פתרון בגרות מכניקה — שאלון 36361

t (s) | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5
v₁ (m/s) | 1.0 | 0.5 | 0 | −0.5 | −1.0 | −1.5
v₂ (m/s) | −0.5 | −1.0 | −1.5 | −2.0 | −2.5 | −3.0
a = ΔvΔt = −5 m/s² ; a = g·sinα ; sinα = 0.5 ; α = 30°

0.000 s

1×

0.000 s

+1.000 m/s

0.125 m

0.063 m

0.000 m

0.000 s

−0.500 m/s

1.271 m

0.635 m

0.000 m

−5.000 m/s²

30°

1.000 m/s

0.500 m/s

(0 , 1.0) (0.1 , 0.5) (0.2 , 0) (0.3 , −0.5) (0.4 , −1.0) (0.5 , −1.5) a = ΔvΔt = −1.5 − 1.00.5 − 0 = −2.50.5 = −5 m/s²

(0 , −0.5) (0.1 , −1.0) (0.2 , −1.5) (0.3 , −2.0) (0.4 , −2.5) (0.5 , −3.0) a = −3.0 − (−0.5)0.5 − 0 = −2.50.5 = −5 m/s²

v₁ = 1.0 m/s ; v₂ = 0.5 m/s

a = g·sinα ; sinα = 510 = 0.5 ; α = 30°

ΣF = m·g·sinα ; a = g·sinα = 5 m/s²

a = −5 m/s² ; a = −g·sinα

v₁(0) = +1.0 m/s ; v₂(0) = −0.5 m/s

v(t) = v₁ + a·t = 0 t_top = v₁|a| = 1.05 = 0.2 s

Δt = 0.5 − 0.2 = 0.3 s

d = 12·|a|·Δt² = 12 · 5 · (0.3)² = 12 · 5 · 0.09 = 0.225 m

v_K = −1.5 m/s ; v_K² = 0 + 2·|a|·d d = v_K²2·|a| = (1.5)²2 · 5 = 2.2510 = 0.225 m

12·m·v_K² = m·g·d·sinα d = v_K²2·g·sinα = 2.252 · 10 · 0.5 = 0.225 m

S = 12 · 0.3 · 1.5 = 0.225 m

0 = v₁² − 2·|a|·d_up d_up = v₁²2·|a| = (1.0)²2 · 5 = 110 = 0.1 m d_up = v₁·t_top − 12·|a|·t_top² = 0.2 − 0.1 = 0.1 m

AK = 0.225 − 0.100 = 0.125 m

Δx = v₁·t + 12·a·t² = (1.0)(0.5) + 12(−5)(0.25) = 0.5 − 0.625 = −0.125 m

S₊ = 12 · 0.2 · 1.0 = +0.100 ; S₋ = −12 · 0.3 · 1.5 = −0.225 Δx = 0.100 − 0.225 = −0.125 m

v₀ = −0.5 m/s ; a = −5 m/s² ; t = 0.62 s

Δx = v₀·t + 12·a·t² = (−0.5)(0.62) + 12(−5)(0.3844) Δx = −0.31 − 0.961 = −1.271 m ; BK = 1.271 m

AB = BK − AK = 1.271 − 0.125 = 1.146 m ≈ 1.15 m

v_K = v₀ + a·t = −0.5 − 5(0.62) = −3.6 m/s BK = v_K² − v₀²2·|a| = 12.96 − 0.2510 = 12.7110 = 1.271 m

12v_K² − 12v₀² = g·sinα·BK 6.48 − 0.125 = 5 · BK ; BK = 6.3555 = 1.271 m

2.5t² + 0.5t − 1.146 = 0 ; t = 0.5844 s v = 0.5 + 5(0.5844) = 3.422 m/s ; Δt = 0.62 − 0.5844 = 0.0356 s d = 3.422(0.0356) + 2.5(0.0356)² = 0.1218 + 0.0032 = 0.125 m

|a_up| = g·sinα + μ·g·cosα ; |a_up| גדול מ 5 m/s²

|a_down| = g·sinα − μ·g·cosα ; |a_down| קטן מ 5 m/s²

μ = 5 − 1.87510 · cos30° = 3.1258.66 = 0.361 a_up = −(5 + 8.66 · 0.361) = −8.125 m/s² t_stop = 1.08.125 = 0.123 s

μ = 0.361 קטן מ tan30° = 0.577

t (s) | 0 – 0.3 | 0.3 – 0.8 | 0.8 – 1.2
T (N) | 10 | 12 | 10
m_A = 3 kg ; g = 10 m/s² ; T_A = T_B = T

