פתרון בגרות מכניקה — שאלון 36361

v_0A = +30 m/s ; a_A = −2 m/s² ; x_A(0) = 0
v_0B = 0 ; a_B1 = −3 m/s² (0 ≤ t ≤ 10 s) ; x_B(0) = AB
t_stop,A = t_stop,B ; x_stop,A = x_stop,B

0.00 s

0.00 s

0.00 m

30.00 m/s

−2.00 m/s²

0.00 m

450.00 m

0.00 m/s

−3.00 m/s²

0.00 m

450.00 m

x_A(0) = 0 m ; v_0A = +30 m/s

a_A = Δv_AΔt = 0 − (+30)Δt ; a_A = −2 m/s²

a_B = Δv_BΔt = (−3) − 01 = −3 m/s²

a_A = −2 m/s² ; a_B1 = −3 m/s² ⟹ a_A · a_B1 = +6 ; (+)

a_A = −2 m/s²

v_A(t) = v_0A + a_A·t = 30 − 2·t 0 = 30 + (−2)·t_stop ⟹ t_stop = 302 = 15 s

30 m/s2 m/s · s = 15 s

v_A² = v_0A² + 2·a_A·Δx_A 0 = 30² + 2·(−2)·Δx_A ⟹ Δx_A = 9004 = 225 m

v̄_A = v_0A + v_A2 = 30 + 02 = 15 m/s Δx_A = v̄_A·t = 15 · 15 = 225 m

Δx_A = 12 · 15 · 30 = 225 m

a_B1 = −3 m/s² ; t₁ = 10 s v_B(t₁) = 0 + (−3)·10 = −30 m/s

t_brake = 15 − 10 = 5 s

a_B2 = Δv_BΔt_brake = 0 − (−30)5 = +6 m/s²

d_B1 = 12 · 3 · 10² = 150 m d_B2 = 30 + 02 · 5 = 75 m 0 = (−30)² + 2·a_B2·(−75) ⟹ a_B2 = 900150 = +6 m/s²

|a_B2||a_B1| = 63 = 2 ; t₁t_brake = 105 = 2

d_B = d_B1 + d_B2 = 150 + 75 = 225 m d_A = 225 m AB = d_A + d_B = 225 + 225 = 450 m

x_A(t) = 30·t − 12·2·t² = 30·t − t² ; x_A(15) = 450 − 225 = 225 m x_B(t) = L − 32·t² ; x_B(10) = L − 150 x_B(15) = (L − 150) − 30·5 + 3·5² = L − 225 L − 225 = 225 ⟹ L = 450 m

12·10·30 + 12·5·30 = 150 + 75 = 225 m AB = 225 + 225 = 450 m D(t) = 4·(t − 15)² , 10 ≤ t ≤ 15 ; D(10) = 100 m ; D(12.5) = 25 m ; D(15) = 0

v_A(t) = 30 − 2·t , 0 ≤ t ≤ 15 a_A = 0 − 3015 − 0 = −2 m/s²

a_B1 = −30 − 010 − 0 = −3 m/s² a_B2 = 0 − (−30)15 − 10 = +6 m/s²

A : (0 , +30) ; (15 , 0) B : (0 , 0) ; (10 , −30) ; (15 , 0)

Δx_A = +225 m ; Δx_B = −225 m ; Δx_A − Δx_B = 450 m

m_A > m_B ; N = 0 ; g = 10 m/s²
S₁ : T₁ | S₂ : T₂ = 2·T₁
m_A = 3 kg ; m_B = 2 kg

3.0 kg

2.0 kg

1.00 m

0.00 s

0.00 m/s²

0.00 m/s

20.00 N

40.00 N

1.000 m

0.000 m

0.00 N

10.00 N

m_B·g = 2 · 10 = 20 N

N = 0

T₂ ; T₁ ; T₁

T₁ − m_B·g = 0 ⟹ T₁ = m_B·g

T₂ − T₁ − T₁ = 0 ⟹ T₂ = 2·T₁ = 2·m_B·g T₂ = 2 · 2 · 10 = 40 N

T₂ + F_yad = (m_A + m_B)·g T₁ + F_yad = m_A·g ⟹ F_yad = m_A·g − m_B·g = 10 N T₂ = (m_A + m_B)·g − (m_A − m_B)·g = 2·m_B·g ✓

|a_A| = |a_B| = a

ΣF_A = m_A·a > m_B·a = ΣF_B 3 · 2 = 6 N ; 2 · 2 = 4 N

T_A = T_B = T₁

ΣF_A = m_A·g − T₁ ; ΣF_B = T₁ − m_B·g

m_A·g − T₁ = m_A·a (1) T₁ − m_B·g = m_B·a (2)

