2.1 Kinematika — Masalalar

Jami: 30 ta | Yechim bilan: ✅

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Asosiy Masalalar (1-10)

Masala 1 ⭐⭐

Avtomobil 20 m/s tezlik bilan harakatlanmoqda. 5 s da qancha masofa bosadi?

Yechim

$x = vt = 20 \times 5 = 100$ m

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Masala 2 ⭐⭐

Velosiped 0 dan 15 m/s gacha 10 s da tezlashdi. Tezlanishni toping.

Yechim

$a = \frac{v - v_0}{t} = \frac{15 - 0}{10} = 1.5$ m/s²

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Masala 3 ⭐⭐

Jism 2 m/s² tezlanish bilan tinch holatdan harakatlandi. 6 s dagi tezligi va bosib o'tgan yo'li?

Yechim

$v = v_0 + at = 0 + 2(6) = 12$ m/s

$x = v_0 t + \frac{1}{2}at^2 = 0 + \frac{1}{2}(2)(36) = 36$ m

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Masala 4 ⭐⭐

Tosh 50 m balandlikdan erkin tushirildi. Yerga tushguncha qancha vaqt o'tadi? ($g = 10$ m/s²)

Yechim

$h = \frac{1}{2}gt^2$

$50 = \frac{1}{2}(10)t^2$

$t^2 = 10$

$t = \sqrt{10} \approx 3.16$ s

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Masala 5 ⭐⭐

Avtomobil 30 m/s tezlikda ketayotib, 4 m/s² tezlanish bilan tormozladi. To'xtaguncha qancha masofa bosadi?

Yechim

$v^2 = v_0^2 + 2ax$

$0 = 900 + 2(-4)x$

$x = \frac{900}{8} = 112.5$ m

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Masala 6 ⭐⭐

To'p 45° burchak ostida 20 m/s tezlik bilan otildi. Maksimal balandlik? ($g = 10$ m/s²)

Yechim

$h_{max} = \frac{v_0^2 \sin^2\theta}{2g} = \frac{400 \times 0.5}{20} = 10$ m

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Masala 7 ⭐⭐

Disk 100 rpm tezlikda aylanmoqda. Burchak tezlikni rad/s da toping.

Yechim

$\omega = \frac{2\pi \times 100}{60} = \frac{200\pi}{60} = \frac{10\pi}{3} \approx 10.47$ rad/s

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Masala 8 ⭐⭐

G'ildirak radiusi 0.3 m, burchak tezligi 20 rad/s. Chiziqli tezlik?

Yechim

$v = r\omega = 0.3 \times 20 = 6$ m/s

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Masala 9 ⭐⭐

Samolyot 200 m/s tezlik bilan 500 m radiusli aylana bo'ylab uchmoqda. Markazga intilma tezlanish?

Yechim

$a_n = \frac{v^2}{r} = \frac{40000}{500} = 80$ m/s²

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Masala 10 ⭐⭐

Proektil 30° burchak ostida 50 m/s bilan otildi. Gorizontal va vertikal tezlik komponentlari?

Yechim

$v_x = v_0 \cos 30° = 50 \times 0.866 = 43.3$ m/s

$v_y = v_0 \sin 30° = 50 \times 0.5 = 25$ m/s

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O'rtacha Masalalar (11-22)

Masala 11 ⭐⭐⭐

Dron A(0, 0, 10) dan B(100, 50, 30) m ga 20 s da uchdi. O'rtacha tezlik vektorini toping.

Yechim

$\Delta \vec{r} = (100-0, 50-0, 30-10) = (100, 50, 20)$ m

$\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t} = \frac{(100, 50, 20)}{20} = (5, 2.5, 1)$ m/s

$|\vec{v}_{avg}| = \sqrt{25 + 6.25 + 1} = \sqrt{32.25} \approx 5.68$ m/s

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Masala 12 ⭐⭐⭐

Robot g'ildiragi 0.1 m radiusli, motor 500 rpm beradi. Robotning chiziqli tezligi?

Yechim

$\omega = \frac{2\pi \times 500}{60} = 52.36$ rad/s

$v = r\omega = 0.1 \times 52.36 = 5.24$ m/s

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Masala 13 ⭐⭐⭐

To'p 60° burchak ostida 40 m/s bilan otildi. Uchish masofasi va vaqti? ($g = 10$ m/s²)

Yechim

$R = \frac{v_0^2 \sin 2\theta}{g} = \frac{1600 \times \sin 120°}{10} = \frac{1600 \times 0.866}{10} = 138.6$ m

$T = \frac{2v_0 \sin\theta}{g} = \frac{2 \times 40 \times 0.866}{10} = 6.93$ s

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Masala 14 ⭐⭐⭐

Jism $\vec{r}(t) = (3t^2, 4t, 5)$ m qonun bo'yicha harakat qilmoqda. $t = 2$ s dagi tezlik va tezlanish?

