פתרון בגרות · מכניקה

✦ בנוי על ידי פהד גאנם ✦


t(s)  0.00  0.04  0.08  0.12  0.16  0.20  0.24  0.28  0.32  0.36  0.40
y(m)  0.00  0.11  0.20  0.29  0.35  0.40  0.43  0.45  0.45  0.43  0.40
3.1
−10.3
0.45
0.48

t₁ = 0.04 s , y₁ = 0.11 m  ·  t₂ = 0.12 s , y₂ = 0.29 m
v = ΔyΔt = y₂ − y₁t₂ − t₁ = 0.29 − 0.110.12 − 0.04 = 0.180.08 = 2.25 m/s

v(0.08) = 0.29 − 0.110.08 = +2.25 m/s v(0.12) = 0.35 − 0.200.08 = +1.88 m/s v(0.20) = 0.43 − 0.350.08 = +1.00 m/s v(0.28) = 0.45 − 0.430.08 = +0.25 m/s v(0.32) = 0.43 − 0.450.08 = −0.25 m/s v(0.36) = 0.40 − 0.450.08 = −0.63 m/s

v₀ ≈ 3.1 m/s   כלפי מעלה
a = ΔvΔt = −0.63 − 2.250.36 − 0.08 = −2.880.28 ≈ −10.3 m/s²
h = v₀²2g = 3.1²2 · 10 ≈ 0.48 m

|aעלייה| = mg + fm = g + fm  >  g
|aירידה| = mg − fm = g − fm  <  g


m₁ > m₂  ·  g = 10 m/s²
(i)   a = m₁m₂ g       (ii)   a = m₁m₂² g
1.50
1.50
2.00
12.0
1.00

m₁g > T > m₂g

kgkg · ms² = ms²
m₁m₂ > 1  ⟹  a > g
kgkg² · ms² = ms²·kg  ≠  ms²

m₁g − T = m₁a
T − m₂g = m₂a
m₁g − m₂g = m₁a + m₂a (m₁ − m₂)·g = (m₁ + m₂)·a a = (m₁ − m₂)·gm₁ + m₂
T = 2·m₁·m₂·gm₁ + m₂

s = 12·a·t² a = 2·st² = 2 · 11² = 2 m/s²
(m₁ − m₂)·gm₁ + m₂ = 2 m₁ − m₂m₁ + m₂ = 210 = 0.2
k = m₁m₂  ⟹  k − 1k + 1 = 0.2
k − 1 = 0.2·k + 0.2 0.8·k = 1.2 k = 1.20.8 = 1.5
a = (1.5 − 1)·101.5 + 1 = 52.5 = 2 m/s²

r = 10 cm  ·  h = 30 cm  ·  vE = 4 m/s  ·  α = 37°  ·  g = 10 m/s²
1.10 m
4.31
186
7.07
2.2
4.24
17.0
4.00
4.69
47
0.59

h = 30 cm = 0.3 m  ·  r = 10 cm = 0.1 m
EA = m·g·H   ·   EE = 12·m·vE² + m·g·h
m·g·H = 12·m·vE² + m·g·h H = h + vE²2·g = 0.3 + 4²2 · 10 = 0.3 + 1620 = 0.3 + 0.8 = 1.1 m

ar = vC²r  >  0
a = g + Nm

vx = vE·cos α = 4 · 0.8 = 3.2 m/s vy0 = vE·sin α = 4 · 0.6 = 2.4 m/s
vx = 3.2 m/s
vy² = vy0² + 2·g·h = 2.4² + 2 · 10 · 0.3 = 5.76 + 6 = 11.76 m²/s² vy = √11.76 = 3.43 m/s
v = √vx² + vy² = √10.24 + 11.76 = √22 = 4.69 m/s
tan β = vyvx = 3.433.2 = 1.07  →  β = 47°
v = √2·g·H = √2 · 10 · 1.1 = √22 = 4.69 m/s

N + m·g = m·vD²r

m·g = m·vD²r  →  vD² = g·r = 10 · 0.1 = 1 m²/s²
m·g·H1 = 12·m·vD² + m·g·(2r) H1 = 2r + vD²2·g = 2r + g·r2·g = 2r + r2 = 2.5 · r
H1 = 2.5 · 0.1 = 0.25 m = 25 cm

m·g·H = 12·m·vx² + m·g·hmax  →  hmax = H − vx²2·g  <  H
hmax = h + vy0²2·g = 0.3 + 2.4²2 · 10 = 0.3 + 0.288 = 0.588 m hmax = 1.1 − 3.2²2 · 10 = 1.1 − 0.512 = 0.588 m

m = 100 gr = 0.1 kg  ·  v0 = 4 m/s  ·  g = 10 m/s2
0.00
0.00
0.00
0.00
0.00

m = 100 gr = 0.1 kg  ·  v₀ = 4 m/s  ·  g = 10 m/s²
Wmg = ΔEk  ⟹  −m·g·h = 0 − m·v₀²2
h = v₀²2·g = 4²2 · 10 = 1620 = 0.80 m

h₁ = 0.80 · h = 0.80 · 0.80 = 0.64 m
ΔEk = 0 − m·v₀²2 = − 0.1 · 4²2 = −0.80 J
Wmg = −m·g·h₁ = −0.1 · 10 · 0.64 = −0.64 J
Wmg + Wf = ΔEk Wf = ΔEk − Wmg = −0.80 − (−0.64) = −0.16 J