0.00 s

0.00 s

10.00 N

40.00 N

0.00 m/s²

0.00 m/s

0.000 m

10.00 N

0.00 N

m_A·g = 3 · 10 = 30 N

m_B·g = 1 · 10 = 10 N

F₁ − T − m_A·g = m_A·a T − m_B·g = m_B·a

F₁ − m_A·g − m_B·g = (m_A + m_B)·a a = F₁ − (m_A + m_B)·gm_A + m_B = F₁m_A + m_B − g

T = m_B·(g + a) = m_B·F₁m_A + m_B

0 < t < 0.3 s ; a = 0 ; T = 10 N

T − m_B·g = m_B · 0 ⟹ T = m_B·g m_B = Tg = 1010 = 1 kg

F₁ = T + m_A·g = 10 + 30 = 40 N F₁ = (m_A + m_B)·g ; 40 = (3 + m_B) · 10 ⟹ m_B = 1 kg

TF₁ = m_Bm_A + m_B = 1040 = 0.25 ⟹ m_B = 0.25 · 4 = 1 kg

a = T − m_B·gm_B ; F₁ = (m_A + m_B)·(g + a) m_A + m_B = 3 + 1 = 4 kg

T = 10 N ; a = 10 − 1 · 101 = 0 m/s² F₁ = 4 · (10 + 0) = 40 N

T = 12 N ; a = 12 − 1 · 101 = 2 m/s² F₁ = 4 · (10 + 2) = 4 · 12 = 48 N

T = 10 N ; a = 10 − 1 · 101 = 0 m/s² F₁ = 4 · (10 + 0) = 40 N

F₁ = T + m_A·g + m_A·a 10 + 30 + 3 · 0 = 40 N ; 12 + 30 + 3 · 2 = 48 N ; 10 + 30 + 0 = 40 N

F₁ = 4 · T ; 4 · 10 = 40 ; 4 · 12 = 48 ; 4 · 10 = 40 N

ΣF = 12 − 10 = 2 N ; a = 21 = 2 m/s²

Δt = 0.8 − 0.3 = 0.5 s v = v₀ + a·Δt = 0 + 2 · 0.5 = 1 m/s Δy = 12 · a · Δt² = 12 · 2 · 0.5² = 0.25 m

ΣF = 10 − 10 = 0 ; a = 0 m/s² Δy = v·Δt = 1 · (1.2 − 0.8) = 0.4 m Δy_total = 0 + 0.25 + 0.4 = 0.65 m

ΣF = (m_A + m_B)·a = 4 · 2 = 8 N ; W = 8 · 0.25 = 2 J ΔE_k = 12 · 4 · 1² − 0 = 2 J W_F₁ = 40 · 0.4 = 16 J ; W_g = −40 · 0.4 = −16 J ; W = 0

F₁ − (m_A + m_B)·g = (m_A + m_B)·a₁ F₁ = (m_A + m_B)·(g + a₁)

F₁ − F₂ − m_A·g = m_A·a₁ ⟹ F₂ = F₁ − m_A·(g + a₁)

F₂ = (m_A + m_B)·(g + a₁) − m_A·(g + a₁) = m_B·(g + a₁) a₁ > 0 ⟹ g + a₁ > g ⟹ F₂ = m_B·(g + a₁) > m_B·g

F₂ = T ; T − m_B·g = m_B·a₁ ⟹ T = m_B·(g + a₁)

F₁ = 4 · (10 + 2) = 48 N ; T = 1 · (10 + 2) = 12 N F₂ = 48 − 3 · 12 = 48 − 36 = 12 N ; m_B·g = 10 N F₂m_B·g = 1210 = 1.2 > 1

h = 6 m ; v₀ = 3 m/s ; Δt = 0.5 s ; g = 10 m/s²
x = v₀·t ; y = h − 12·g·t² ; v_y = g·t
v = √v_x² + v_y² ; v² = v₀² + 2·g·h

6.00 m

3.00 m/s

0.50 s

3

×0.25

0.000 s

6.00 m

3.00 m/s

0.00 m/s

3.00 m/s

0.00 °

1.0954 s

10.9545 m/s

11.3578 m/s

74.68 °

3.2863 m

1.50 m

v_0x = v₀ = 3 m/s ; v_0y = 0 ; a_y = g = 10 m/s²

h = v_0y·t + 12·g·t² = 12·g·t² t = √2·hg = √2 · 610 = √1.2 = 1.0954 s

v_y² = v_0y² + 2·g·h = 0 + 2 · 10 · 6 = 120 v_y = √120 = 10.9545 m/s h = v_y2·t ; t = 2·hv_y = 1210.9545 = 1.0954 s

