פתרון בגרות · מכניקה

✦ בנוי על ידי פהד גאנם ✦

t (s) | 0 1 2 3 4 5 6 7 8 9 10 11 12 13
v (m/s) | 0 2 4 6 8 10 12 13 13.5 13.8 14 14 14 14
v₀ = 14 m/s ; t_r = 0.75 s ; |a| = 3.5 m/s²
v_man = 100 km/h ; Δt_man = 2.6 s ; Δv_ad = 10 km/h

0.00 s

0.00 s

0.00 m/s

2.00 m/s²

0.0 m

0.00 m/s

a = ΔvΔt

a₁ = v(6) − v(0)6 − 0 = 12 − 06 = 2 m/s² v(3) − v(1)3 − 1 = 6 − 22 = 2 m/s²

13 − 127 − 6 = 1 ; 13.5 − 138 − 7 = 0.5 ; 13.8 − 13.59 − 8 = 0.3 ; 14 − 13.810 − 9 = 0.2 m/s²

a₃ = 14 − 1413 − 10 = 0 m/s² ; v = 14 m/s

v = 100 km/h = 100 · 10003600 = 2509 = 27.78 m/s

a_man = v − v₀Δt = 27.78 − 02.6 = 10.68 m/s²

a_max = a₁ = 2 m/s²

n = a_mana_max = 10.682 = 5.34

t = va₁ = 27.782 = 13.89 s ; 13.892.6 = 5.34

v̄ = ΔxΔt ; Δt = 13 s

x₁ = 12 · Δt · v = 12 · 6 · 12 = 36 m

x₂ = 12 + 132 + 13 + 13.52 + 13.5 + 13.82 + 13.8 + 142 x₂ = 12.5 + 13.25 + 13.65 + 13.9 = 53.3 m

x₃ = v · Δt = 14 · 3 = 42 m

x = 36 + 53.3 + 42 = 131.3 m v̄ = 131.313 = 10.1 m/s = 36.4 km/h

x_low = 0 + 2 + 4 + 6 + 8 + 10 + 12 + 13 + 13.5 + 13.8 + 14 + 14 + 14 = 124.3 m x_high = 2 + 4 + 6 + 8 + 10 + 12 + 13 + 13.5 + 13.8 + 14 + 14 + 14 + 14 = 138.3 m 124.3 + 138.32 = 131.3 m 124.313 = 9.56 ; 138.313 = 10.64 m/s

v₀ = 14 m/s ; v = 0 ; a = −3.5 m/s²

v = v₀ + a·t ⟹ 0 = 14 + (−3.5)·t t = v₀|a| = 143.5 = 4 s

d = v₀²2·|a| = 14²2 · 3.5 = 1967 = 28 m v̄ = v₀ + v2 = 14 + 02 = 7 m/s ; t = 287 = 4 s

d_r = v₀ · t_r = 14 · 0.75 = 10.5 m

v² = v₀² + 2·a·d_b ⟹ 0 = 14² − 2 · 3.5 · d_b d_b = v₀²2·|a| = 1967 = 28 m

d = d_r + d_b = 10.5 + 28 = 38.5 m

d_b = v₀·t + 12·a·t² = 14 · 4 − 12 · 3.5 · 4² = 56 − 28 = 28 m

12·v₀² = |a| · d_b ⟹ d_b = 983.5 = 28 m

d = v²2·|a|

d′d = (v − 10)²v² = (v − 10)v²

v = 20 ⟹ 1020² = 0.25 ; 25 % v = 50 ⟹ 4050² = 0.64 ; 64 % v = 100 ⟹ 90100² = 0.81 ; 81 %

v − 10v² = 12 ⟹ v − 10v = 1√2 v = 101 − 1√2 = 10 · (2 + √2) = 34.14 km/h 24.1434.14² = 0.7071² = 0.500

g = 10 m/s² ; a = g4 = 2.5 m/s²
m₁ = M3 ; T₀ = m₁·g = M·g3 ; T = M·a = M·g4
sinα = ag = 14 = 0.25 ; α = 14.4775 °

