פתרון בגרות מכניקה — שאלון 36361

MN = 189 km ; P = MN2 ; MP = PN = 94.5 km
A : 6 ● ⟹ 5 · Δt | B : 7 ● ⟹ 6 · Δt
t_B (M → N) = 3 h

0.00 h

0.00 h

0 min

0.00 km

96.00 km/h

63.00 km/h

0.00 km

—

96.00 km/h

63.00 km/h

1.566 h

94.50 km/h

63.00 km/h

−31.50 km/h

0.50 h

−63.00 km/h²

MP = PN = 1892 = 94.5 km

Δx_B = 1896 = 31.5 km 3 · 31.5 = 94.5 km

x_A (km) : 0 , 48 , 94.5 , 141.75 , 157.5 , 189 Δx_A (km) : 48 , 46.5 , 47.25 , 15.75 , 31.5 48 + 46.5 = 94.5 km

Δt = 36 = 0.5 h = 1800 s

v = ΔxΔt

v_B = 1893 = 63 km/h

v_B = 63 · 10003600 = 630003600 = 17.5 m/s

v_B = 31.50.5 = 63 km/h

t_B = 6 · Δt ; t_A = 5 · Δt

t_A = 5 · Δt < 6 · Δt = t_B

v_A = MNt_A ; v_B = MNt_B ; t_A < t_B ⟹ v_A > v_B

t_A = 5 · 0.5 = 2.5 h ⟹ v_A = 1892.5 = 75.6 km/h = 21 m/s v_Av_B = 75.663 = 65 = 1.2

t_MP = 2 · Δt = 2 · 0.5 = 1 h

MP = 1892 = 94.5 km

v_A,MP = MPt_MP = 94.51 = 94.5 km/h v_A,MP = 94.5 · 10003600 = 945003600 = 26.25 m/s

480.5 = 96 km/h ; 46.50.5 = 93 km/h 96 + 932 = 94.5 km/h

a = v₂ − v₁Δt

v₁ = 94.5 km/h = 26.25 m/s

v₂ = 63 km/h = 17.5 m/s

Δt = 0.5 h = 0.5 · 3600 = 1800 s a = 17.5 − 26.251800 = −8.751800 = −4.861 · 10⁻³ m/s²

a = 63 − 94.50.5 = −63 km/h² −63 · 10003600² = −6300012960000 = −4.861 · 10⁻³ m/s²

15.750.5 = 31.5 km/h = 8.75 m/s ; 26.25 + 17.52 = 21.875 m/s

2.0 h ≤ t ≤ 2.5 h : v_A = 63 = v_B km/h

v_A = 94.5 > 63 = v_B km/h v_A ≤ 31.5 < 63 = v_B km/h v_A − v_B : + ⟹ − ⟹ ∃ t : v_A = v_B

#  |  t (h)   |  Δx_A (km) |  v_A (km/h) |  Δx_B (km) |  v_B (km/h)
1  | 0.0 – 0.5 |   48.00    |   96.0     |   31.50    |   63.0
2  | 0.5 – 1.0 |   46.50    |   93.0     |   31.50    |   63.0
3  | 1.0 – 1.5 |   47.25    |   94.5     |   31.50    |   63.0
4  | 1.5 – 2.0 |   15.75    |   31.5     |   31.50    |   63.0
5  | 2.0 – 2.5 |   31.50    |   63.0     |   31.50    |   63.0
6  | 2.5 – 3.0 |    0.00    |    0.0     |   31.50    |   63.0
Σ  |  A : 189 km , 2.5 h , 1892.5 = 75.6 km/h  |  B : 189 km , 3.0 h , 1893 = 63.0 km/h

h (m) | 0.5 | 1.0 | 1.5
t (s) | 1.01 | 1.40 | 1.72
a (m/s²) | 0.98 | 1.02 | 1.01
g = 10 m/s²

(m_A − m_B) (kg) | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06
a (m/s²) | 0.26 | 0.34 | 0.54 | 0.87 | 1.07 | 1.24
a = 20 · (m_A − m_B) ; (0 , 0) ; (0.06 , 1.2)