Wmg = +m·g·h₁ = +0.64 J  ·  Wf = −0.16 J
m·v²2 − 0 = 0.64 − 0.16 = 0.48 J
v = √2 · 0.480.1 = √9.6 ≈ 3.1 m/s

ΔEk = m·4²2 − m·4²2 = 0
WF + Wmg + Wf = ΔEk = 0
WF = −Wmg − Wf = 0.64 + 0.16 = 0.80 J
WF = F · (BC + BA₁)  ⟹  F = 0.80BC + BA₁

W₃ = F · s  ·  W₄ = F · cos α · s  <  F · s
N₃ = m·g·cos α  <  N₄ = m·g·cos α + F·sin α

1 :  m1 = m   ·   v1 = v
2 :  m2 = 2m   ·   v2 = 0.5v
3 :  m3 = 0.5m   ·   v3 = 2v
2.0 kg
0.50 kg
1.60
4.27
16.0
3.20
80

m · v = (M + m) · u1
u1 = m · vM + m
E₀ = m · v²2 E₁ = (M + m) · u1²2 = M + m2 · m² · v²(M + m)² = m² · v²2 · (M + m)
E₁E₀ = mM + m  <  1 ΔE = E₀ − E₁ = M · m · v²2 · (M + m)  >  0
u1 = 0.5 · 82 + 0.5 = 1.60 m/s E₀ = 16.0 J  ·  E₁ = 2.5 · 1.60²2 = 3.20 J

p1 = m · v  ·  p2 = 2m · 0.5v = m · v  ·  p3 = 0.5m · 2v = m · v
u1 = m · vM + m u2 = 2m · 0.5vM + 2m = m · vM + 2m u3 = 0.5m · 2vM + 0.5m = m · vM + 0.5m
u2 < u1 < u3
E₀(1) = m · v²2  ·  E₀(2) = m · v²4  ·  E₀(3) = m · v²

T = (M + mk) · u²L
T1 = M + mL · m² · v²(M + m)² = m² · v²(M + m) · L T2 = m² · v²(M + 2m) · L T3 = m² · v²(M + 0.5m) · L
T = p²(M + mk) · L  ,  p = m · v

M + 0.5m  <  M + m  <  M + 2m T3  >  T1  >  T2
T = 2 · E1L  ,  E1 = p²2 · (M + mk)

ΣF = T  ≠  0
|Δp| = √2 · (M + mk) · u
Δt = π · L2 · u F = |Δp|Δt = 2√2π · T ≈ 0.90 · T

R2 = 2 · R1   ·   G = 6.674 · 10−11 N·m²/kg²
0.00
0.00
1.00
1.51
3.96
5.9
6.5
1.1

tA→B = tB→C  ⟹  SA→B = SB→C
S ≈ 12 · r̄ · s r̄AB · sAB = r̄BC · sBC  ⟹  r̄ABr̄BC = sBCsAB < 1

G·M·mr² = m·v²r  ⟹  v² = G·Mr Ek = 12·m·v² = G·M·m2·r  ·  Ep = − G·M·mr E = Ek + Ep = − G·M·m2·r
Ek1 = G·M·m2·R1  ·  Ek2 = G·M·m2·(2·R1) = G·M·m4·R1 = 12·Ek1
E1 = − G·M·m2·R1  ·  E2 = − G·M·m4·R1 E2 − E1 = + G·M·m4·R1 > 0  ⟹  E2 > E1
F1 = G·M·mR1²  ·  F2 = G·M·m(2·R1)² = G·M·m4·R1² F2F1 = 14

g* = G·Mr² = (G·M) · 1r²
(0 , 0)  ·  (2.4 · 10−14 m−2 , 9.5 N/kg)
G·M = Δg*Δ1r² = 9.5 − 02.4 · 10−14 − 0 = 3.96 · 1014 N·m²/kg
M = 3.96 · 1014G = 3.96 · 10146.674 · 10−11 = 5.93 · 1024 kg

E1 = 12·m·v² − G·M·mR
E2 = 0 + 0 = 0
12·m·v² − G·M·mR = 0  ⟹  v² = 2·G·MR v = √2·G·MR
v = √2 · 3.96 · 10146.45 · 106 = 1.11 · 104 m/s

1R² = 2.4 · 10−14 m−2  ·  g*R = 9.5 N/kg
R² = 12.4 · 10−14 = 4.17 · 1013 m² R = √4.17 · 1013 = 6.45 · 106 m
g*R = G·MR² = 3.96 · 10144.17 · 1013 = 9.5 N/kg ✓
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