgan energiya?

Yechim

Impuls saqlanishi: $m_1v_1 + m_2v_2 = (m_1+m_2)v_f$

$1000(20) + 1500(-10) = 2500 v_f$

$20000 - 15000 = 2500 v_f$

$v_f = 2$ m/s

Yo'qotilgan energiya: $KE_i = \frac{1}{2}(1000)(400) + \frac{1}{2}(1500)(100) = 200000 + 75000 = 275000$ J

$KE_f = \frac{1}{2}(2500)(4) = 5000$ J

$\Delta KE = 275000 - 5000 = 270000$ J = 270 kJ

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Masala 15 ⭐⭐⭐

Dron (2 kg) 15 m balandlikda hovering qilmoqda. 3 m/s tezlik bilan yuqoriga ko'tarila boshladi. Qo'shimcha kinetik energiya?

Yechim

$\Delta KE = \frac{1}{2}mv^2 = \frac{1}{2}(2)(9) = 9$ J

Motor bu energiyani berishi kerak (+ hovering uchun quvvat).

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Masala 16 ⭐⭐⭐

Mayatnik (1 kg, 2 m ip) 45° dan qo'yib yuborildi. Eng past nuqtadagi tezlik?

Yechim

Balandlik o'zgarishi: $h = L - L\cos 45° = L(1 - \cos 45°) = 2(1 - 0.707) = 0.586$ m

$mgh = \frac{1}{2}mv^2$

$v = \sqrt{2gh} = \sqrt{2(10)(0.586)} = 3.42$ m/s

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Masala 17 ⭐⭐⭐

Robot (20 kg) 2 m/s tezlikda harakatlanmoqda. 100 N kuch 3 m da to'xtatadi. Ishqalanish kuchi?

Yechim

Boshlang'ich KE: $\frac{1}{2}(20)(4) = 40$ J

Motor ishi: $W_{motor} = 100 \times 3 = 300$ J

Ish-energiya: $W_{net} = \Delta KE$ $W_{motor} - W_f = 0 - 40$ $300 - W_f = -40$ $W_f = 340$ J

Ishqalanish kuchi: $f = \frac{W_f}{d} = \frac{340}{3} = 113.3$ N

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Masala 18 ⭐⭐⭐

Avtomobil (1200 kg) 25 m/s tezlikda. Tormoz quvvati 50 kW. To'xtash vaqti va masofasi?

Yechim

Boshlang'ich KE: $\frac{1}{2}(1200)(625) = 375000$ J

To'xtash vaqti: $t = \frac{KE}{P} = \frac{375000}{50000} = 7.5$ s

O'rtacha tezlik: $\bar{v} = \frac{25}{2} = 12.5$ m/s

Masofa: $d = \bar{v} \times t = 12.5 \times 7.5 = 93.75$ m

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Murakkab Masalalar (19-25)

Masala 19 ⭐⭐⭐⭐

Raketa (1000 kg) vertikal uchmoqda. Exhaust tezligi 2000 m/s, yonilg'i sarfi 10 kg/s. 50 s dan keyin tezlik? (Tsiolkovsky)

Yechim

$m_0 = 1000$ kg, $m_f = 1000 - 10(50) = 500$ kg

Tsiolkovsky (gravitatsiyasiz): $\Delta v = v_e \ln\frac{m_0}{m_f} = 2000 \ln\frac{1000}{500} = 2000 \ln 2 = 1386$ m/s

Gravitatsiya ta'siri: $\Delta v_{actual} = v_e \ln\frac{m_0}{m_f} - gt = 1386 - 10(50) = 886$ m/s

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Masala 20 ⭐⭐⭐⭐

Quadcopter (2 kg) hovering qilmoqda (balandlik 50 m). Batareya 100 Wh. Uchish vaqti? (Hovering quvvati $P = \frac{(mg)^{3/2}}{\sqrt{2\rho A}}$, $\rho = 1.2$ kg/m³, A = 0.1 m²)

Yechim

$P = \frac{(2 \times 10)^{3/2}}{\sqrt{2 \times 1.2 \times 0.1}} = \frac{89.44}{\sqrt{0.24}} = \frac{89.44}{0.49} = 182.5$ W

Batareya energiyasi: $100$ Wh = $100 \times 3600 = 360000$ J

Uchish vaqti: $t = \frac{360000}{182.5} = 1973$ s ≈ 33 min

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Masala 21 ⭐⭐⭐⭐

Ballistik mayatnik: 10 g o'q 500 m/s tezlikda 2 kg blokka kirib qoldi. Blok qancha balandlikka ko'tariladi?

