פתרון בגרות מכניקה — שאלון 36361

v_A(0) = 12 m/s | v_A(10 s) = 8 m/s | v_B(0) = 0 | v_B(10 s) = 8 m/s
a_A = 8 − 1210 − 0 = −0.4 m/s² ; a_B = 8 − 010 − 0 = +0.8 m/s²
x_A(10 s) = 100 m ; x_B(10 s) = 40 m ; x_station = 180 m ; a_B,2 = −835 m/s²

0.00 s

×1

2.5 s

0.00 s

12.00 m/s

0.00 m/s

0.0 m

1

180.0 m

v_A(0) = 12 m/s ; v_A(10 s) = 8 m/s v_B(0) = 0 ; v_B(10 s) = 8 m/s

a = ΔvΔt = v(10 s) − v(0)10 − 0 a_A = 8 − 1210 − 0 = −410 = −0.4 m/s²

a_B = 8 − 010 − 0 = 810 = +0.8 m/s²

v_A(t) = 12 − 0.4·t → v_A(10 s) = 12 − 4 = 8 m/s v_B(t) = 0.8·t → v_B(10 s) = 8 m/s

v_A(10 s) = +8 m/s ; v_B(10 s) = +8 m/s

x_A = v₀ + v2·t = 12 + 82·10 = 10 · 10 = 100 m x_B = 0 + 82·10 = 4 · 10 = 40 m

x_A(t) = 12t − 12(0.4)t² = 12t − 0.2t² → x_A(10 s) = 120 − 20 = 100 m x_B(t) = 12(0.8)t² = 0.4t² → x_B(10 s) = 40 m x_Ax_B = 10040 = 2.5

v² = v₁₀² + 2a·Δx → 0 = 8² + 2(−0.4)·Δx Δx = 8²2 · 0.4 = 640.8 = 80 m x_station = x_A(10 s) + Δx = 100 + 80 = 180 m

0 = 12² + 2(−0.4)·x → x = 1440.8 = 180 m

v = v₁₀ + a·Δt → 0 = 8 + (−0.4)·Δt Δt = 80.4 = 20 s t_total = 10 + 20 = 30 s

v̄ = v₀ + v2 = 12 + 02 = 6 m/s t = xv̄ = 1806 = 30 s x_A(30 s) = 12 · 30 − 0.2 · 30² = 360 − 180 = 180 m v_A(30 s) = 12 − 0.4 · 30 = 0

d = x_station − x_B(10 s) = 180 − 40 = 140 m

d = v₁₀ + 02·Δt_B → 140 = 8 + 02·Δt_B = 4·Δt_B Δt_B = 1404 = 35 s

t_B,stop = 10 + 35 = 45 s ; t_A,stop = 30 s Δt = t_B,stop − t_A,stop = 45 − 30 = 15 s

a_B,2 = 0 − v₁₀²2d = −8²2 · 140 = −64280 = −835 ≈ −0.229 m/s² Δt_B = 0 − 8−835 = 8 · 358 = 35 s d = 8 · 35 − 12 · 835 · 35² = 280 − 140 = 140 m

Δx = v̄ · Δt_drop ; Δt_drop = const

t (s) : 0 2.5 5 7.5 10 12.5 15 17.5 20 x (m) : 0 2.5 10 22.5 40 59.29 77.14 93.57 108.57 Δx (m) : 2.5 7.5 12.5 17.5 19.29 17.86 16.43 15.00

F (N) | 3.5 | 4.0 | 4.5 | 5.0 | 5.5
a (m/s²) | 1.9 | 2.7 | 3.4 | 4.2 | 5.0
T = F ; N = m·g ; f = μ·N = μ·m·g ; g = 10 m/s²

3.50 N

3.50 N

6.50 N

2.275 N

1.225 N

1.88 m/s²

0.00 m/s

0.00 m

1.94 m/s

T = F

N = m·g

ΣF_y = N − m·g = 0 N = m·g

f = μ·N = μ·m·g

ΣF_x = T − f = m·a F − μ·m·g = m·a a = F − μ·m·gm = Fm − μ·g

(3.5 , 1.9) (4.0 , 2.7) (4.5 , 3.4) (5.0 , 4.2) (5.5 , 5.0)

slope = a₂ − a₁F₂ − F₁ = 5.0 − 1.95.5 − 3.5 = 3.12.0 = 1.55 b = 1.9 − 1.55 · 3.5 = −3.525 m/s² a = 1.55 · F − 3.53