1.0 m

0.050 kg

0.500 kg

0.00 s

1.000 m

0.000 m/s

1.000 m/s²

1.414 s

0.275 kg

0.225 kg

0.050 kg

0.500 kg

2.475 N

4.950 N

m_A·g − T = m_A·a ; T − m_B·g = m_B·a (m_A − m_B)·g = (m_A + m_B)·a a = (m_A − m_B)·gm_A + m_B

a = ΣFm

h = v₀·t + 12·a·t² ; v₀ = 0 ⟹ h = 12·a·t² a = 2·ht²

a₁ = 2 · 0.51.01² = 11.0201 = 0.98030 ≈ 0.98 m/s² a₂ = 2 · 1.01.40² = 21.96 = 1.02041 ≈ 1.02 m/s² a₃ = 2 · 1.51.72² = 32.9584 = 1.01406 ≈ 1.01 m/s²

v_end2 = ht ⟹ v_end = 2·ht a = v_endt = 2·ht²

a₁ = 0.98 ; a₂ = 1.02 ; a₃ = 1.01 m/s² ā = 0.98 + 1.02 + 1.013 = 1.00333 m/s² Δa_max = 0.02333 m/s² ; 0.023331.00333 = 0.0233 = 2.33 %

t = √2·ha ; a = 1.00 m/s² h = 0.5 ⟹ t = 1.0000 s ; Δt = 0.010 s h = 1.0 ⟹ t = 1.4142 s ; Δt = 0.014 s h = 1.5 ⟹ t = 1.7321 s ; Δt = 0.012 s

t²h = 2a 1.02010.5 = 2.040 ; 1.961.0 = 1.960 ; 2.95841.5 = 1.972 s²/m

m_A·g > T

T > m_B·g

m_A·g > T > m_B·g

T = m_A·(g − a) = 0.275 · (10 − 1) = 2.475 N T = m_B·(g + a) = 0.225 · (10 + 1) = 2.475 N m_A·g = 2.75 N ; m_B·g = 2.25 N ; 2.75 > 2.475 > 2.25

ΣF_A = m_A·g − T = m_A·a (1)

ΣF_B = T − m_B·g = m_B·a (2)

m_A·g − T + T − m_B·g = m_A·a + m_B·a (m_A − m_B)·g = (m_A + m_B)·a a = gm_A + m_B · (m_A − m_B)

slope = gm_A + m_B (m/s²)/kg

Δ(m_A − m_B) = 2·δ = 0.01 kg ⟹ δ = 0.005 kg

slope = ΔaΔ(m_A − m_B) = 1.2 − 00.06 − 0 = 20 (m/s²)/kg 1.0 − 0.20.05 − 0.01 = 0.80.04 = 20 (m/s²)/kg

slope = gm_A + m_B ⟹ m_A + m_B = gslope

m_A + m_B = 1020 = 0.5 kg = 500 g

a = 10 · 0.030.5 = 0.6 m/s²

slope = Σ x_i·y_iΣ x_i² = 0.18830.0091 = 20.69 (m/s²)/kg m_A + m_B = 1020.69 = 0.483 kg

m_A − m_B = a·(m_A + m_B)g = 1.00 · 0.510 = 0.05 kg m_A = 0.5 + 0.052 = 0.275 kg ; m_B = 0.5 − 0.052 = 0.225 kg

(m_A − m_B)·g·h = 12·(m_A + m_B)·v² 0.05 · 10 · 0.5 = 0.25 J ; 0.25·v² = 0.25 ⟹ v = 1 m/s v = √2·a·h = √2 · 1 · 0.5 = 1 m/s

| 1    | 2     | 3     | 4     | 5
f (Hz) | 0.45 | 0.5 | 0.6 | 0.7 | 1
α (°)   | 32    | 45   | 63   | 70   | 80
1f² (s²) | ?     | ?     | ?     | ?     | ?
cos α | ?     | ?     | ?     | ?     | ?
g = 10 m/s² ; ω = 2π·f ; r = L·sin α

0.60 Hz · 1.667 s

1.50 m

0.50 kg

62.03 °

0.469

2.78 s²

3.770 rad/s

1.325 m

18.83 m/s²

10.66 N

2.132

0.169 s⁻²

0.411 Hz

63 · 62.03

r = L·sin α ; ω = 2π·f

T·cos α − m·g = 0 ⟹ T·cos α = m·g (1)

T·sin α = m·a_c = m·ω²·r = m·(2π·f)²·L·sin α T = 4π²·f²·L·m (2)