v_x = v₀ = 3 m/s

v_y = g·t = 10 · 1.0954 = 10.9545 m/s

v = √v_x² + v_y² = √3² + 10.9545² = √9 + 120 = √129 v = 11.358 m/s ≈ 11.4 m/s

tan θ = v_yv_x = 10.95453 = 3.6515 θ = 74.68° ≈ 74.7°

12·m·v₀² + m·g·h = 12·m·v² v² = v₀² + 2·g·h = 9 + 120 = 129 ; v = √129 = 11.358 m/s

t = √2·hg = 1.0954 s

x = v_x·t = 3 · 1.0954 = 3.2863 m ≈ 3.29 m

Δx = (x₂ + v·t) − (x₁ + v·t) = x₂ − x₁ = v₀·Δt Δx = 3 · 0.5 = 1.5 m

x₁ = 0 + 3 · 1.0954 = 3.2863 m x₂ = 1.5 + 3 · 1.0954 = 4.7863 m x₂ − x₁ = 4.7863 − 3.2863 = 1.5 m

d = 12·g·t² d₃ = 0 ; d₂ = 12 · 10 · 0.5² = 1.25 m ; d₁ = 12 · 10 · 1.0² = 5.00 m 1.25 − 0 = 1.25 m ; 5.00 − 1.25 = 3.75 m ; 1.25 : 3.75 = 1 : 3

y₃ = 6 m ; y₂ = 6 − 1.25 = 4.75 m ; y₁ = 6 − 5.00 = 1 m

12·m·v₀² + m·g·h = 12·m·v² v² = v₀² + 2·g·h

v = √3² + 2 · 10 · 6 = √9 + 120 = √129 = 11.358 m/s ≈ 11.4 m/s

v_x = v₀·cos α ; v_y² = (v₀·sin α)² + 2·g·h v² = v₀²·cos²α + v₀²·sin²α + 2·g·h = v₀²·(cos²α + sin²α) + 2·g·h = v₀² + 2·g·h v² = 9 + 120 = 129 ; v = 11.358 m/s

h (m) | 0.6 | 0.8 | 1.0 | 1.2 | 1.4
N_C (N) | 0.20 | 0.55 | 0.75 | 0.95 | 1.20
N_C,min = 0.6 N ; g = 10 m/s²

1.000 m

1.000 m

0.00 m/s

4.47 m/s

3.71 m/s

0.73 N

0.0940 J

0.313 m

mg ; N ; N + mg = ΣF_radial

N_C = N

m·g·h = m·g·(2R) + ½·m·v_C² v_C² = 2·g·(h − 2R)

N + m·g = m·v_C²R N = m·v_C²R − m·g = m·2·g·(h − 2R)R − m·g

N = 2·m·gR·h − 4·m·g − m·g N_C = 2·m·gR·h − 5·m·g

a_slope = 2·m·gR ; b = −5·m·g

m·g = m·v_C²R ; v_C² = g·R 2·g·(h − 2R) = g·R ; h = 2.5·R N_C = 2·m·gR·(2.5·R) − 5·m·g = 5·m·g − 5·m·g = 0

(0.6 , 0.20) (0.8 , 0.55) (1.0 , 0.75) (1.2 , 0.95) (1.4 , 1.20)

a_slope = ΔN_CΔh = 1.21 − 0.251.4 − 0.6 = 0.960.8 = 1.20 N/m

b = 0.25 − 1.20 · 0.6 = 0.25 − 0.72 = −0.47 N N_C = 1.20 · h − 0.47

h̄ = 5.05 = 1.000 m ; N̄ = 3.655 = 0.730 N a_slope = Σ(h − h̄)(N − N̄)Σ(h − h̄)² = 0.4800.400 = 1.2000 N/m b = N̄ − a_slope·h̄ = 0.73 − 1.2 · 1.0 = −0.4700 N

N_C = 2·m·gR·h − 5·m·g N_C = 1.20 · h − 0.47

5·m·g = 0.47 N m·g = 0.475 = 0.094 N m = 0.09410 = 0.0094 kg = 9.4 g

2·m·gR = 1.20 N/m R = 2 · 0.0941.20 = 0.1881.20 = 0.1567 m ≈ 0.157 m 2R = 0.3133 m

5·m·g2·m·gR = 2.5·R = 0.471.20 = 0.3917 m R = 0.39172.5 = 0.1567 m

v_C² = 2·g·(h − 2R) = 20 · (1.0 − 0.3133) = 13.733 m²/s² N_C = m·v_C²R − m·g = 0.0094 · 13.7330.1567 − 0.094 = 0.824 − 0.094 = 0.73 N

