התנע ושימורו — פרק 3

p = m·v | J = F·Δt = Δp | F_ממוצע = ΔpΔt

0.15 kg

-30 m/s

2100 N

5.0 ms

0

v₀ = −30 m/s , v₁ = +40 m/s

p₀ = m·v₀ = 0.15 · (−30) = −4.50 kg·m/s p₁ = m·v₁ = 0.15 · (+40) = +6.00 kg·m/s

Δp = p₁ − p₀ = 6.00 − (−4.50) = 10.50 kg·m/s

F = ΔpΔt = 10.500.0050 = 2100 N

p = m·v | Σp = m₁v₁ + m₂v₂ = const | v = m₁v₁ + m₂v₂m₁ + m₂

2.0 kg

3.0 m/s

4.0 kg

0.0 m/s

0.00

6.0

p₁ = m₁·v₁ = 2.0 · 3.0 = 6.0 kg·m/s p₂ = m₂·v₂ = 4.0 · 0 = 0 kg·m/s

Σp = p₁ + p₂ = 6.0 + 0 = 6.0 kg·m/s

Σp = (m₁ + m₂)·v = (2.0 + 4.0)·v = 6.0·v

v = m₁v₁ + m₂v₂m₁ + m₂ = 6.06.0 = 1.00 m/s

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ | e = v₂ − v₁u₁ − u₂ | v₁ = m₁ − m₂m₁ + m₂·u₁ | v₂ = 2m₁m₁ + m₂·u₁

1.0 kg

+4.0 m/s

1.0 kg

0.0 m/s

1.00

0.00

p = m₁u₁ + m₂u₂ = 1.0·4.0 + 3.0·0 = 4.0 kg·m/s Eₖ = ½m₁u₁² + ½m₂u₂² = ½·1.0·4.0² = 8.0 J

v₁ = m₁ − m₂m₁ + m₂·u₁ = 1.0 − 3.01.0 + 3.0·4.0 = −2.0 m/s

v₂ = 2m₁m₁ + m₂·u₁ = 2·1.01.0 + 3.0·4.0 = +2.0 m/s

p = 1.0·(−2.0) + 3.0·(+2.0) = −2.0 + 6.0 = 4.0 kg·m/s Eₖ = ½·1.0·(−2.0)² + ½·3.0·(2.0)² = 2.0 + 6.0 = 8.0 J e = v₂ − v₁u₁ − u₂ = 2.0 − (−2.0)4.0 − 0 = 1.00

Σpx : m₁v₁ = m₁v₁′·cosθ₁ + m₂v₂′·cosθ₂ | Σpy : 0 = m₁v₁′·sinθ₁ − m₂v₂′·sinθ₂
tanθ₁ = p₁yp₁x | v₁′ = v₁ − 2m₂m₁ + m₂·(v₁·n̂)·n̂

0.50 kg

4.0 m/s

0.50

p₁x

0.000

p₁y

0.000

p₂x

0.000

p₂y

0.000

0.0

0.000

x : 4.0 = u·cos30° + w·cosφ y : 0 = u·sin30° − w·sinφ 4.0² = u² + w²

w·cosφ = 4.0 − u·cos30° ; w·sinφ = u·sin30° w² = 16 − 8·u·cos30° + u²

16 = u² + 16 − 8·u·cos30° + u² → 2u² = 8·u·cos30° u = 8·cos30°2 = 6.9282 = 3.46 m/s w = √(16 − 12) = 2.00 m/s tanφ = u·sin30°4.0 − u·cos30° = 1.7321.000 = 1.732 → φ = 60.0°

Σpx : 0.50·4.0 = 0.50·3.46·cos30° + 0.50·2.00·cos60° = 1.500 + 0.500 = 2.000 kg·m/s Σpy : 0 = 0.50·3.46·sin30° − 0.50·2.00·sin60° = 0.866 − 0.866 = 0 θ₁ + φ = 30° + 60° = 90°

m·v + M·V = 0 | V = m·vM | Ek = p²2m

5.0 kg

0.500 kg

200 m/s

0.0

0.00

0

p = 0 → m·v + M·V = 0

m·v = 0.010 · 700 = 7.0 kg·m/s

V = m·vM = 0.010 · 7004.0 = 1.75 m/s

EmEM = 24506.125 = 400 EmEM = Mm = 4.00.010 = 400