3.0 kg

1.000 kg

0.00 s

0.000 m

0.00 m/s

0.00 m/s²

10.00 N

1.333

N = M·g ; f = 0

m₁·g > T ; m₁·gT = 43

T = M·a m₁·g − T = m₁·a

m₁·g − M·a = m₁·a ⟹ m₁·(g − a) = M·a m₁ = M·ag − a

a = g4 = 2.5 m/s² ; g − a = 3·g4 m₁ = M · g43·g4 = M3

a = m₁·gM + m₁ = M·g34·M3 = g4 a = 1 · 103 + 1 = 2.5 m/s²

T₀ = m₁·g = M3 · g = M·g3 T₀ = 3.333 · M N

T = M·a = M · g4 = M·g4 = 2.5 · M N

T₀T = M·g3M·g4 = 43 ≈ 1.33

T = m₁·(g − a) = M3 · 3·g4 = M·g4 T₀ = 1 · 10 = 10 N ; T = 3 · 2.5 = 7.5 N ; 107.5 = 1.333

N = M·g·cosα ; cosα = 0.9682

M·g·sinα = M·a ⟹ a = g·sinα sinα = ag = 2.510 = 14 = 0.25

α = arcsin(0.25) = 14.4775 ° ≈ 14.5 °

d = v₀²2·a ; h = v₀²2·g sinα = hd = ag = 0.25 d = 255 = 5 m ; h = 2520 = 1.25 m ; 1.255 = 0.25

a = g·sinα = g4 = 2.5 m/s²

ΣF = M·g·sinα = M·g4 ≠ 0 ⟹ a = ΣFM = 2.5 m/s²

t_up = v₀a = 52.5 = 2 s ; t_total = 4 s ; v_end = 5 m/s

a = g·sinα = 10 · 0.25 = 2.5 m/s²

m·g·sinα − T = m·a m·g·sinα − T = m·g·sinα ⟹ T = 0

M·g·sinα ± T = M·a_box ; T ≠ 0 ⟹ a_box ≠ a_body

m_A = 0.1 kg ; m_B = 0.3 kg ; ℓ = 0.4 m ; g = 10 m/s²
ω = 2·π·f ; F_c = m·ω²·ℓ ; f_s ≤ μ_s·N

0.33

0.46 Hz

2.89 rad/s

2.18 s

1.15 m/s

3.33 m/s²

1.00 N

0.00 N

3.00 N

1.00 N

m_B·g = 0.3 · 10 = 3 N

N − m_B·g = 0 ⟹ N = m_B·g = 3 N

m_A·g = 0.1 · 10 = 1 N

T − m_A·g = 0 ⟹ T = m_A·g = 0.1 · 10 = 1 N

T = m_B·ω²·ℓ ; ω = 2·π·f ⟹ T = 4·π²·f²·m_B·ℓ

f = 12·π√m_A·gm_B·ℓ ω² = 0.1 · 100.3 · 0.4 = 10.12 = 8.3333 rad²/s² ω = 2.8868 rad/s ; f = 2.88682·π = 0.4594 Hz

T₀ = 2·π·√m_B·ℓm_A·g = 2·π·√0.12 = 2.1766 s f = 1T₀ = 12.1766 = 0.4594 Hz

v = ω·ℓ = 2.8868 · 0.4 = 1.1547 m/s F_c = m_B·v²ℓ = 0.3 · 1.33330.4 = 1 N = m_A·g

