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g = 10 m/s²  |  vx = 74 km/h = 20.56 m/s  |  θ = 42°
t = 0 → vy = 0  |  H₂ = H₁ − 1 m  |  L₂ = L₁
74 km/h
42°
1.0 m
21.18 m/s
20.56
21.18
17.13
16.13
1.851
1.796
18.51
17.96
27.66
27.77
38.04
38.04
42.0
40.3
0.00
0.0
0.0

a = g = 10 m/s²
ax = 0   ;   vx = 743.6 = 20.56 m/s   ;   x(t) = vx·t
ay = g = 10 m/s²   ;   vy(0) = 0   ;   vy(t) = g·t y(t) = H − ½·g·t²
v(t) = √vx² + (g·t)²   ;   tan θ(t) = g·tvx

vx = 743.6 = 20.56 m/s
cos 42° = vxv   →   v = vxcos 42°
v = 20.560.7431 = 27.66 m/s
tan 42° = vyvx   →   vy = 20.56 · 0.9004 = 18.51 m/s v = √20.56² + 18.51² = √422.5 + 342.6 = √765.1 = 27.66 m/s
½mv² = ½mvx² + mgH   →   v² = vx² + 2gH v² = 422.5 + 2 · 10 · 17.13 = 765.1   →   v = 27.66 m/s

vy = vx · tan 42° = 20.56 · 0.9004 = 18.51 m/s
vy² = v0y² + 2gH = 2gH   →   H = vy²2g H = 18.51²2 · 10 = 342.620 = 17.13 m
t₁ = vyg = 18.5110 = 1.851 s H = ½·g·t₁² = 5 · 3.426 = 17.13 m
L = vx · t₁ = 20.56 · 1.851 = 38.04 m

H₂ = H₁ − 1 = 17.13 − 1 = 16.13 m L₂ = L₁ = 38.04 m
H₂ = ½·g·t₂²   →   t₂ = √2H₂g = √32.2610 = 1.796 s
vx2 = Lt₂ = 38.041.796 = 21.18 m/s
L = vx · √2Hg   →   vx2 = vx1 · √H₁H₂ vx2 = 20.56 · √17.1316.13 = 20.56 · 1.0305 = 21.18 m/s

tan θ = vyvx
vy = √2gH   ;   H₂ < H₁   →   vy2 < vy1
t₂ < t₁   ;   L₂ = L₁   →   vx2 = Lt₂ > vx1
tan θ₂ < tan θ₁   →   θ₂ < θ₁
vy2 = g·t₂ = 10 · 1.796 = 17.96 m/s tan θ₂ = 17.9621.18 = 0.848   →   θ₂ = 40.3°

μk = 0.24  |  α₃ = 22°  |  h = 0.3 m  |  g = 10 m/s²
sin 22° = 0.3746   ;   cos 22° = 0.9272   ;   tan 22° = 0.4040
22.0°
0.40
0.24
0.30 m
1.0 kg
3.0 m/s
0.375
0.927
0.404
21.8
9.27
3.71
3.71
1.52
5.97
1.52
0.801
0.00
1.56
3.000
0.000
0.000

W = m·g
N = m·g·cos α₀
fs = m·g·sin α₀
√(m·g·sin α₀)² + (m·g·cos α₀)² = m·g

tan α₁ < tan α₀ ≤ μs
fs = m·g·sin α
fs(α₁) = m·g·sin α₁ < m·g·sin α₀ = fs(α₀)

N − m·g·cos α₂ = 0   ;   N = m·g·cos α₂
m·g·sin α₂ − fs,max = 0   ;   fs,max = μs·N
m·g·sin α₂ = μs·m·g·cos α₂ sin α₂cos α₂ = μs   ;   tan α₂ = μs
α₂ = arctan(μs)

N = m·g·cos α₃   ;   fk = μk·m·g·cos α₃
a = m·g·sin α₃ − μk·m·g·cos α₃m = g·(sin α₃ − μk·cos α₃) a = 10 · (0.3746 − 0.24 · 0.9272) = 10 · (0.3746 − 0.2225) = 1.52 m/s²
L = hsin α₃ = 0.30.3746 = 0.801 m
v² = v₀² + 2·a·L = 2 · 1.5208 · 0.8009 = 2.436 v = √2·a·L = √2.436 = 1.56 m/s
m·g·h − μk·m·g·cos α₃·L = 12·m·v² 12·v² = 10 · 0.3 − 0.24 · 10 · 0.7425 = 3 − 1.782 = 1.218 v² = 2.436   ;   v = 1.56 m/s
v²2·g·h = 2.4366 ≈ 0.41 t = va = 1.561.52 ≈ 1.03 s