Yechim

$\vec{v} = \frac{d\vec{r}}{dt} = (6t, 4, 0)$

$t = 2$: $\vec{v}(2) = (12, 4, 0)$ m/s

$|\vec{v}| = \sqrt{144 + 16} = \sqrt{160} \approx 12.65$ m/s

$\vec{a} = \frac{d\vec{v}}{dt} = (6, 0, 0)$ m/s² (doimiy)

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Masala 15 ⭐⭐⭐

Ikki mashina A va B bir nuqtadan bir vaqtda harakatlandi. A shimolga 60 km/h, B sharqqa 80 km/h. 1 soatdan keyin ular orasidagi masofa?

Yechim

1 soatda:

• A: 60 km shimolga
• B: 80 km sharqqa

Masofa: $d = \sqrt{60^2 + 80^2} = \sqrt{3600 + 6400} = \sqrt{10000} = 100$ km

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Masala 16 ⭐⭐⭐

Raketa vertikal ko'tarilmoqda. Tezlanish $a = 20$ m/s². 10 s da qanday balandlikka ko'tariladi va tezligi qancha?

Yechim

$h = \frac{1}{2}at^2 = \frac{1}{2}(20)(100) = 1000$ m

$v = at = 20 \times 10 = 200$ m/s

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Masala 17 ⭐⭐⭐

Differensial drive robot: chap g'ildirak 1 m/s, o'ng 1.5 m/s, g'ildiraklar orasidagi masofa 0.5 m. Robot tezligi va burchak tezligi?

Yechim

$v = \frac{v_R + v_L}{2} = \frac{1.5 + 1}{2} = 1.25$ m/s

$\omega = \frac{v_R - v_L}{L} = \frac{1.5 - 1}{0.5} = 1$ rad/s

Robot chapga burilmoqda.

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Masala 18 ⭐⭐⭐

Quadcopter propeller 0 dan 10000 rpm gacha 2 s da tezlashdi. Burchak tezlanish va bu vaqtda necha marta aylandi?

Yechim

$\omega_0 = 0$, $\omega = \frac{2\pi \times 10000}{60} = 1047.2$ rad/s

$\alpha = \frac{\omega - \omega_0}{t} = \frac{1047.2}{2} = 523.6$ rad/s²

$\theta = \omega_0 t + \frac{1}{2}\alpha t^2 = 0 + \frac{1}{2}(523.6)(4) = 1047.2$ rad

Aylanishlar: $n = \frac{\theta}{2\pi} = \frac{1047.2}{2\pi} = 166.7$ aylanish

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Masala 19 ⭐⭐⭐

Jism 5 m radiusli aylana bo'ylab harakat qilmoqda. Tezligi $v = 2t$ m/s. $t = 3$ s da tangensial va normal tezlanish?

Yechim

$t = 3$ s da: $v = 6$ m/s

Tangensial: $a_t = \frac{dv}{dt} = 2$ m/s²

Normal: $a_n = \frac{v^2}{r} = \frac{36}{5} = 7.2$ m/s²

Umumiy: $|a| = \sqrt{4 + 51.84} = \sqrt{55.84} \approx 7.47$ m/s²

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Masala 20 ⭐⭐⭐

Kema A shimolga 30 km/h, kema B sharqqa 40 km/h harakat qilmoqda. B dan A ning nisbiy tezligi?

Yechim

$\vec{v}_A = (0, 30)$ km/h (shimolga) $\vec{v}_B = (40, 0)$ km/h (sharqqa)

$\vec{v}_{A/B} = \vec{v}_A - \vec{v}_B = (-40, 30)$ km/h

$|\vec{v}_{A/B}| = \sqrt{1600 + 900} = 50$ km/h

Yo'nalish: $\theta = \arctan(\frac{30}{-40}) = 143.1°$ (shimoli-g'arbga)

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Masala 21 ⭐⭐⭐

Snaryad 45° burchak ostida 100 m/s bilan otildi, lekin shamol gorizontal 10 m/s tezlikda esmoqda (qarshi). Uchish masofasi qanchaga kamayadi?