m_A·g − m_B·g = (m_A + m_B)·a a = m_A − m_Bm_A + m_B · g

m_A·g·h − m_B·g·h = 12 · (m_A + m_B)·v² v² = 2·(m_A − m_B)·g·hm_A + m_B 2·a·h = 2·(m_A − m_B)·g·hm_A + m_B ⟹ a = m_A − m_Bm_A + m_B · g ✓

a = 3 − 23 + 2 · 10 = 15 · 10 = 2 m/s²

T₁ = m_A·(g − a) = 3 · (10 − 2) = 24 N T₁ = m_B·(g + a) = 2 · (10 + 2) = 24 N ✓

ag = 210 = 15 ; m_A − m_Bm_A + m_B = 15 ✓

T₁ = m_B·(g + a) = 2 · (10 + 2) = 24 N

T₂ = 2·T₁ = 2 · 24 = 48 N

(m_A + m_B)·g − T₂ = 50 − 48 = 2 N 3 · 2 − 2 · 2 = 6 − 4 = 2 N ✓

40 < 48 < 50 N

T₁ = 2·m_A·m_Bm_A + m_B · g ; T₂ = 4·m_A·m_Bm_A + m_B · g T₂ = 4 · 3 · 25 · 10 = 245 · 10 = 48 N ✓

h (m) | 10 | 20 | 30 | 40 | 50
d (m) | 21.2 | 31.6 | 38.1 | 43.6 | 47.4
d² (m²) | 449.4 | 998.6 | 1451.6 | 1901.0 | 2246.8
g = 10 m/s² ; v_0y = 0 ; a_x = 0

25 m

15.0 m/s

0.00 s

25.00 m

15.00 m/s

0.00 m/s

15.00 m/s

0.00 °

0.00 m

0.00 m²

a_x = ΣF_xm = 0m = 0 ; a_y = g = 10 m/s²

v_x = v₀ = const ; x = v₀·t

v_y = v_0y + g·t = 0 + g·t = 10·t

h = 12·g·t² ⟹ t = √2·hg

d = v_x·t = v₀·√2·hg

d² = v₀²·2·hg = 2·v₀²g·h m = 2·v₀²g

2·v₀²g ⟶ m²/s²m/s² = m

(10 , 449.4) ; (20 , 998.6) ; (30 , 1451.6) (40 , 1901.0) ; (50 , 2246.8) 21.2² = 449.44 ; 47.4² = 2246.76

d² ≈ 45·h

449.410 = 44.94 ; 2246.850 = 44.94 m 998.620 = 49.93 ; 1451.630 = 48.39 ; 1901.040 = 47.53 m

m = Δ(d²)Δh = 2246.8 − 449.450 − 10 = 1797.440 = 44.94 m

m = 2·v₀²g ⟹ v₀² = m·g2 = 44.94 · 102 = 224.7 m²/s² v₀ = √224.7 = 14.99 ≈ 15 m/s

t₍₁₎ = √2 · 1010 = 1.414 s ; v₀ = 21.21.414 = 14.99 m/s t₍₅₎ = √2 · 5010 = 3.162 s ; v₀ = 47.43.162 = 14.99 m/s

2 · 15²10 = 45 m d(10) = 15·√2 = 21.21 m ; d(50) = 15·√10 = 47.43 m

h = 25 m ; v_x = v₀ = 15 m/s

v_y² = v_0y² + 2·g·h = 0 + 2 · 10 · 25 = 500 m²/s² v_y = √500 = 22.36 m/s

v = √v_x² + v_y² = √225 + 500 = √725 = 26.93 ≈ 26.9 m/s

tan θ = v_yv_x = 22.3615 = 1.491 θ = arctan(1.491) = 56.1° ≈ 56°

12·m·v₀² + m·g·h = 12·m·v² v² = v₀² + 2·g·h = 225 + 500 = 725 m²/s² ⟹ v = 26.93 m/s

t = √2 · 2510 = √5 = 2.236 s ; v_y = g·t = 22.36 m/s d = v₀·t = 15 · 2.236 = 33.54 m ; d² = 1125 = 45 · 25 m²

m = 0.5 kg ; α = 30° ; h = 2 m ; g = 10 m/s²
L_AB = hsin α = 4 m ; d = htan α = 2√3 ≈ 3.46 m
E_k(0) = 10 J ; E_k(5 m) = 0 ; Δ = −2 J/m

2.00 m

30°

0.50 kg

0.400

×1.00

0.00 m/s

0.00 m

0.00 J

4.00 m

5.00 m

2.00 N

6.32 m/s

L_AB = hsin α = 20.5 = 4 m d = htan α = 2√3 ≈ 3.46 m

W = F · s · cos θ ; θ = 90° ; cos 90° = 0 W_N = 0 J N = m·g·cos α = 0.5 · 10 · √32 = 2.5√3 ≈ 4.33 N