m·au = m·g·sin α + μk·m·g·cos α au = g·(sin α + μk·cos α) = 10 · (0.3746 + 0.2225) = 5.97 m/s²
m·ad = m·g·sin α − μk·m·g·cos α ad = g·(sin α − μk·cos α) = 10 · (0.3746 − 0.2225) = 1.52 m/s²
au − ad = 2·μk·g·cos α > 0 auad = 5.971.52 ≈ 3.9
μk = 0   ;   au = ad = g·sin α

m = 0.2 kg  |  h = 30 cm = 0.30 m  |  θ = 37°  |  TB = 3 N  |  g = 10 m/s²
sin 37° = 0.6  ;  cos 37° = 0.8  ;  tan 37° = 0.75  ;  m·g = 2 N
LA = hcos 37° = 0.300.8 = 0.375 m   ;   r = LA·sin 37° = h·tan 37° = 0.225 m
1.59 Hz
0.20 kg
0.30 m
37°
0.225
0.375
10.00
2.25
0.63
2.50
3.00
4.50
0.000
0.000
0.000

LA·cos 37° = 0.30 m LA = 0.30cos 37° = 0.300.8 = 0.375 m r = LA·sin 37° = 0.375 · 0.6 = 0.225 m m·g = 0.2 · 10 = 2 N
ΣFx = TA·sin 37° + TB = Fc

ay = 0   ;   ax = ac = ω²·r
ΣFy = m·ay = 0 TA·cos 37° − m·g = 0 TA = m·gcos 37° = 0.2 · 100.8 = 20.8 = 2.5 N
m·g·tan 37° = 2 · 0.75 = 1.5 N TA·sin 37° = 2.5 · 0.6 = 1.5 N

r = h·tan 37° = 0.30 · 0.75 = 0.225 m
ΣFx = m·ac TA·sin 37° + TB = m·ω²·r 2.5 · 0.6 + 3 = 0.2 · ω² · 0.225 4.5 = 0.045 · ω² ω² = 4.50.045 = 100  →  ω = 10 rad/s
f = ω2π = 102 · 3.1416 = 1.59 Hz
v = ω·r = 10 · 0.225 = 2.25 m/s Fc = m·v²r = 0.2 · 5.06250.225 = 4.5 N T = 2π·rv = 2π · 0.2252.25 = 0.628 s f = 1T = 10.628 = 1.59 Hz

W = F·d·cos 90° = 0
Wtotal = ΔEk = 0 W(m·g) + W(TB) + W(TA) = 0

r = 0.225 m   ;   TA = m·gcos 37° = 2.5 N
TA·sin 37° = m·ω′²·r 1.5 = 0.2 · ω′² · 0.225 = 0.045 · ω′² ω′² = 1.50.045 = 33.33  →  ω′ = √33.33 = 5.77 rad/s f′ = ω′2π = 5.776.283 = 0.92 Hz
tan 37° = ω′²·rg  →  ω′² = g·tan 37°r = 10 · 0.750.225 = 33.33 f′f = √1.54.5 = 1√3 = 0.577  →  1.59 · 0.577 = 0.92 Hz

TA·cos 37° − m·g = 0 TA = m·gcos 37° = 20.8 = 2.5 N
TB = m·ω²·r − 1.5 N  →  TB > 3 N

mA = 0.1 kg  |  mB = 0.2 kg  |  v = 0.6 m/s  |  g = 10 m/s²
p = (mA + mB)·v = 0.18 kg·m/s   ;   uB = 0.180.2 = 0.9 m/s
WA = −0.018 J   ;   u′B = −0.45 m/s   ;   uC = 0.45 m/s   ;   mC = 0.6 kg
0.10 kg
0.20 kg
0.60 kg
0.60 m/s
0.027 J
0.60
0.60
0.00
0.06
0.12
0.00
0.018
0.036
0.000
0.180
0.054
0.027
−0.018
0.045

FA→B = −FB→A  →  ΣFin = 0
N − (mA + mB)·g = 0  →  N = (mA + mB)·g
f = 0  →  ΣFx,ext = 0
ΣFext = 0  →  pbefore = pafter

pbefore = (mA + mB)·v = (0.1 + 0.2)·0.6 = 0.18 kg·m/s
pafter = mA·uA + mB·uB = 0 + 0.2·uB
(mA + mB)·v = mB·uB uB = (mA + mB)·vmB = 0.180.2 = 0.9 m/s
Ek,before = 12·0.3·(0.6)² = 0.054 J Ek,after = 12·0.2·(0.9)² = 0.081 J ΔEk = 0.081 − 0.054 = 0.027 J

