dane:
v₀ = 1 m/s
V₁ =11 m/s
s = 1 km = 1000 m
x = s/2 = 500 m
szukane:
v = ?
a = (V₁ - v₀)/t => t = (V₁ - v₀)/a
s = v₀t + at²/2
s = v₀(v₁ - v₀)/a + [a(v₁ - v₀)²/a²]/2
s = v₀(v₁ - v₀)/a + (v₁ - v₀)²/2a
2a s = 2v₀(v₁ - v₀) + (v₁ - v₀)²
2a s = (v₁ - v₀)(2v₀ + v₁ - v₀)
2a s = (v₁ - v₀)(v₀ + v₁)
a = (v₁² - v₀²)/2s
x = v₀t₁ + at₁²/2
x = v₀t₁ + [t₁²(v₁² - v₀²)/2s]/2
x = v₀t₁ + t₁²(v₁² - v₀²)/4s
s/2 = v₀t₁ + t₁²(v₁² - v₀²)/4s
2s² = v₀t4s + t²(v₁² - v₀²)
t²(v₁² - v₀²) + tv₀4s - 2s² = 0
Δ = 16s²v₀² + 8s²(v₁² - v₀²) = 8[2s²v₀² + s²v₁² - s²v₀²] = 8[s²v₀² + s²v₁²]
√Δ = 2√[2(s²v₀² + s²v₁²)] = 2s√[2(v₀² + v₁²)]
t = {v₀4s ± 2s√[2(v₀² + v₁²)]}/2(v₁² - v₀²) = {2v₀s ± s√[2(v₀² + v₁²)]}/(v₁² - v₀²)
v = v₀ + at = v₀ + (v₁² - v₀²)/2s * {2v₀s ± s√[2(v₀² + v₁²)]}/(v₁² - v₀²) = v₀ + {2v₀s ± s√[2(v₀² + v₁²)]}/2s = v₀ + v₀ ± √[2(v₀² + v₁²)]/2 = v₀ ± √[2(v₀² + v₁²)]/2
ze względu na sensowność odpowiedzi mamy:
v = v₀ + √[2(v₀² + v₁²)]/2