T′ = 2·m_A·g = 2 · 0.1 · 10 = 2 N

F_c = m_B·ω²·ℓ = 0.3 · 8.3333 · 0.4 = 1 N

T′ + f_s = m_B·ω²·ℓ ; 2 + f_s = 1 ⟹ f_s = −1 N

f = 12·π√20.3 · 0.4 = 0.6497 Hz 2 − 1 = 1 N

T′ − |f_s| = 2 − 1 = 1 N = m_B·ω²·ℓ

N = m_B·g = 0.3 · 10 = 3 N

f_s ≤ μ_s·N ⟹ μ_s ≥ f_sm_B·g = 13 = 0.3333

f_s,max = μ_s·N = 13 · 3 = 1 N

μ_s,min = 2·m_A·g − m_B·ω²·ℓm_B·g = 2 − 13 = 0.3333

T′ = 2·m_A·g = 2 N

F_c = (m_B + m_C)·ω²·ℓ = (0.3 + 0.3) · 8.3333 · 0.4 = 0.6 · 3.3333 = 2 N

T′ + f_s = (m_B + m_C)·ω²·ℓ ; 2 + f_s = 2 ⟹ f_s = 0

f_C = m_C·ω²·ℓ = 0.3 · 3.3333 = 1 N 2 − 1 + f_s = m_B·ω²·ℓ = 1 ⟹ f_s = 0

T′ = 2·m_A·g = 2 N ; F_c = (m_B + m_C)·ω²·ℓ

(m_B + m_C)·ω²·ℓ > (m_B + m_B)·ω²·ℓ = 2 N = T′

f_s = (m_B + m_C)·ω²·ℓ − 2·m_A·g = 3.3333·(m_C − 0.3) N

3.3333 · (0.3 − 0.3) = 0 3.3333 · (0 − 0.3) = −1 N

m = 0.1 kg ; T = 1.00 N ; F_c = 1.00 N ; f_s = 0

m = 0.2 kg ; T = 2.00 N ; F_c = 1.00 N ; f_s = 1.00 N

m = 0.2 kg ; T = 2.00 N ; F_c = 2.00 N ; f_s = 0

m = 0.2 kg ; T = 2.00 N ; F_c > 2.00 N ; f_s = F_c − T′

n | 1 | 2 | 3 | 4 | 5
h_n (m) | 1.30 | 1.12 | 0.88 | 0.73 | 0.53
ℓ = 0.25 m ; g = 10 m/s²

1.491 m

0.386

0.25 m

1.00

1.491 m

0.00 m/s

1.491 m

0.000 m

0.193 m

14.910 J/kg

0.000 J/kg

ΣF_y = N − m·g = 0 ⟹ N = m·g f = μ·N = μ·m·g

a = −μ·m·gm = −μ·g = −0.386 · 10 = −3.86 m/s²

W_f = −f · (2·ℓ) = −μ·m·g · 2·ℓ = −2·μ·m·g·ℓ

W_f = m·g·h₁ − m·g·h₀ = −m·g·(h₀ − h₁) h₀ − h₁ = 2·μ·ℓ

−2·μ·g·ℓ = −2 · 0.386 · 10 · 0.25 = −1.93 J/kg −g·(h₀ − h₁) = −10 · (1.491 − 1.298) = −1.93 J/kg

ΔE = 2·μ·m·g·ℓ

m·g·h_n = m·g·h₀ − n · 2·μ·m·g·ℓ h_n = m·g·h₀ − 2·n·μ·m·g·ℓm·g = h₀ − 2·μ·ℓ·n

h₁ = h₀ − 2·μ·ℓ ; h₂ = h₁ − 2·μ·ℓ = h₀ − 4·μ·ℓ ; h_n = h_n−1 − 2·μ·ℓ

n_max = h₀2·μ·ℓ = 1.4910.193 = 7.73

(1 , 1.30) ; (2 , 1.12) ; (3 , 0.88) ; (4 , 0.73) ; (5 , 0.53)

n̄ = 1 + 2 + 3 + 4 + 55 = 3 ; h̄ = 4.565 = 0.912 m Σ(n − n̄)·(h − h̄) = (−2)·0.388 + (−1)·0.208 + 0·(−0.032) + 1·(−0.182) + 2·(−0.382) = −1.930 Σ(n − n̄)² = 4 + 1 + 0 + 1 + 4 = 10 שיפוע = −1.93010 = −0.193 m חיתוך = h̄ − שיפוע · n̄ = 0.912 + 0.193 · 3 = 0.912 + 0.579 = 1.491 m h_n = 1.491 − 0.193·n

שיפוע = 0.526 − 1.2985 − 1 = −0.7724 = −0.193 m

1.298 ; 1.105 ; 0.912 ; 0.719 ; 0.526 m · קו המגמה 1.30 ; 1.12 ; 0.88 ; 0.73 ; 0.53 m · המדוד

h_n = h₀ + (−2·μ·ℓ)·n ; h_n = 1.491 − 0.193·n

h₀ = 1.491 ≈ 1.5 m

−2·μ·ℓ = −0.193 ⟹ μ = 0.1932 · ℓ = 0.1930.5 = 0.386 ≈ 0.39

2·μ·ℓ = h₁ − h₅4 = 1.30 − 0.534 = 0.1925 m μ = 0.19250.5 = 0.385 ; h₀ = h₁ + 2·μ·ℓ = 1.30 + 0.1925 = 1.4925 m

v_B = √2·g·h₀ = √29.82 = 5.46 m/s v_C² = v_B² − 2·μ·g·ℓ = 29.82 − 1.93 = 27.89 ⟹ v_C = 5.28 m/s v_B′² = 27.89 − 1.93 = 25.96 ⟹ v_B′ = 5.10 m/s h₁ = v_B′²2·g = 25.9620 = 1.298 m