WA = ΔEk,A = 12·mA·uA² − 12·mA·v²
WA = 12·0.1·(0)² − 12·0.1·(0.6)² WA = 0 − 0.018 = −0.018 J
WB = 12·0.2·(0.9)² − 12·0.2·(0.6)² = 0.081 − 0.036 = +0.045 J WA + WB = −0.018 + 0.045 = 0.027 J

u′B = −12·uB = −0.92 = −0.45 m/s
uB − uC,before = −(u′B − uC)  →  uC − u′B = uB − uC,before uC + 0.45 = 0.9 − 0  →  uC = 0.45 m/s
mB·uB = mB·u′B + mC·uC  →  0.18 + 0.09 = mC·uC = 0.27   (1) 12mBuB² = 12mBu′B² + 12mCuC² 0.081 = 0.02025 + 12mCuC²  →  mC·uC² = 0.1215   (2)
uC = mC·uC²mC·uC = 0.12150.27 = 0.45 m/s

mC·uC = 0.27 kg·m/s mC = 0.27uC = 0.270.45 = 0.6 kg
u′B = mB − mCmB + mC·uB = 0.2 − 0.60.8·0.9 = −0.45 m/s uC = 2·mBmB + mC·uB = 0.40.8·0.9 = 0.45 m/s
12·0.2·(0.45)² + 12·0.6·(0.45)² = 0.02025 + 0.06075 = 0.081 J 0.2·(−0.45) + 0.6·(0.45) = −0.09 + 0.27 = 0.18 kg·m/s
mC = 0.6 kg > mB = 0.2 kg

m = 0.5 kg  |  A = 60 cm = 0.6 m  |  ΣF = −c·x
x (cm)  :  60    40    20    0    −10    −30    −50
F (N)  :  −9.4    −6.5    −3.0    0    1.7    5.0    7.8
15.85 N/m
0.50 kg
0.60 m
6.76 m/s
1.0 ×
0.600
0.00
-19.02
-9.51
5.63
1.12
0.90
3.38
0.00
2.85
2.85
0.60

ΣF = −c·x   ,   c > 0
ΣF = m·g = const   ≠   −c·x
ΣF = −m·g·sinθ   ;   ΣF = G·M·mr²

x (m) = 0.60 , 0.40 , 0.20 , 0 , −0.10 , −0.30 , −0.50 F (N) = −9.4 , −6.5 , −3.0 , 0 , 1.7 , 5.0 , 7.8
n = 7  ;  Σx = 0.300  ;  ΣF = −4.4  ;  Σx² = 0.910  ;  ΣxF = −14.41
a = n·ΣxF − Σx·ΣFn·Σx² − (Σx)² = 7·(−14.41) − 0.300·(−4.4)7·0.910 − (0.300)² = −99.556.28 = −15.85 N/m
b = ΣF − a·Σxn = −4.4 + 15.85·0.3007 = +0.051 N ≈ 0 F = −15.9 · x
a = 7.8 − (−9.4)−0.50 − 0.60 = 17.2−1.10 = −15.6 N/m

ΣF = Fspring = −k·x
c = k = |a| = 15.85 N/m
m·at = −k·x     at = − km · x at = −ω²·x     ω² = km
ω = √km = √15.850.5 = √31.7 = 5.63 rad/s
T = 2πω = 2π5.63 = 1.12 s f = 1T = 11.12 = 0.90 Hz

E = Ek + Ep = 12m·v² + 12k·x² = const
12k·A² = 12m·vmax² vmax = A · √km = A · ω
vmax = 0.6 · 5.63 = 3.38 m/s ≈ 3.4 m/s
E = 12k·A² = 12·15.85·(0.6)² = 2.85 J v² = 2·Em = 2·2.850.5 = 11.4    v = 3.38 m/s
x(t) = A·cos(ωt)     v(t) = −A·ω·sin(ωt)

v₀ = 2 · vmax = 2 · 3.38 = 6.76 m/s
f = ω2π = 12π · √km
f₁ = f₂ = 5.632π = 0.90 Hz   ;   T = 1.12 s
A₂ = v₀ω = 6.765.63 = 1.2 m = 2·A₁
E₂ = 12m·v₀² = 12·0.5·(6.76)² = 11.4 J = 4·E₁ 12k·A₂² = 12·15.85·(1.2)² = 11.4 J

