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Revisão das 17h09min de 5 de junho de 2013
colisões elásticas
Antes da colisão
Momentum linear antes da colisão:
P
→
i
=
p
→
1
,
i
+
p
→
2
,
i
{\displaystyle {\vec {P}}_{i}={\vec {p}}_{1,i}+{\color {red}{\vec {p}}_{2,i}}}
P
i
=
m
1
v
1
,
i
{\displaystyle {P}_{i}=m_{1}{v}_{1,i}}
Energia cinética antes da colisão:
K
i
=
1
2
m
1
v
i
2
{\displaystyle K_{i}={\frac {1}{2}}m_{1}v_{i}^{2}}
Após a colisão:
Momentum linear após a colisão:
P
→
f
=
p
→
1
,
f
+
p
→
2
,
f
{\displaystyle {\vec {P}}_{f}={\vec {p}}_{1,f}+{\color {red}{\vec {p}}_{2,f}}}
P
f
=
m
1
v
1
,
f
+
m
2
v
2
,
f
{\displaystyle {P}_{f}=m_{1}{v}_{1,f}+{\color {red}m_{2}{v}_{2,f}}}
Energia cinética após a colisão:
K
f
=
1
2
m
1
v
1
,
f
2
+
1
2
m
2
v
2
,
f
2
{\displaystyle K_{f}={\frac {1}{2}}m_{1}v_{1,f}^{2}+{\color {red}{\frac {1}{2}}m_{2}v_{2,f}^{2}}}
Conservação de Momentum Linear:
P
i
=
P
f
{\displaystyle {P}_{i}={P}_{f}}
m
1
v
1
,
i
=
m
1
v
1
,
f
+
m
2
v
2
,
f
{\displaystyle m_{1}{v}_{1,i}=m_{1}{v}_{1,f}+{\color {red}m_{2}{v}_{2,f}}}
Reacrupando os termos
m
1
(
v
1
,
i
−
v
1
,
f
)
=
m
2
v
2
,
f
{\displaystyle m_{1}({v}_{1,i}-{v}_{1,f})={\color {red}m_{2}{v}_{2,f}}}
Conservação da energia cinética:
K
i
=
K
f
{\displaystyle K_{i}=K_{f}}
1
2
m
1
v
1
,
i
2
=
1
2
m
1
v
1
,
f
2
+
1
2
m
2
v
2
,
f
2
{\displaystyle {\frac {1}{2}}m_{1}v_{1,i}^{2}={\frac {1}{2}}m_{1}v_{1,f}^{2}+{\color {red}{\frac {1}{2}}m_{2}v_{2,f}^{2}}}
Reacrupando os termos
1
2
m
1
(
v
1
,
i
2
−
v
1
,
f
2
)
=
1
2
m
2
v
2
,
f
2
{\displaystyle {\frac {1}{2}}m_{1}(v_{1,i}^{2}-v_{1,f}^{2})={\color {red}{\frac {1}{2}}m_{2}v_{2,f}^{2}}}
m
1
(
v
1
,
i
2
−
v
1
,
f
2
)
=
m
2
v
2
,
f
2
{\displaystyle m_{1}(v_{1,i}^{2}-v_{1,f}^{2})=m_{2}v_{2,f}^{2}}
m
1
(
v
1
,
i
−
v
1
,
f
)
(
v
1
,
i
+
v
1
,
f
)
=
m
2
v
2
,
f
2
{\displaystyle m_{1}(v_{1,i}-v_{1,f})(v_{1,i}+v_{1,f})=m_{2}v_{2,f}^{2}}
Usando a equação oriunda da conservação do momentum na equação anterior:
m
1
(
v
1
,
i
−
v
1
,
f
)
(
v
1
,
i
+
v
1
,
f
)
=
m
2
v
2
,
f
2
{\displaystyle m_{1}(v_{1,i}-v_{1,f})(v_{1,i}+v_{1,f})=m_{2}v_{2,f}^{2}}
m
2
v
2
,
i
(
v
1
,
i
+
v
1
,
f
)
=
m
2
v
2
,
f
2
{\displaystyle {\color {red}m_{2}{v}_{2,i}}(v_{1,i}+v_{1,f})=m_{2}v_{2,f}^{2}}
m
2
v
2
,
i
(
v
1
,
i
+
v
1
,
f
)
=
m
2
v
2
,
f
2
{\displaystyle {\color {red}{\cancel {m_{2}}}{v}_{2,i}}(v_{1,i}+v_{1,f})={\cancel {m_{2}}}v_{2,f}^{2}}
v
2
,
f
(
v
1
,
i
+
v
1
,
f
)
=
v
2
,
f
2
{\displaystyle {\color {red}{\cancel {{v}_{2,f}}}}(v_{1,i}+v_{1,f})=v_{2,f}^{\cancel {2}}}
(
v
1
,
i
+
v
1
,
f
)
=
v
2
,
f
{\displaystyle (v_{1,i}+v_{1,f})=v_{2,f}}
Substituindo na equação do momentum:
m
1
v
1
,
i
=
m
1
v
1
,
f
+
m
2
v
2
,
f