W_grav = −m·g·h = −0.5 · 10 · 2 = −10 J

m·g·sin α = 0.5 · 10 · 0.5 = 2.5 N W_grav = −(m·g·sin α) · L_AB = −2.5 · 4 = −10 J

W_tot = ΔE_k = E_k(A) − E_k(B)

E_k(B) = 12·m·0² = 0 ; E_k(A) = 12·m·0² = 0 W_tot = 0 − 0 = 0 J

W_tot = W_F + W_grav + W_N

0 = W_F + (−10) + 0 W_F = +10 J

W_F = ΔE_mech = ΔE_p + ΔE_k = m·g·h + 0 = 0.5 · 10 · 2 = 10 J

m·g·h + 0 = 0 + 12·m·v_B² v_B = √2·g·h = √2 · 10 · 2 = √40 ≈ 6.3 m/s E_k(B) = m·g·h = 0.5 · 10 · 2 = 10 J

a = g·sin α = 10 · 0.5 = 5 m/s² v_B² = v₀² + 2·a·L_AB = 0 + 2 · 5 · 4 = 40 m²/s² v_B = √40 = 6.3246 m/s

N = m·g ; f_k = μ_k·N = μ_k·m·g

W_fric = −f_k·x = −μ_k·m·g·x

E_k(x) − E_k(B) = W_fric ; E_k(B) = m·g·h E_k(x) = m·g·h − μ_k·m·g·x = m·g·(h − μ_k·x)

E_k(x) = 0.5 · 10 · 2 − μ_k · 0.5 · 10 · x = 10 − 5·μ_k·x J

E_k(0) = m·g·h = 0.5 · 10 · 2 = 10 J

ΔE_kΔx = 0 − 105 − 0 = −2 J/m

−μ_k·m·g = −2 ; m·g = 0.5 · 10 = 5 N μ_k = 2m·g = 25 = 0.4

m·g·h − μ_k·m·g·x_stop = 0 x_stop = hμ_k = 20.4 = 5 m

a = μ_k·g = 0.4 · 10 = 4 m/s² x_stop = v_B²2·a = 408 = 5 m

m₁ = 2 kg ; F_max = 10⁴ N ; Δt = 1 · 10⁻³ s ; g = 10 m/s²
t (10⁻³ s) | 0 | 0.5 | 1
F (N) | 0 | 10⁴ | 0

1.875 m/s

4.00 kg

0.050 m

0.0000 s

0.00 N

0.000 N·s

1.875 m/s

0.000 m/s

1.250 m/s

3.750 kg·m/s

3.516 J

0.000 J

J = ∫ F · dt = Δp₂ = m₂·u₂ − m₂ · 0

J = 12 · Δt · F_max = 12 · (1 · 10⁻³) · (10⁴) J = 5 N·s

F_avg = F_max2 = 10⁴2 = 5 · 10³ N J = F_avg · Δt = 5 · 10³ · 1 · 10⁻³ = 5 N·s ✓

a_2,max = F_maxm₂ = 10⁴4 = 2500 m/s²

J = Δp₂ = m₂·u₂ − m₂ · 0 = m₂·u₂

m₂ = Ju₂ = 51.25

u₂ = 2·m₁m₁ + m₂ · v₁ = 2 · 22 + 4 · 1.875 = 46 · 1.875 = 1.25 m/s ✓

m₁·v₁ = 2 · 1.875 = 3.75 kg·m/s m₁·u₁ + m₂·u₂ = 2 · (−0.625) + 4 · 1.25 = −1.25 + 5 = 3.75 kg·m/s ✓

m₁ = 2 kg ; m₂ = 4 kg ; u₂ = 1.25 m/s

m₁·v₁ + m₂ · 0 = m₁·u₁ + m₂·u₂ 2 · v₁ = 2 · u₁ + 4 · 1.25 ; 2·v₁ = 2·u₁ + 5

12·m₁·v₁² + 0 = 12·m₁·u₁² + 12·m₂·u₂² 12 · 2 · v₁² = 12 · 2 · u₁² + 12 · 4 · (1.25)² v₁² = u₁² + 3.125

v₁ = 1.875 m/s ; u₁ = −0.625 m/s u₁ = m₁ − m₂m₁ + m₂ · v₁ = 2 − 42 + 4 · 1.875 = −13 · 1.875 = −0.625 m/s

2 · 1.875 = 3.75 ; 2 · (−0.625) + 4 · 1.25 = 3.75 kg·m/s ✓ 12 · 2 · 1.875² = 3.515625 J 12 · 2 · 0.625² + 12 · 4 · 1.25² = 0.390625 + 3.125 = 3.515625 J ✓

Δp₁ = m₁·u₁ − m₁·v₁ = 2 · (−0.625) − 2 · 1.875 = −5 N·s ✓

F_2,1(t) = − F_1,2(t)