F₀ = 3.5251.55 = 2.27 N

slope = ΔaΔF = 1m = 1.55 m = 1slope = 11.55 = 0.645 ≈ 0.65 kg

1.9 = 1.55 · 3.5 + b ; b = 1.9 − 5.425 = −3.525 −μ·g = −3.525 ; μ = 3.52510 = 0.3525 ≈ 0.35

F₀ = 3.5251.55 = 2.274 N μ = F₀m·g = 2.2740.645 · 10 = 0.3525 ≈ 0.35

F̄ = 4.5 N ; ā = 17.25 = 3.44 m/s² slope = Σ(F − F̄)(a − ā)Σ(F − F̄)² = 3.852.5 = 1.54 m = 11.54 = 0.649 kg ; b = 3.44 − 1.54 · 4.5 = −3.49 ; μ = 0.349

a = F0.65 − 3.5 1.88 | 2.65 | 3.42 | 4.19 | 4.96 m/s² (מחושב) 1.9 | 2.7 | 3.4 | 4.2 | 5.0 m/s² (נמדד)

W_T = F · d = 5.0 · 1 = 5.0 J W_f = −μ·m·g·d = −0.35 · 0.65 · 10 · 1 = −2.275 J W = 5.0 − 2.275 = 2.725 J v = √2 · 2.7250.65 = √8.385 = 2.90 m/s v = √2·a·d = √2 · 4.19 · 1 = 2.90 m/s

f_max = μ·N = μ·m·g = 0.35 · 0.65 · 10 = 2.275 N

F = 1.5 N ; f_max = 2.275 N ; f_max − F = 0.775 N f = T = F = 1.5 N ; ΣF = 0 ; a = 0

a = 1.55 · 1.5 − 3.525 = −1.2 m/s²

ΣF = F − μ·m·g = 1.5 − 2.275 = −0.775 N

m = 2 kg | a = 3 m | ℓ = 1 m | α = 30° | g = 10 m/s² | T_max = 45 N
sin 30° = 0.5 ; cos 30° = 0.866 ; tan 30° = 0.577 ; m·g = 20 N
r = a + ℓ·sin α = 3 + 1 · 0.5 = 3.5 m ; T = m·gcos α ; T·sin α = m·ω²·r

0.204 Hz

2.0 kg

3.0 m

1.0 m

45 N

30.00°

3.50 m

1.28 rad/s

4.89 s

4.50 m/s

23.09 N

11.55 N

20.00 N

1.15

r = a + ℓ·sin α = 3 + 1 · 0.5 = 3.5 m m·g = 2 · 10 = 20 N

F_c = T·sin α ; ΣF_x = m·ω²·r

ΣF_y = m·a_y = 0 ; T·cos α = m·g

Tm·g = 1cos 30° = 2√3 = 1.155

a_y = 0 ; a_x = a_c = ω²·r

ΣF_y = m·a_y = 0 T·cos α − m·g = 0 T = m·gcos α = 2 · 10cos 30° = 40√3 = 23.09 N

F_c = m·g·tan α = 20 · 0.5774 = 11.547 N T = √(m·g)² + F_c² = √400 + 133.33 = √533.33 = 23.09 N