W_קיר = ΔE_k = 12·m·v_אחרי² − 12·m·v_לפני²

12·m·v_לפני² = m·g·h₀ − μ·m·g·ℓ

m·g·h₁′ = 12·m·v_אחרי² − μ·m·g·ℓ v_אחרי < v_לפני ⟹ W_קיר = 12·m·(v_אחרי² − v_לפני²) < 0

W_קיר = −m·g·(h₁ − h₁′)

m = 400 g = 0.4 kg ; h₀ = 1 m ; h_max = 6 m ; g = 10 m/s²
m₁ = m₂ = m ; v₂ = 0 ; ΔE_k = 0

0.000 s

0.25×

6.00 m

0.000 s

1.000 m

10.000 m/s

4.000 kg·m/s

20.000 J

4.000 J

24.000 J

m = 400 g = 0.4 kg ; h_max = 6 m ; g = 10 m/s²

m·g·h_max = 12·m·v² v = √2·g·h_max = √2 · 10 · 6 = √120 = 10.95 m/s

v² = v₀² + 2·g·Δh = 0 + 2 · 10 · 6 = 120 m²/s² t = √2·h_maxg = √1.2 = 1.095 s ; v = g·t = 10 · 1.095 = 10.95 m/s

v² = 120 ; E_k = 12 · 0.4 · 120 = 24.00 J

12·m·v₀² + m·g·h₀ = 12·m·v² + 0 12·m·v² = 12·m·v₀² + m·g·h₀ > 12·m·v₀² ⟹ v₀ < v

6 − 1 = 5 m ; 6 − 0 = 6 m

v₀ = √2·g·(h_max − h₀) = √2 · 10 · 5 = √100 = 10.00 m/s

E = 12 · 0.4 · 10² + 0.4 · 10 · 1 = 20 + 4 = 24.00 J E = m·g·h_max = 0.4 · 10 · 6 = 24.00 J E = 12 · 0.4 · 120 = 24.00 J

J = Δp = m·v_after − m·v_before

v_before = −√120 m/s ; v_after = +√120 m/s J = m·(v_after − v_before) = m·(√120 + √120) = 2·m·√120 J = 2 · 0.4 · 10.95 = 0.8 · 10.95 = 8.76 N·s

p_before = 0.4 · 10.95 = 4.38 kg·m/s ; p_after = 4.38 kg·m/s Δp = 4.38 + 4.38 = 8.76 N·s

W = ΔE_k = 12·m·v_after² − 12·m·v_before² W = 12 · 0.4 · 120 − 12 · 0.4 · 120 = 24 − 24 = 0 J

E_k = 12·m·v² ; ΔE_k = 0 ; Δp = 8.76 N·s

W = F · Δr ; Δr ≈ 0 ⟹ W ≈ 0 ; ΔE_p = 0

12·m·v² = 12·m·v₁² + m·g·h₀ v₁ = √v² − 2·g·h₀ = √120 − 20 = √100 = 10.00 m/s v₂ = 0 m/s

m·v₁ + m · 0 = m·u₁ + m·u₂ ⟹ v₁ = u₁ + u₂ 12·m·v₁² = 12·m·u₁² + 12·m·u₂² ⟹ v₁² = u₁² + u₂²

v₁² = u₁² + (v₁ − u₁)² = 2·u₁² − 2·v₁·u₁ + v₁² 0 = 2·u₁·(u₁ − v₁) u₁ = 0 ; u₂ = v₁ = 10.00 m/s

p_before = 0.4 · 10 + 0.4 · 0 = 4.00 kg·m/s ; p_after = 0.4 · 0 + 0.4 · 10 = 4.00 kg·m/s E_k = 12 · 0.4 · 10² = 20.00 J ; E_k = 0 + 12 · 0.4 · 10² = 20.00 J v₁ − v₂ = 10 − 0 = 10 ; u₂ − u₁ = 10 − 0 = 10 m/s