4π²·f²·L·m·cos α = m·g ⟹ 4π²·f²·L·cos α = g cos α = g4π²·L · 1f²

k = g4π²·L

T·sin αT·cos α = m·ω²·L·sin αm·g ⟹ 1cos α = 4π²·f²·Lg cos α = g4π²·L·f²

[k] = m/s²m = s⁻² ; s⁻² · s² = 1

10.45² = 10.2025 = 4.94 s² 10.5² = 10.25 = 4.00 s² 10.6² = 10.36 = 2.78 s² 10.7² = 10.49 = 2.04 s² 11² = 1.00 s²

cos 32° = 0.848 ; cos 45° = 0.707 ; cos 63° = 0.454 cos 70° = 0.342 ; cos 80° = 0.174

f (Hz) | 0.45 | 0.5 | 0.6 | 0.7 | 1 α (°) | 32 | 45 | 63 | 70 | 80 1f² (s²) | 4.94 | 4.00 | 2.78 | 2.04 | 1.00 cos α | 0.848 | 0.707 | 0.454 | 0.342 | 0.174

(1.00 , 0.174) ; (2.04 , 0.342) ; (2.78 , 0.454) ; (4.00 , 0.707) ; (4.94 , 0.848)

(1.00 s² , 0.170) ; (5.00 s² , 0.845) k = 0.845 − 0.1705.00 − 1.00 = 0.6754.00 = 0.169 s⁻²

k = g4π²·L ⟹ L = g4π²·k L = 1039.478 · 0.169 = 106.672 = 1.4988 m

L = g4π²·f²·cos α 1.475 ; 1.433 ; 1.550 ; 1.511 ; 1.459 m L̄ = 7.4285 = 1.486 ≈ 1.49 m

cos α = 0.1689 · 1f² 33.5° | 47.5° | 62.0° | 69.8° | 80.3° 32° | 45° | 63° | 70° | 80°

cos α ≤ 1

1 = k · 1f² ⟹ 1f² = 10.169 = 5.92 s² f_min = 1√5.92 = 12.433 = 0.411 Hz

1 = g4π²·L·f_min² ⟹ f_min = 12π · √gL f_min = 12π · √101.5 = 2.5826.283 = 0.411 Hz

T_max = 2π · √Lg = 2π · 0.3873 = 2.433 s ⟹ f = 12.433 = 0.411 Hz

θ = 30° ; g = 10 m/s² ; m = 0.2 kg
h_B₁ = 0.45 m ; h_B₂ = 0.30 m
| v_A | = | v_A₁ | ; | v_A | ≠ | v_A₂ | ; sin 30° = 0.5

0.00 s

0.00 s

0.00 m

3.00 m/s

−7.50 m/s²

0.00 m

0.90 J

0.00 J

0.90 J

0.00 J

7.50 m/s²

2.50 m/s²

| v_A₁ | = | v_A | ⟹ E_k,A₁ = E_k,A

E_k,A = 12·m·v_A² ; E_p,A = 0 ; E_k,B₁ = 0 ; E_p,B₁ = m·g·h_B₁ 12·m·v_A² = m·g·h_B₁ v_A = √2·g·h_B₁

v_A = √2 · 10 · 0.45 = √9 = 3 m/s

a = g·sin 30° = 10 · 0.5 = 5 m/s² L₁ = h_B₁sin 30° = 0.450.5 = 0.9 m 0 = v_A² − 2·a·L₁ ⟹ v_A = √2 · 5 · 0.9 = √9 = 3 m/s

E_k,A = 12·m·v_A² = 12 · 0.2 · 3² = 0.9 J E_p,A = m·g·0 = 0 J

v_B₂ = 0 ⟹ E_k,B₂ = 0 J E_p,B₂ = m·g·h_B₂ = 0.2 · 10 · 0.3 = 0.6 J

W_f = (E_k,B₂ + E_p,B₂) − (E_k,A + E_p,A) W_f = (0 + 0.6) − (0.9 + 0) = 0.6 − 0.9 = −0.3 J

sin 30° = h_B₂L ⟹ L = h_B₂sin 30° = 0.30.5 = 0.6 m

W_f = −f·L ⟹ f = | W_f |L = 0.30.6 = 0.5 N

a₁ = g·sin 30° + fm = 5 + 0.50.2 = 5 + 2.5 = 7.5 m/s² 0 = v_A² − 2·a₁·L ⟹ v_A = √2 · 7.5 · 0.6 = √9 = 3 m/s

N = m·g·cos 30° = 0.2 · 10 · √32 = √3 ≈ 1.73 N μ = fN = 0.5√3 = √36 ≈ 0.289

ms · s = m

S = v_A·t₁2 = L = 0.6 m t₁ = 2·Lv_A = 2 · 0.63 = 1.23 = 0.4 s

a₁ = g·sin 30° + fm = 5 + 2.5 = 7.5 m/s² t₁ = v_Aa₁ = 37.5 = 0.4 s

a₂ = g·sin 30° − fm = 5 − 2.5 = 2.5 m/s² ; a₁a₂ = 3 | v_A₂ | = √2·a₂·L = √2 · 2.5 · 0.6 = √3 ≈ 1.73 m/s t₂ = t₁ + | v_A₂ |a₂ = 0.4 + √32.5 ≈ 0.4 + 0.693 = 1.093 s