m = 14 kg  |  RE = 6.37×10⁶ m = 6,370 km  |  ME = 5.98×10²⁴ kg
G = 6.67×10⁻¹¹ N·m²/kg²  |  G·ME = 3.99×10¹⁴ m³/s²
hmax = 2,515 km  |  hmin = 354 km  |  T = 100 min = 6,000 s  |  Δh = 1,000 km
×1.00
0.138
8,885 km
2,515 km
6,220 m/s
2.71×10⁸ J
−6.29×10⁸ J
−3.58×10⁸ J
7.74×10¹¹ kg·m²/s
0 J

rA = RE + 2,515 = 6,370 + 2,515 = 8,885 km = 8.885×10⁶ m rB = RE + 354 = 6,370 + 354 = 6,724 km = 6.724×10⁶ m
EA = EB 12m·vA² − G·ME·mrA = 12m·vB² − G·ME·mrB
U = − G·ME·mr rA > rB ⇒ UA > UB ⇒ Ek,A < Ek,B ⇒ vA < vB
m·vA·rA = m·vB·rB vAvB = rBrA = 6,7248,885 = 0.757 < 1
a = rA + rB2 = 8.885×10⁶ + 6.724×10⁶2 = 7.80×10⁶ m v = √G·ME(2r − 1a) vA = 6,220 m/s  ;  vB = 8,220 m/s  ;  6,2208,220 = 0.757

Wgrav = ΔEk = 12m·vB² − 12m·vA² > 0
Wgrav = −ΔU = UA − UB = G·ME·m(1rB − 1rA)
G·ME·m = 3.99×10¹⁴ · 14 = 5.586×10¹⁵ W = 5.586×10¹⁵ (1.487×10⁻⁷ − 1.125×10⁻⁷) = 5.586×10¹⁵ · 3.62×10⁻⁸ ≈ 2.0×10⁸ J
ΔEk = 12 · 14 · (8,220² − 6,220²) = 7 · 2.89×10⁷ ≈ 2.0×10⁸ J

T = 100 min = 100 · 60 = 6,000 s
G·ME·mr² = m · 4π²·rT²
r³ = G·ME·T²4π² r = 3√G·ME·T²4π²
r = 3√3.99×10¹⁴ · (6,000)²4π² = 3√1.436×10²²39.48 r = 3√3.64×10²⁰ = 7.14×10⁶ m = 7,140 km h = r − RE = 7.14×10⁶ − 6.37×10⁶ = 7.7×10⁵ m ≈ 770 km
G·ME = g·RE² = 10 · (6.37×10⁶)² = 4.06×10¹⁴ r = 3√3.70×10²⁰ = 7.18×10⁶ m
v = 2π·rT = 2π · 7.14×10⁶6,000 ≈ 7,480 m/s v = √G·MEr = √5.59×10⁷ ≈ 7,480 m/s

r₂ = r₁ + 1,000 km = 7.14×10⁶ + 1.00×10⁶ = 8.14×10⁶ m
v² = G·MEr  ;  Ek = G·ME·m2r E = Ek + U = G·ME·m2r − G·ME·mr = − G·ME·m2r
W = E₂ − E₁ = − G·ME·m2r₂ + G·ME·m2r₁ = G·ME·m2(1r₁ − 1r₂)
G·ME·m2 = 3.99×10¹⁴ · 142 = 2.79×10¹⁵ W = 2.79×10¹⁵ (1.401×10⁻⁷ − 1.229×10⁻⁷) = 2.79×10¹⁵ · 1.72×10⁻⁸ ≈ 4.8×10⁷ J
ΔU = G·ME·m(1r₁ − 1r₂) = 5.586×10¹⁵ · 1.72×10⁻⁸ = +9.6×10⁷ J Ek1 = 3.91×10⁸ J  ;  Ek2 = 3.43×10⁸ J  ;  ΔEk = −4.8×10⁷ J W = ΔU + ΔEk = 9.6×10⁷ − 4.8×10⁷ = 4.8×10⁷ J

T² = 4π²G·ME · r³ T = 2π√r³√G·ME
v = √G·MEr  ;  T = 2π·rv
T₂T₁ = (r₂r₁)1.5 = (8.14×10⁶7.14×10⁶)1.5 = (1.140)1.5 = 1.217 T₂ = 100 · 1.217 ≈ 122 min
T₂ = 2π√r₂³G·ME = 2π√5.39×10²⁰3.99×10¹⁴ = 2π√1.352×10⁶ T₂ = 2π · 1,163 ≈ 7,307 s = 7,30760 ≈ 122 min
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