F_min = −10⁴ N ; Δt = 1 · 10⁻³ s

a_1,max = 10⁴2 = 5000 m/s² ; a_2,max = 10⁴4 = 2500 m/s²

J₁ = −12 · (1 · 10⁻³) · (10⁴) = −5 N·s Δp₁ = m₁·(u₁ − v₁) = 2 · (−0.625 − 1.875) = −5 N·s ✓

J₁ + J₂ = (−5) + (+5) = 0 N·s ✓

u₂ = 1.25 m/s ; a = 0

a = −g·sin θ = const

12·m·u₂² = 12·m·v_CD² + m·g·h v_CD = √u₂² − 2·g·h

h < u₂²2·g = 1.25²2 · 10 = 1.562520 = 0.078125 m v_CD = √1.25² − 2 · 10 · 0.05 = √0.5625 = 0.75 m/s ✓

W₀ = 700 N ; g = 10 m/s² ; m = 70 kg
R = 6.37·10⁶ m ; h = 3200 km = 3.2·10⁶ m
r = R + h = 9.57·10⁶ m ; T_⊕ = 86400 s

3200 km

×2000

9570 km

310.14 N

3.54 N

306.59 N

4.4305 m/s²

695.95 m/s

9234 s

2.565 h

6511.5 m/s

ΣF = 0 ; a = 0 ; v = 0

F_g = m·GMr² ;

N ;

N − F_g = 0 ⟹ N = F_g

m = W₀g = 70010 = 70 kg r = R + h = 6.37·10⁶ + 3.2·10⁶ = 9.57·10⁶ m

N = F_g(r)

F_g(r)W₀ = G·M·mr²G·M·mR² = (Rr)²

Rr = 6.37·10⁶9.57·10⁶ = 0.66562 F_g(r) = 700 · (0.66562)² = 700 · 0.44305 = 310.14 N

GM = g·R² = 10 · (6.37·10⁶)² = 4.0577·10¹⁴ m³/s² g(r) = GMr² = 4.0577·10¹⁴9.15849·10¹³ = 4.4305 m/s² N = m·g(r) = 70 · 4.4305 = 310.14 N

ω = 2πT_⊕ = 2π86400 = 7.2722·10⁻⁵ rad/s

F_g − N = m·ω²·r ⟹ N = F_g − m·ω²·r

m·ω²·r ⟹ N = F_g − m·ω²·r F_g

ω² = (7.2722·10⁻⁵)² = 5.2885·10⁻⁹ 1/s² m·ω²·r = 70 · 5.2885·10⁻⁹ · 9.57·10⁶ = 3.54 N N = 310.14 − 3.54 = 306.59 N ≈ 3.07·10² N

a_c = ω²·r = 5.2885·10⁻⁹ · 9.57·10⁶ = 0.0506 m/s² N = m·(g(r) − a_c) = 70 · (4.4305 − 0.0506) = 306.59 N

r = R + h = 9.57·10⁶ m

GM·m_kr² = m_k·4π²T²·r T² = 4π²·r³GM ⟹ T = 2π√r³GM

r³ = (9.57·10⁶)³ = 8.7647·10²⁰ m³ r³GM = 8.7647·10²⁰4.0577·10¹⁴ = 2.1600·10⁶ s² T = 2π · 1469.70 = 9234 s ≈ 9.23·10³ s T = 92343600 = 2.565 h

v = √GMr = √4.2400·10⁷ = 6511.5 m/s T = 2π·rv = 6.0130·10⁷6511.5 = 9234 s

T = 2π√rg(r) = 2π√9.57·10⁶4.4305 = 2π · 1469.70 = 9234 s

F_g − 0 = m·ω²·r₁ ⟹ GM·mr₁² = m·ω²·r₁ r₁³ = GMω² ⟹ r₁ = ∛GMω²

ω = 2π86400 = 7.2722·10⁻⁵ rad/s ; ω² = 5.2885·10⁻⁹ 1/s² r₁³ = 4.0577·10¹⁴5.2885·10⁻⁹ = 7.6727·10²² m³ r₁ = ∛7.6727·10²² = 4.2493·10⁷ m

h₁ = r₁ − R = 4.2493·10⁷ − 6.37·10⁶ = 3.6123·10⁷ m h₁ ≈ 3.61·10⁷ m = 3.61·10⁴ km ≈ 36100 km

T = 2π√7.6727·10²²4.0577·10¹⁴ = 2π√1.8909·10⁸ = 2π · 13751 = 86400 s

r₁r = (864009234)^2÷3 = (9.3574)^2÷3 = 4.4402 r₁ = 4.4402 · 9.57·10⁶ = 4.2493·10⁷ m

v = ω·r₁ = 7.2722·10⁻⁵ · 4.2493·10⁷ = 3090 m/s g(r₁) = GMr₁² = 0.2247 m/s² ; F_g = 70 · 0.2247 = 15.73 N

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