E_k,A' = m·g·h_B₂ − f·L = 0.6 − 0.5 · 0.6 = 0.3 J | v_A₂ | = √2·E_k,A'm = √2 · 0.30.2 = √3 ≈ 1.73 m/s

v(t) = v_A − a₁·t = 3 − 7.5·t E_k(t) = 12·m·v²(t) = 12 · 0.2 · (3 − 7.5·t)² = 0.1·(3 − 7.5·t)² E_k(t) = 0.9 − 4.5·t + 5.625·t²

dE_kdt = −(m·g·sin 30° + f)·v(t)

t (s) | 0 | 0.1 | 0.2 | 0.3 | 0.4 v (m/s) | 3.00 | 2.25 | 1.50 | 0.75 | 0 E_k (J) | 0.900 | 0.506 | 0.225 | 0.056 | 0

m_A = 0.1 kg ; m_B = 0.2 kg ; m_C = 0.25 kg
v_A = 0.7 m/s ; g = 10 m/s²
t (s) | 0 | 0.04 | 0.08
F (N) | 0 | F_max | 0

0.28 m/s

×0.25

0.00 s

0.00 m/s

0.000 kg·m/s

0.000 N·s

3.50 N

0.000 kg·m/s

0.000 m/s

0.00000 J

m_A = 0.1 kg ; m_B = 0.2 kg ; m_C = 0.25 kg

p_before = m_A · 0 + m_B · 0 = 0

ΣF_ext,x = 0

p_after = p_before = 0

m_A·v_A + m_B·v_B = 0 ⟹ v_B = −m_A·v_Am_B

v_B = −0.1 · 0.70.2 = −0.070.2 = −0.35 m/s

E_k,A = 12 · 0.1 · 0.7² = 0.0245 J E_k,B = 12 · 0.2 · 0.35² = 0.01225 J E_spring = 0.0245 + 0.01225 = 0.03675 J E_k,AE_k,B = m_Bm_A = 0.20.1 = 2 ; 0.02450.01225 = 2

p_A = 0.1 · 0.7 = 0.07 kg·m/s ; p_B = 0.2 · 0.35 = 0.07 kg·m/s

12·m_A·v_before² = 12·m_A·v_after² ⟹ |v_after| = |v_before| = 0.7 m/s

v_A,after = −0.7 m/s

v_A,after = m_A − Mm_A + M · v_A ⟹ v_A,after = −v_A = −0.7 m/s E_k = 12 · 0.1 · 0.7² = 0.0245 J

J = Δp = m_A · (v_after − v_before)

J = 0.1 · (−0.7 − 0.7) = 0.1 · (−1.4) = −0.14 N·s

p_before = +0.07 kg·m/s ; p_after = −0.07 kg·m/s Δp = −0.07 − 0.07 = −0.14 kg·m/s

J = ∫ F dt = S

J = Δp_A

N · s = kg · ms² · s = kg · ms

Δt = 0.08 s ; t_peak = 0.04 s

J = S = 12 · Δt · F_max ⟹ F_max = 2 · JΔt F_max = 2 · 0.140.08 = 0.280.08 = 3.5 N

m_up = 3.5 − 00.04 − 0 = 87.5 N/s ; F(t) = 87.5·t m_down = 0 − 3.50.08 − 0.04 = −87.5 N/s ; F(t) = 7 − 87.5·t

F_avg = JΔt = 0.140.08 = 1.75 N ; F_max = 2 · 1.75 = 3.5 N

S = 12·0.04·3.5 + 12·0.04·3.5 = 0.07 + 0.07 = 0.14 N·s

E_k,after = 12·(m_B + m_C)·u² = 0 ⟹ u = 0 ⟹ p_after = 0

m_B·|v_B| = m_C·|v_C| ⟹ |v_C| = m_B·|v_B|m_C

|v_C| = 0.2 · 0.350.25 = 0.070.25 = 0.28 m/s

E_k,before = 12·0.2·0.35² + 12·0.25·0.28² = 0.01225 + 0.0098 = 0.02205 J ΔE_k = 0.02205 − 0 = 0.02205 J

p_B = 0.2 · 0.35 = 0.07 kg·m/s ; p_C = 0.25 · 0.28 = 0.07 kg·m/s

p_total = 0.2 · 0.35 − 0.25 · 0.2 = 0.07 − 0.05 = 0.02 kg·m/s u = 0.020.45 = 0.044 m/s E_k,before = 0.01225 + 0.005 = 0.01725 J ; E_k,after = 0.00044 J