α = 0 ; T₀ = m·g = 2 · 10 = 20 N

T = m·gcos α = m·g0.866 = 1.1547 · m·g = 1.1547 · T₀ TT₀ = 1cos 30° = 1.1547

20 N ; 23.09 N ; ΔT = 3.09 N

r = a + ℓ·sin α = 3 + 1 · sin 30° = 3 + 0.5 = 3.5 m

ΣF_x = m·a_c ; T·sin α = m·ω²·r 23.094 · 0.5 = 2 · ω² · 3.5 11.547 = 7 · ω² ω² = 11.5477 = 1.6496 s⁻² ; ω = 1.2844 rad/s

f = ω2π = 1.28446.2832 = 0.204 Hz T_מחזור = 1f = 4.89 s

tan α = ω²·rg ; ω = √g·tan αr = √10 · 0.57743.5 = 1.2844 rad/s

v = ω·r = 1.2844 · 3.5 = 4.4952 m/s T_מחזור = 2π·rv = 21.9914.4952 = 4.892 s ; f = 0.2044 Hz F_c = m·v²r = 2 · 20.2073.5 = 11.547 N = T·sin α

T_max·cos α_max = m·g cos α_max = m·gT_max = 2045 = 49 = 0.4444 ; α_max = 63.61° sin α_max = √659 = 0.8958 ; tan α_max = √654 = 2.0156

r_max = a + ℓ·sin α_max = 3 + 1 · 0.8958 = 3.8958 m

T_max·sin α_max = m·ω_max²·r_max 45 · 0.8958 = 2 · ω_max² · 3.8958 ; 40.311 = 7.7916 · ω_max² ω_max² = 40.3117.7916 = 5.1737 s⁻² ; ω_max = 2.2746 rad/s f_max = ω_max2π = 2.27466.2832 = 0.362 Hz

ω_max = √g·tan α_maxr_max = √10 · 2.01563.8958 = 2.2746 rad/s v_max = ω_max·r_max = 8.8613 m/s ; T_מחזור = 2π·r_maxv_max = 2.762 s F_c = m·v_max²r_max = 40.311 N = T_max·sin α_max

1 : 500

0.040 kg

+3.0 m/s

−1.5 m/s

5 ms

0.00 ms

+3.00 m/s

−1.50 m/s

+0.420 kg·m/s

−0.060 kg·m/s

0.360 kg·m/s

2.000 m/s

0.675 J

0.000 N·s

0.0 N

v_A > 0 ; v_B < 0

v_A = +3 m/s ; v_B = −1.5 m/s

|Δv₁| = 2 m/s ; |Δv₂| = 7 m/s

m_A·v_A + m_B·v_B = m_A·v′_A + m_B·v′_B 0.14 · 3 + m_B · (−1.5) = 0.14 · 1 + m_B · 5.5 0.42 − 0.14 = 5.5·m_B + 1.5·m_B 0.28 = 7·m_B m_B = 0.287 = 0.04 kg

m_A·|Δv_A| = m_B·|Δv_B| 0.14 · 2 = m_B · 7 m_B = 0.287 = 0.04 kg

E_k = ½·0.14·3² + ½·0.04·1.5² = 0.63 + 0.045 = 0.675 J E′_k = ½·0.14·1² + ½·0.04·5.5² = 0.07 + 0.605 = 0.675 J

v_A − v_B = 3 − (−1.5) = 4.5 m/s v′_B − v′_A = 5.5 − 1 = 4.5 m/s v_cm = 0.14·3 + 0.04·(−1.5)0.18 = 0.360.18 = 2 m/s v′_cm = 0.14·1 + 0.04·5.50.18 = 0.360.18 = 2 m/s

J = Δp_A = m_A·v′_A − m_A·v_A = m_A·(v′_A − v_A) J = 0.14 · (1 − 3) = 0.14 · (−2) = −0.28 N·s

J_B = m_B·(v′_B − v_B) = 0.04 · (5.5 − (−1.5)) = 0.04 · 7 = +0.28 N·s Δp_A + Δp_B = (−0.28) + (+0.28) = 0

Δt = 10 − 5 = 5 ms = 5·10⁻³ s = 0.005 s

F = JΔt = Δp_AΔt F = −0.285·10⁻³ = −56 N

a_A = 1 − 3(10 − 5)·10⁻³ = −25·10⁻³ = −400 m/s² F = m_A·a_A = 0.14 · (−400) = −56 N

a_B = 5.5 − (−1.5)5·10⁻³ = 75·10⁻³ = +1400 m/s² F_B = m_B·a_B = 0.04 · 1400 = +56 N