Yechim

1-qadam: Noelastik to'qnashuv (impuls saqlanishi) $m_{oq}v_{oq} = (m_{oq} + m_{blok})v_f$

$0.01(500) = (2.01)v_f$

$v_f = \frac{5}{2.01} = 2.49$ m/s

2-qadam: Energiya saqlanishi (ko'tarilish) $\frac{1}{2}mv^2 = mgh$

$h = \frac{v^2}{2g} = \frac{6.2}{20} = 0.31$ m

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Masala 22 ⭐⭐⭐⭐

Robot qo'li (5 kg) gorizontaldan vertikal holatga 90° aylandi. Motor qancha ish bajaradi? (Uzunlik 0.8 m, massa markazi o'rtada)

Yechim

Massa markazining balandlik o'zgarishi: $\Delta h = \frac{L}{2} = 0.4$ m

Gravitatsiya ishiga qarshi: $W = mgh = 5 \times 10 \times 0.4 = 20$ J

(Bu minimal ish — haqiqatda samaradorlik va dinamik effektlar hisobga olinadi)

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Masala 23 ⭐⭐⭐⭐

2D elastik to'qnashuv: 2 kg jism (4, 0) m/s, 3 kg jism (0, 3) m/s. 2 kg jism to'qnashuvdan keyin (1, 2) m/s bo'lsa, 3 kg jismning tezligi?

Yechim

Impuls saqlanishi (vektor): $m_1\vec{v}_1 + m_2\vec{v}_2 = m_1\vec{v}_1' + m_2\vec{v}_2'$

X: $2(4) + 3(0) = 2(1) + 3v_{2x}'$ $8 = 2 + 3v_{2x}'$ $v_{2x}' = 2$ m/s

Y: $2(0) + 3(3) = 2(2) + 3v_{2y}'$ $9 = 4 + 3v_{2y}'$ $v_{2y}' = \frac{5}{3}$ m/s

$\vec{v}_2' = (2, 1.67)$ m/s

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Masala 24 ⭐⭐⭐⭐

Figure skater (60 kg, I₁ = 3 kg·m²) qo'llarini yig'di (I₂ = 1.5 kg·m²). Boshlang'ich ω₁ = 2 rad/s. Yangi burchak tezlik va kinetik energiya o'zgarishi?

Yechim

Burchak impuls saqlanishi: $I_1\omega_1 = I_2\omega_2$

$\omega_2 = \frac{I_1\omega_1}{I_2} = \frac{3 \times 2}{1.5} = 4$ rad/s

Kinetik energiya: $KE_1 = \frac{1}{2}I_1\omega_1^2 = \frac{1}{2}(3)(4) = 6$ J

$KE_2 = \frac{1}{2}I_2\omega_2^2 = \frac{1}{2}(1.5)(16) = 12$ J

$\Delta KE = 12 - 6 = 6$ J (ortdi!)

Energiya skater mushak kuchidan keldi.

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Masala 25 ⭐⭐⭐⭐

Kosmik kema (10000 kg, v = 100 m/s) 1000 kg yuk qismini 50 m/s nisbiy tezlikda orqaga uloqtirdi. Kemaning yangi tezligi?

Yechim

Impuls saqlanishi: $(m_{kema} + m_{yuk})v_0 = m_{kema}v_1 + m_{yuk}v_2$

Yuk tezligi (kema sanoq sistemasida): $v_2 = v_1 - 50$

$10000(100) = 9000 v_1 + 1000(v_1 - 50)$

$1000000 = 9000v_1 + 1000v_1 - 50000$

$1050000 = 10000v_1$

$v_1 = 105$ m/s

Kema 5 m/s tezlashdi.

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✅ Tekshirish Ro'yxati

• 1-10: Asosiy energiya va impuls
• 11-18: O'rtacha masalalar
• 19-25: Murakkab masalalar

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Keyingi Qadam

📖 2.4 Tebranishlar