Yechim

Shamolsiz: $R_0 = \frac{v_0^2 \sin 90°}{g} = \frac{10000}{10} = 1000$ m

Shamol bilan (gorizontal tezlik kamayadi): $v_x = v_0 \cos 45° - 10 = 70.7 - 10 = 60.7$ m/s

Uchish vaqti (vertikal komponent o'zgarmaydi): $T = \frac{2 v_0 \sin 45°}{g} = \frac{2 \times 70.7}{10} = 14.14$ s

$R = v_x \times T = 60.7 \times 14.14 = 858.3$ m

Farq: $1000 - 858.3 = 141.7$ m kamaydi

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Masala 22 ⭐⭐⭐

Lift 3 m/s² tezlanish bilan yuqoriga ko'tarilmoqda. Unda turgan odam (70 kg) o'z og'irligini qanday sezadi?

Yechim

Effektiv tezlanish: $g_{eff} = g + a = 10 + 3 = 13$ m/s²

Sezilgan og'irlik: $W = mg_{eff} = 70 \times 13 = 910$ N

Oddiy og'irlik: $W_0 = 70 \times 10 = 700$ N

30% ko'proq og'irlik seziladi.

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Murakkab Masalalar (23-30)

Masala 23 ⭐⭐⭐⭐

Dron $\vec{r}(t) = (10\cos(0.5t), 10\sin(0.5t), 2t)$ m traektoriya bo'ylab uchmoqda. $t = 4$ s da tezlik va tezlanish vektorlarini toping.

Yechim

$\vec{v} = \frac{d\vec{r}}{dt} = (-5\sin(0.5t), 5\cos(0.5t), 2)$

$t = 4$: $0.5 \times 4 = 2$ rad

$\vec{v}(4) = (-5\sin 2, 5\cos 2, 2) = (-4.55, -2.08, 2)$ m/s

$|\vec{v}| = \sqrt{20.7 + 4.33 + 4} = 5.39$ m/s

$\vec{a} = \frac{d\vec{v}}{dt} = (-2.5\cos(0.5t), -2.5\sin(0.5t), 0)$

$\vec{a}(4) = (-2.5\cos 2, -2.5\sin 2, 0) = (1.04, -2.27, 0)$ m/s²

Bu spiral (helix) traektoriya.

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Masala 24 ⭐⭐⭐⭐

Robot qo'li ikkita bo'g'inli: $L_1 = 0.5$ m, $L_2 = 0.3$ m. Burchaklar: $\theta_1 = 45°$, $\theta_2 = 30°$. Burchak tezliklar: $\dot{\theta}_1 = 2$ rad/s, $\dot{\theta}_2 = 1$ rad/s. End effector tezligini toping.

Yechim

Forward kinematika: $x = L_1\cos\theta_1 + L_2\cos(\theta_1+\theta_2)$ $y = L_1\sin\theta_1 + L_2\sin(\theta_1+\theta_2)$

Tezlik (Jacobian usuli): $\dot{x} = -L_1\sin\theta_1 \dot{\theta}_1 - L_2\sin(\theta_1+\theta_2)(\dot{\theta}_1+\dot{\theta}_2)$ $\dot{y} = L_1\cos\theta_1 \dot{\theta}_1 + L_2\cos(\theta_1+\theta_2)(\dot{\theta}_1+\dot{\theta}_2)$

$\theta_1 = 45°$, $\theta_1+\theta_2 = 75°$

$\dot{x} = -0.5(0.707)(2) - 0.3(0.966)(3) = -0.707 - 0.869 = -1.576$ m/s

$\dot{y} = 0.5(0.707)(2) + 0.3(0.259)(3) = 0.707 + 0.233 = 0.940$ m/s

$|\vec{v}| = \sqrt{2.48 + 0.88} = 1.83$ m/s

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Masala 25 ⭐⭐⭐⭐

Raketa ko'tarilishda. Massasi $m(t) = 1000 - 10t$ kg (yonilg'i sarfi). Itarish kuchi $F = 15000$ N. $t = 50$ s dagi tezlanish? (Havo qarshiligini hisobga olmang)

Yechim

$t = 50$ s da massa: $m = 1000 - 500 = 500$ kg

Natijaviy kuch: $F_{net} = F - mg = 15000 - 500(10) = 10000$ N

Tezlanish: $a = \frac{F_{net}}{m} = \frac{10000}{500} = 20$ m/s²

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Masala 26 ⭐⭐⭐⭐

Dron A va B bir-birini kuzatmoqda. A(100, 50, 30) m da turgan, B (0, 0, 20) dan (10, 5, 0.5) m/s tezlik bilan harakatlanmoqda. Qancha vaqtdan keyin eng yaqin bo'ladi?