{\displaystyle m_{1}{v}_{1,i}=m_{1}{v}_{1,f}+m_{2}{\color {red}{v}_{2,f}}}
m
1
v
1
,
i
=
m
1
v
1
,
f
+
m
2
(
v
1
,
i
+
v
1
,
f
)
{\displaystyle m_{1}{v}_{1,i}=m_{1}{v}_{1,f}+m_{2}{\color {red}(v_{1,i}+v_{1,f})}}
Reagrupando de maneira a escrever a velocidade final em termos da velocidade
inicial:
m
1
v
1
,
i
−
m
2
v
1
,
i
=
m
1
v
1
,
f
+
m
2
v
1
,
f
{\displaystyle m_{1}{v}_{1,i}-m_{2}v_{1,i}=m_{1}{v}_{1,f}+m_{2}{v_{1,f}}}
logo:
v
1
,
f
=
v
1
,
i
(
m
1
−
m
2
)
m
1
+
m
2
{\displaystyle {v}_{1,f}={\frac {{v}_{1,i}(m_{1}-m_{2})}{m_{1}+m_{2}}}}
Substituindo na equação que relaciona as velocidades teremos:
(
v
1
,
i
+
v
1
,
f
)
=
v
2
,
f
{\displaystyle (v_{1,i}+{\color {red}v_{1,f}})=v_{2,f}}
(
v
1
,
i
+
v
1
,
i
(
m
1
−
m
2
)
m
1
+
m
2
)
=
v
2
,
f
{\displaystyle (v_{1,i}+{\color {red}{\frac {{v}_{1,i}(m_{1}-m_{2})}{m_{1}+m_{2}}}})=v_{2,f}}
Reescrevendo:
(
v
1
,
i
(
m
1
+
m
2
)
+
v
1
,
i
(
m
1
−
m
2
)
m
1
+
m
2
=
v
2
,
f
{\displaystyle {\frac {(v_{1,i}({m_{1}+m_{2}})+{v}_{1,i}(m_{1}-m_{2})}{m_{1}+m_{2}}}=v_{2,f}}
v
2
,
f
=
2
m
1
v
1
,
i
m
1
+
m
2
{\displaystyle v_{2,f}={\frac {2m_{1}{v}_{1,i}}{m_{1}+m_{2}}}}
Resumindo:
Antes da colisão: após a colisão:
v
1
,
f
=
v
1
,
i
(
m
1
−
m
2
)
m
1
+
m
2
v
:<
m
a
t
h
>
v
2
,
f
=
2
m
1
v
1
,
i
m
1
+
m
2
{\displaystyle {v}_{1,f}={\frac {{v}_{1,i}(m_{1}-m_{2})}{m_{1}+m_{2}}}v:<math>v_{2,f}={\frac {2m_{1}{v}_{1,i}}{m_{1}+m_{2}}}}
Alguns casos particulares
m
1
=
m
2
{\displaystyle m_{1}=m_{2}}
v
1
,
f
=
v
1
,
i
(
m
1
−
m
2
)
m
1
+
m
2
=
v
1
,
i
(
m
2
−
m
2
)
m
2
+
m
2
=
0
{\displaystyle {v}_{1,f}={\frac {{v}_{1,i}(m_{1}-m_{2})}{m_{1}+m_{2}}}={\frac {{v}_{1,i}(m_{2}-m_{2})}{m_{2}+m_{2}}}=0}
v
2
,
f
=
2
m
1
v
1
,
i
m
1
+
m
2
=
2
m
2
v
1
,
i
m
2
+
m
2
=
2
m
2
v
1
,
i
2
m
2
=
v
1
,
i
{\displaystyle v_{2,f}={\frac {2m_{1}{v}_{1,i}}{m_{1}+m_{2}}}={\frac {2m_{2}{v}_{1,i}}{m_{2}+m_{2}}}={\frac {2m_{2}{v}_{1,i}}{2m_{2}}}={v}_{1,i}}
m
1
<<
m
2
{\displaystyle m_{1}<<m_{2}}
v
1
,
f
=
v
1
,
i
(
m
1
−
m
2
)
m
1
+
m
2
≈
v
1
,
i
(
−
m
2
)
m
2
=
−
v
1
,
i
{\displaystyle {v}_{1,f}={\frac {{v}_{1,i}(m_{1}-m_{2})}{m_{1}+m_{2}}}\approx {\frac {{v}_{1,i}(-m_{2})}{m_{2}}}=-{v}_{1,i}}
v
2
,
f
=
2
m
1
v
1
,
i
m
1
+
m
2
≈
2
m
1
v
1
,
i
m
2
≈
0
{\displaystyle v_{2,f}={\frac {2m_{1}{v}_{1,i}}{m_{1}+m_{2}}}\approx {\frac {2m_{1}{v}_{1,i}}{m_{2}}}\approx 0}
m
1
>>
m
2
{\displaystyle m_{1}>>m_{2}}
v
1
,
f
=
v
1
,
i
(
m
1
−
m
2
)
m
1
+
m
2
≈
v
1
,
i
(
m
1
)
m
1
=
v
1
,
i
{\displaystyle {v}_{1,f}={\frac {{v}_{1,i}(m_{1}-m_{2})}{m_{1}+m_{2}}}\approx {\frac {{v}_{1,i}(m_{1})}{m_{1}}}={v}_{1,i}}
v
2
,
f
=
2
m
1
v
1
,
i
m
1
+
m
2
≈
2
m
1
v
1
,
i
m
1
=
2
v
1
,
i
{\displaystyle v_{2,f}={\frac {2m_{1}{v}_{1,i}}{m_{1}+m_{2}}}\approx {\frac {2m_{1}{v}_{1,i}}{m_{1}}}=2{v}_{1,i}}