F_AB = − F_BA

J_AB = F_AB·Δt = − F_BA·Δt = − J_BA

|J_A| = 0.14 · |1 − 3| = 0.28 N·s |J_B| = 0.04 · |5.5 − (−1.5)| = 0.28 N·s

Δv = Jm ; |Δv_A| = 0.280.14 = 2 m/s ; |Δv_B| = 0.280.04 = 7 m/s

m = 2 kg | μ = 0.2 | g = 10 m/s² | x₀ = 0 | x_A = 5 m
N = m·g = 2 · 10 = 20 N ; f = μ·N = 0.2 · 20 = 4 N
F = 5 N (0 ≤ x ≤ 3 m) ; F = 5 + 5·(x − 3) N (3 m ≤ x ≤ 5 m)
k = 15 − 55 − 3 = 102 = 5 N/m

0.20

2.0 kg

30°

0.00 m

0.00 m/s

0.00 m²/s²

0.00 J

5.00 N

4.00 N

20.00 N

0.00 J

8.75 m

17.11 N

3.42 N

19.04 J

9.76 m

ΔW = F·Δx ; W_F = Σ F·Δx = S

S₁ = F·Δx = 5 · 3 = 15 J S₂ = (5 + 15) · 22 = 402 = 20 J W_F = S₁ + S₂ = 15 + 20 = 35 J

k = 15 − 55 − 3 = 102 = 5 N/m F(x) = 5 + 5·(x − 3) N (3 ≤ x ≤ 5) W_F = ∫₀³ 5 dx + ∫₃⁵ [5 + 5(x − 3)] dx = 15 + (10 + 10) = 35 J

ΣF_y = 0 ; N = m·g = 2 · 10 = 20 N f = μ·N = 0.2 · 20 = 4 N

W_f = f·d·cos 180° = −f·x_A = −μ·m·g·x_A W_f = −0.2 · 2 · 10 · 5 = −4 · 5 = −20 J

|W_f| = 4 · 5 = 20 J ; W_f = −20 J

W_total = W_F + W_f = 35 + (−20) = 15 J

W_total = ΔE_k = 12·m·v_A² − 0 15 = 12 · 2 · v_A² v_A² = 2 · 152 = 15 m²/s² v_A = √15 = 3.87 m/s

a₁ = F − fm = 5 − 42 = 0.5 m/s² v₃² = 2·a₁·Δx = 2 · 0.5 · 3 = 3 m²/s² → v₃ = 1.73 m/s 3 → 5 m : W = 20 − 4 · 2 = 12 J 12·2·v_A² = 3 + 12 = 15 → v_A² = 15 → v_A = 3.87 m/s

−f·d = 0 − 12·m·v_A² d = m·v_A²2·f = m·v_A²2·μ·m·g = v_A²2·μ·g d = 152 · 0.2 · 10 = 154 = 3.75 m

x_B = x_A + d = 5 + 3.75 = 8.75 m

a = −fm = −42 = −2 m/s² 0 = v_A² + 2·a·d = 15 − 4·d → d = 154 = 3.75 m t = v_A|a| = 3.8732 = 1.94 s ; d = v_A·t2 = 3.75 m

W(F₁) = ∫ F₁·cos α dx = ∫ F dx = 35 J

N₁ = m·g − F₁·sin α < m·g f₁ = μ·N₁ = μ·(m·g − F₁·sin α) < μ·m·g = 4 N

E_k(A) = 35 − μ·(m·g − F₁·sin α)·5 > 35 − 20 = 15 J

d = E_k(A)f ; E_k,C(A) > E_k,B(A) → d_C > d_B = 3.75 m x_C = 5 + d_C > 5 + 3.75 = x_B = 8.75 m

G = 6.67·10⁻¹¹ N·m²/kg² | M = 86.98·10²⁴ kg = 8.698·10²⁵ kg | R = 26.1·10⁶ m = 2.61·10⁷ m
r_M = 13·10⁷ m = 1.3·10⁸ m ; r_A = 19·10⁷ m = 1.9·10⁸ m ; M_E = 5.974·10²⁴ kg ; R_E = 6.38·10⁶ m
G·M = 6.67·10⁻¹¹ · 8.698·10²⁵ = 5.801566·10¹⁵ m³/s² ; R² = 6.8121·10¹⁴ m²