0.60 = 1.20 · h_min − 0.47 1.20 · h_min = 1.07 h_min = 1.071.20 = 0.8917 m ≈ 0.89 m

h_min = R·(N_C,min + 5·m·g)2·m·g = 0.1567 · (0.60 + 0.47)0.188 = 0.8917 m

v_C² = 2·g·(h_min − 2R) = 20 · (0.8917 − 0.3133) = 11.567 m²/s² v_C = √11.567 = 3.40 m/s N_C = m·v_C²R − m·g = 0.0094 · 11.5670.1567 − 0.094 = 0.694 − 0.094 = 0.600 N

0 = 1.20 · h₁ − 0.47 ; h₁ = 0.471.20 = 0.3917 m ≈ 0.39 m

m·g = m·v_C²R ; v_C² = g·R

v_C = √g·R = √10 · 0.1567 = √1.5667 = 1.25 m/s

v_C² = 2·g·(h₁ − 2R) = 20 · (0.3917 − 0.3133) = 20 · 0.07833 = 1.5667 m²/s² v_C = √1.5667 = 1.25 m/s

g = 10 m/s² ; m_A = 0.4 kg ; m_B = 1.2 kg
h = 0.6 m ; x_A = 1.5 m ; v_B = 4 m/s
m·g·h = 12·m·v² ; W_f = −μ·m·g·x ; J = Δp

0.600 m

0.400

30°

0.400 kg

3.464 m/s

4.000 m/s²

1.500 m

0.866 s

0.000 m/s

0.000 m

0.000 s

0.000 J

2.400 J

0.000 J

W_N = N · d · cos90° = 0 ΔE_mech = W_non-cons = 0 ; E_k + E_p = const

W_f = −f_k · x = −μ·m·g·x

ΔpΔt = ΣF ΣF = m·g·sinθ ≠ 0 ΣF = −μ·m·g ≠ 0

m_A·g·h = 12·m_A·v² v = √2·g·h = √2 · 10 · 0.6 = √12 = 3.46 m/s

12·m_A·v² = μ·m_A·g·x_A μ = v²2 · g · x_A = 122 · 10 · 1.5 = 1230 = 0.4

m_A·g·h = μ·m_A·g·x_A μ = hx_A = 0.61.5 = 0.4

a = −μ·g = −0.4 · 10 = −4 m/s² v_f² = v² + 2·a·x_A = 12 + 2·(−4)·1.5 = 12 − 12 = 0 t = vμ·g = 3.46414 = 0.866 s x = v · t2 = 3.4641 · 0.8662 = 1.5 m

m_B·g·h = μ·m_B·g·x_B x_B = hμ = 0.60.4 = 1.5 m

E_p = m·g·h ; f_k = μ·m·g ; x = hμ

v = √2·g·h = √12 m/s ; a = μ·g = 4 m/s² x = v²2 · a = 128 = 1.5 m

m_B·v_B = m_A·v′_A + m_B·v′_B ; 1.2 · 4 = 0.4·v′_A + 1.2·v′_B 12·m_B·v_B² = 12·m_A·v′_A² + 12·m_B·v′_B²

v′_B = m_B − m_Am_A + m_B·v_B = 1.2 − 0.41.6·4 = 2 m/s v′_A = 2·m_Bm_A + m_B·v_B = 2.41.6·4 = 6 m/s

J_B = Δp_B = m_B·v′_B − m_B·v_B = 1.2·2 − 1.2·4 = 2.4 − 4.8 = −2.4 N·s

J_A = m_A·v′_A − 0 = 0.4 · 6 = 2.4 N·s J_A + J_B = 2.4 − 2.4 = 0

E_k = 12·1.2·4² = 9.6 J ; 12·1.2·2² + 12·0.4·6² = 2.4 + 7.2 = 9.6 J v_B − v_A = 4 − 0 = 4 m/s ; v′_A − v′_B = 6 − 2 = 4 m/s

x_A = 6²2 · 4 = 4.5 m ; x_B = 2²2 · 4 = 0.5 m H_B = v_B²2 · g = 1620 = 0.8 m

J = Δp = m_A·v − m_A·0 = m_A·v

m_A·g·H = 12·m_A·v² ; v = √2·g·H J₁ = m_A·√2·g·H = J₂

t = vg · sinθ ; J_grav = m·g·t

v = √2 · 10 · 0.6 = 3.4641 m/s a₁ = 10·sin30° = 5 m/s² ; t₁ = 3.46415 = 0.6928 s ; J_grav = 2.7713 N·s a₂ = 10·sin60° = 8.6603 m/s² ; t₂ = 3.46418.6603 = 0.4 s ; J_grav = 1.6 N·s J₁ = 0.4 · 3.4641 = 1.3856 N·s ; J₂ = 0.4 · 3.4641 = 1.3856 N·s

r (10⁹ m) | b = 1.73 | c = 2.36 | d = ?
T (86400 s) | b = 1.51 | c = ? | d = 4.05
G = 6.67·10⁻¹¹ N·m²/kg² ; M_☉ = 2·10³⁰ kg