T_P = 0.3189 d ; r_P = 9.377·10⁶ m ; T_D = 1.262 d ; T_m = 27.3 d
R = 3.4·10⁶ m ; M_s = 53 kg ; h = 20 m ; m = 1.3 kg ; u = 12.5 m/s
G = 6.67·10⁻¹¹ N·m²/kg² ; 1 d = 86400 s

2.3460 ·10⁷ m

0.000 d

0.00

3.9574

2.3460 ·10⁷ m

1.262 d

1.0861 ·10¹² m³/s²

6.4282 ·10²³ kg

2138 m/s

1352 m/s

r_P³T_P² = r_D³T_D² = G·M4π²

r_D³ = r_P³ · T_D²T_P² r_D = r_P · (T_DT_P)²³

T_DT_P = 1.2620.3189 = 3.9574 (3.9574)²³ = 2.5019

r_D = 9.377·10⁶ · 2.5019 = 2.346·10⁷ m

T_P = 0.3189 · 86400 = 2.7553·10⁴ s ; T_D = 1.262 · 86400 = 1.0904·10⁵ s r_P³T_P² = 8.245·10²⁰7.5917·10⁸ = 1.0861·10¹² m³/s² r_D³T_D² = 1.2912·10²²1.1889·10¹⁰ = 1.0861·10¹² m³/s²

r_PR = 9.377·10⁶3.4·10⁶ = 2.76 ; r_DR = 2.346·10⁷3.4·10⁶ = 6.90

r³T² = G·M4π²

G·M4π² = r_P³T_P² = 1.0861·10¹² m³/s²

r_m = (G·M_E·T_m²4π²)¹³

G·M·m_Pr_P² = m_P·ω²·r_P = m_P·4π²T_P²·r_P

M = 4π²·r_P³G·T_P²

T_P = 0.3189 · 86400 = 2.7553·10⁴ s r_P³ = (9.377·10⁶)³ = 8.245·10²⁰ m³

M = 4π² · 8.245·10²⁰6.67·10⁻¹¹ · (2.7553·10⁴)² = 3.2550·10²²5.0636·10⁻² M = 6.43·10²³ kg

M = 4π² · 1.2912·10²²6.67·10⁻¹¹ · (1.0904·10⁵)² = 5.0979·10²³0.79300 = 6.43·10²³ kg

ρ = M43·π·R³ = 6.43·10²³1.646·10²⁰ = 3.9·10³ kg/m³

m = 1.3 kg ; M_s = 53 kg ; m + M_s = 54.3 kg

m·u + M_s·0 = (m + M_s)·v v = m·um + M_s

v = 1.3 · 12.51.3 + 53 = 16.2554.3 = 0.299 m/s

E_k = 12 · 1.3 · 12.5² = 101.56 J E_k = 12 · 54.3 · 0.2993² = 2.43 J ΔE_k = 2.43 − 101.56 = −99.1 J

G·M·mR² = m·g ⟹ g = G·MR²

hR = 203.4·10⁶ = 5.9·10⁻⁶

g = 6.67·10⁻¹¹ · 6.43·10²³(3.4·10⁶)² = 4.288·10¹³1.156·10¹³ = 3.71 m/s²

g_Mg_E = 3.7110 = 0.371

G·Mr_P² = 0.4876 m/s² ; 4π²·r_PT_P² = 0.4876 m/s²

v_0x = 0.2993 m/s ; v_0y = 0 ; h = 20 m ; g = 3.709 m/s²

h = v_0y·t + 12·g·t² = 12·g·t² t = √2·hg

t = √2 · 203.709 = √10.784 = 3.28 s

12·v_y² = g·h ⟹ v_y = √2·g·h = √2 · 3.709 · 20 = 12.18 m/s t = v_yg = 12.183.709 = 3.28 s

t_Mt_E = √g_Eg_M = √103.709 = 1.64 ; 2.0 · 1.64 = 3.28 s

x = v_x·t = 0.2993 · 3.284 = 0.98 m v = √v_x² + v_y² = √0.2993² + 12.18² = 12.18 m/s

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