1.30·10⁸ m

1.90·10⁸ m

8.698·10²⁵ kg

2.61·10⁷ m

8.52 m/s²

0.343 m/s²

0.161 m/s²

1.223·10⁵ s

33.96 h

2.160·10⁵ s

60.01 h

6.68·10³ m/s

5.53·10³ m/s

4.616·10⁵ s

8.156·10⁵ s

1.767

0.00

G·M·mR² = m·g g = G·MR²

M = 86.98·10²⁴ = 8.698·10²⁵ kg ; R = 26.1·10⁶ = 2.61·10⁷ m g = 6.67·10⁻¹¹ · 8.698·10²⁵(2.61·10⁷)² = 5.801566·10¹⁵6.8121·10¹⁴ = 8.5166 m/s² g ≈ 8.5 m/s²

g_Ug_E = MM_E · (R_ER)² = 86.985.974 · (6.3826.1)² = 14.5598 · 0.059753 = 0.86999 g_E = 6.67·10⁻¹¹ · 5.974·10²⁴(6.38·10⁶)² = 3.984658·10¹⁴4.07044·10¹³ = 9.7893 m/s² g_U = 0.86999 · 9.7893 = 8.5166 m/s²

G·M·m_Mr_M² = m_M·a_r a_r = G·Mr_M²

r_M = 13·10⁷ = 1.3·10⁸ m ; r_M² = 1.69·10¹⁶ m² a_r = 5.801566·10¹⁵1.69·10¹⁶ = 0.34329 m/s² ≈ 0.34 m/s²

a_r = g · (Rr_M)² = 8.5166 · (2.61·10⁷1.3·10⁸)² = 8.5166 · (0.200769)² a_r = 8.5166 · 0.0403083 = 0.34329 m/s²

a_r = ω²·r_M = 4π²·r_MT_M² T_M = 2π√r_Ma_r

T_M = 2π√1.3·10⁸0.34329 = 2π√3.786908·10⁸ = 2π · 19459.98 T_M = 1.22271·10⁵ s ≈ 1.22·10⁵ s

a_r = v²r_M → v = √a_r·r_M = √0.34329 · 1.3·10⁸ = √4.46274·10⁷ = 6680.4 m/s T_M = 2π·r_Mv = 8.168141·10⁸6680.4 = 1.22271·10⁵ s

T_M = 2π√r_M³G·M = 2π√2.197·10²⁴5.801566·10¹⁵ = 2π√3.786908·10⁸ T_M = 1.22271·10⁵ s

G·M·mr² = m · 4π²·rT² T² = 4π²·r³G·M → T = 2π√r³G·M

T_A²T_M² = r_A³r_M³ → T_AT_M = (r_Ar_M)^1.5 T_AT_M = (19·10⁷13·10⁷)^1.5 = (1.461538)^1.5 = √68592197 = √3.121985 = 1.7669 T_AT_M ≈ 1.77

T_A = 2π√(1.9·10⁸)³5.801566·10¹⁵ = 2π√1.1822669·10⁹ = 2π · 34384.11 T_A = 2.16042·10⁵ s T_AT_M = 2.16042·10⁵1.22271·10⁵ = 1.7669

T = 2π√r³G·M_E T_בT_א = √r_ב³r_א³ = (r_בr_א)^1.5

r_בr_א = r_Ar_M = 1913 → T_בT_א = (1913)^1.5 = 1.7669

G·M_E = g·R_E² = 10 · (6.38·10⁶)² = 4.07044·10¹⁴ m³/s² T_א = 2π√2.197·10²⁴4.07044·10¹⁴ = 2π · 73467.3 = 4.6161·10⁵ s T_ב = 2π√6.859·10²⁴4.07044·10¹⁴ = 2π · 129810.5 = 8.1562·10⁵ s T_בT_א = 8.1562·10⁵4.6161·10⁵ = 1.7669

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