×4.37·10⁴

4.50·10⁹ m

2.406 ימים

3.340 ·10⁹ m

8.33 ·10⁴ m/s

7.13 ·10⁴ m/s

6.00 ·10⁴ m/s

4.013 m/s²

3.287 ·10⁻¹⁸ s²/m³

1.800 ·10²⁹ kg

11.1

0.00 ימים

4.50 ·10⁹ m

6.335 ימים

5.17 ·10⁴ m/s

0.593 m/s²

3.287 ·10⁻¹⁸ s²/m³

1.800 ·10²⁹ kg

14.82 ·10¹² J

1.334 ·10¹² J

G·M·mr² = m · 4π²·rT² T²r³ = 4π²G·M

T_b = 1.51 · 86400 = 130464 s ; T_d = 4.05 · 86400 = 349920 s k = T_b²r_b³ = 1.7021·10¹⁰5.1777·10²⁷ = 3.2873·10⁻¹⁸ s²/m³

T_c²T_b² = r_c³r_b³ T_c = T_b · √1.3144·10²⁸5.1777·10²⁷ = 1.51 · √2.53861 = 1.51 · 1.59330 T_c = 2.406 ·86400 s = 2.0787·10⁵ s

r_d³ = r_b³ · T_d²T_b² = 5.1777·10²⁷ · 1.2244·10¹¹1.7021·10¹⁰ = 5.1777·10²⁷ · 7.19366 r_d³ = 3.7247·10²⁸ m³ ; r_d = ∛3.7247·10²⁸ = 3.3396·10⁹ m

T_c² = k · r_c³ = 3.2873·10⁻¹⁸ · 1.3144·10²⁸ = 4.3210·10¹⁰ s² T_c = 2.0787·10⁵ s = 2.406 ·86400 s r_d³ = T_d²k = 1.2244·10¹¹3.2873·10⁻¹⁸ = 3.7247·10²⁸ m³

G·M·mr² = m·v²r ; v = √G·Mr

v = 2π·rT ; T ∝ r^1.5 ; v ∝ rr^1.5 = r^−0.5

v_b = 2π · 1.73·10⁹130464 = 8.33·10⁴ m/s v_c = 2π · 2.36·10⁹2.0787·10⁵ = 7.13·10⁴ m/s v_d = 2π · 3.3396·10⁹349920 = 6.00·10⁴ m/s

v_b = √6.67·10⁻¹¹ · 1.80·10²⁹1.73·10⁹ = √6.942·10⁹ = 8.33·10⁴ m/s

g_b = v²r ; v = 2π·rT

g_b = 4π²·r²T²·r = 4π²·rT²

g_b = 4π² · 1.73·10⁹(130464)² = 6.8298·10¹⁰1.7021·10¹⁰ = 4.013 m/s²

g_b = G·Mr_b² = 6.67·10⁻¹¹ · 1.8005·10²⁹2.9929·10¹⁸ = 4.013 m/s²

g_b = v_b²r_b = 6.9418·10⁹1.73·10⁹ = 4.013 m/s²

W = G·M_b·mR_b²

G·M·m_br_b² = m_b · 4π²·r_bT_b² ; M = 4π²·r_b³G·T_b²

M = 39.4784 · 5.1777·10²⁷6.67·10⁻¹¹ · 1.7021·10¹⁰ = 2.04408·10²⁹1.135291 = 1.8005·10²⁹ kg

M = g_b · r_b²G = 4.0126 · 2.9929·10¹⁸6.67·10⁻¹¹ = 1.8005·10²⁹ kg

M = 39.4784 · 3.7247·10²⁸6.67·10⁻¹¹ · 1.2244·10¹¹ = 1.47052·10³⁰8.16702 = 1.8005·10²⁹ kg

1.80·10²⁹2·10³⁰ = 0.090

E_k = G·M·m_s2R ; E_p = −G·M·m_sR E = E_k + E_p = −G·M·m_s2R

ΔE = 0 − E = G·M·m_s2R

ΔE_IΔE_II = M_☉M = 2·10³⁰1.8005·10²⁹ = 11.1

v_esc = √2·G·MR ; ΔE = m_s·v_esc²2 − m_s·v²2 = G·M·m_s2R

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