h = 1 + 10²2 · 10 = 1 + 5 = 6.00 m t = √2 · 110 = 0.447 s ; v = √20 = 4.47 m/s

v = u₁ + u₂ ; v² = u₁² + u₂² u₁ · u₂ = 0 ⟹ θ = 90°

u₁ + u₂ ≈ 0 ; θ ≈ 180°

u₁² + u₂² = v² ; θ = 90° ; u₁ + u₂ ∥ v

v = √120 = 10.95 m/s v₀ = 10.00 m/s < 10.95 m/s J = 8.76 N·s W = 0 J u₁ = 0 ; u₂ = 10.00 m/s

G = 6.67·10⁻¹¹ N·m²/kg² ; M_E = 5.98·10²⁴ kg ; M_M = 7.35·10²² kg
d = 3.84·10⁸ m ; R_M = 1.74·10⁶ m ; h = 2.60·10⁵ m
G·M_E = 3.98866·10¹⁴ ; G·M_M = 4.90245·10¹² ; 9·T₁ = 10·T₂

3.0000·10⁸ m

3.0000 ·10⁸ m

8.4000 ·10⁷ m

4.4318 ·10⁻³ N/kg

0.6948 ·10⁻³ N/kg

6.3787

G·M_E·mx² = G·M_M·m(d − x)²

x²(d − x)² = M_EM_M xd − x = √M_EM_M

M_EM_M = 5.98·10²⁴7.35·10²² = 81.3605 xd − x = √81.3605 = 9.0200 x = d · 9.02001 + 9.0200 = 3.84·10⁸ · 0.90020

x = 3.457·10⁸ m d − x = 3.84·10⁸ − 3.457·10⁸ = 3.832·10⁷ m

F_Em = 3.98866·10¹⁴(3.4568·10⁸)² = 3.98866·10¹⁴1.1949·10¹⁷ = 3.338·10⁻³ N/kg F_Mm = 4.90245·10¹²(3.8323·10⁷)² = 4.90245·10¹²1.4687·10¹⁵ = 3.338·10⁻³ N/kg

t = 9·T₁ = 10·T₂ ⟹ T₁T₂ = 109 = 1.1111

G·M_M·mr² = m · 4π²·rT² ⟹ T² = 4π²G·M_M · r³ r₁³r₂³ = T₁²T₂² ⟹ r₁r₂ = T₁T₂^⅔

r₁r₂ = 109^⅔ = ∛10081 = ∛1.2346 = 1.073

G·M_Mr² = ω²·r ⟹ ω = √G·M_Mr³ ω₁ω₂ = 0.9 ⟹ r₂r₁ = 0.9^⅔ = 0.9322 ⟹ r₁r₂ = 10.9322 = 1.073

1.0728³ = 1.2346 = 10081

h = 260 km = 2.60·10⁵ m r₂ = R_M + h = 1.74·10⁶ + 2.60·10⁵ = 2.00·10⁶ m

G·M_M·mr₂² = m·a ⟹ a = G·M_Mr₂² a = 6.67·10⁻¹¹ · 7.35·10²²(2.00·10⁶)² = 4.90245·10¹²4.00·10¹² = 1.226 m/s²

v₂ = √G·M_Mr₂ = √2.4512·10⁶ = 1.566·10³ m/s T₂ = 2π·r₂v₂ = 2π · 2.00·10⁶1565.6 = 8.03·10³ s a = v₂²r₂ = 2.4512·10⁶2.00·10⁶ = 1.226 m/s² a = 4π²·r₂T₂² = 7.896·10⁷6.442·10⁷ = 1.226 m/s²

g_M = G·M_MR_M² = 4.90245·10¹²3.0276·10¹² = 1.619 m/s² 1.6191.321 = 1.226 m/s²

F = G·m₁·m₂r²

a₁ = Fm₁ ; a₂ = Fm₂

F_thrust − m·g_M = m·a ; a > 0 ⟹ F_thrust > m·g_M

a = 2.519 − 1.619 = 0.900 m/s² ; t = 300.900 = 33.33 s Δy = 30²2 · 0.900 = 500 m