Yechim

B pozitsiyasi: $\vec{r}_B(t) = (10t, 5t, 20 + 0.5t)$

Orasidagi vektor: $\vec{d}(t) = \vec{r}_A - \vec{r}_B = (100-10t, 50-5t, 10-0.5t)$

Masofa kvadrati: $d^2 = (100-10t)^2 + (50-5t)^2 + (10-0.5t)^2$

Minimum uchun: $\frac{d(d^2)}{dt} = 0$

$-20(100-10t) - 10(50-5t) - 1(10-0.5t) = 0$

$-2000 + 200t - 500 + 50t - 10 + 0.5t = 0$

$250.5t = 2510$

$t = 10.02$ s

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Masala 27 ⭐⭐⭐⭐

G'ildirak radiusi 0.2 m, burchak tezlanish $\alpha = 5$ rad/s². Boshlang'ich $\omega_0 = 10$ rad/s. 4 s da g'ildirak chetidagi nuqtaning umumiy tezlanishi?

Yechim

$t = 4$ s da burchak tezlik: $\omega = \omega_0 + \alpha t = 10 + 5(4) = 30$ rad/s

Tangensial tezlanish: $a_t = r\alpha = 0.2 \times 5 = 1$ m/s²

Normal tezlanish: $a_n = r\omega^2 = 0.2 \times 900 = 180$ m/s²

Umumiy: $|a| = \sqrt{1^2 + 180^2} = \sqrt{32401} \approx 180$ m/s²

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Masala 28 ⭐⭐⭐⭐

Yo'ldosh Yer atrofida 400 km balandlikda aylanmoqda. Orbital tezlik va davr? (Yer radiusi 6371 km, $g = 9.81$ m/s²)

Yechim

Orbital radius: $r = 6371 + 400 = 6771$ km = $6.771 \times 10^6$ m

Orbital tezlik: $v = \sqrt{\frac{gR^2}{r}} = \sqrt{\frac{9.81 \times (6.371 \times 10^6)^2}{6.771 \times 10^6}}$

$v = \sqrt{\frac{398.2 \times 10^{12}}{6.771 \times 10^6}} = \sqrt{58.8 \times 10^6} = 7670$ m/s ≈ 7.67 km/s

Davr: $T = \frac{2\pi r}{v} = \frac{2\pi \times 6.771 \times 10^6}{7670} = 5545$ s ≈ 92.4 min

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Masala 29 ⭐⭐⭐⭐

Avtomobil egri yo'lda harakat qilmoqda. Egrilik radiusi $r = 100$ m. Tezlik $v = 20$ m/s va 2 m/s² bilan ortmoqda. Rul burchagi qanday o'zgarmoqda?

Yechim

Tangensial: $a_t = 2$ m/s² (tezlik ortishi)

Normal: $a_n = \frac{v^2}{r} = \frac{400}{100} = 4$ m/s²

Umumiy tezlanish burchagi (yo'nalishdan): $\phi = \arctan(\frac{a_n}{a_t}) = \arctan(\frac{4}{2}) = \arctan(2) = 63.4°$

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Masala 30 ⭐⭐⭐⭐

Quadcopter GPS'dan pozitsiya olmoqda (10 Hz). Ikki o'lchov: $P_1 = (100.0, 50.0, 30.0)$ m, $P_2 = (100.5, 50.2, 30.1)$ m. Tezlik va yo'nalish (heading)?

Yechim

$\Delta t = 0.1$ s (10 Hz)

$\Delta \vec{r} = (0.5, 0.2, 0.1)$ m

Tezlik: $\vec{v} = \frac{\Delta \vec{r}}{\Delta t} = (5, 2, 1)$ m/s

$|\vec{v}| = \sqrt{25 + 4 + 1} = \sqrt{30} \approx 5.48$ m/s

Heading (gorizontal): $\psi = \arctan(\frac{v_y}{v_x}) = \arctan(\frac{2}{5}) = 21.8°$ (shimoli-sharqqa)

Elevatsiya: $\theta = \arctan(\frac{v_z}{\sqrt{v_x^2+v_y^2}}) = \arctan(\frac{1}{\sqrt{29}}) = 10.5°$

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✅ Tekshirish Ro'yxati

• 1-10: Asosiy kinematika
• 11-22: O'rtacha masalalar
• 23-30: Murakkab masalalar

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