Origem: Wikilivros, livros abertos por um mundo aberto.
Antes da colisão
Momentum linear antes da colisão:
![{\displaystyle {\vec {P}}_{i}={\vec {p}}_{1,i}+{\color {red}{\vec {p}}_{2,i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7007f30c6ddc7b49ab9703e549f3e2291493ab3)
![{\displaystyle {P}_{i}=m_{1}{v}_{1,i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/488fccfeaee5e4ac9238258821e5623064f72357)
Energia cinética antes da colisão:
Após a colisão:
Momentum linear após a colisão:
![{\displaystyle {\vec {P}}_{f}={\vec {p}}_{1,f}+{\color {red}{\vec {p}}_{2,f}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be9a56cf9305c28fbe37cac45a70cece38a7c35)
![{\displaystyle {P}_{f}=m_{1}{v}_{1,f}+{\color {red}m_{2}{v}_{2,f}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/531325c2a51cafbfbdf3a001780e1cde265ffcef)
Energia cinética após a colisão:
![{\displaystyle K_{f}={\frac {1}{2}}m_{1}v_{1,f}^{2}+{\color {red}{\frac {1}{2}}m_{2}v_{2,f}^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60c2dfcc15e2b6dcae018c4530b307215aacb4b8)
Conservação de Momentum Linear:
![{\displaystyle {P}_{i}={P}_{f}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55f9e8515fbd21c47c53d1dc22a1d99cf305734b)
![{\displaystyle m_{1}{v}_{1,i}=m_{1}{v}_{1,f}+{\color {red}m_{2}{v}_{2,f}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9234cfc3d2c9d44ba4a7e4cf0adf2b90e89a989f)
Reacrupando os termos
Conservação da energia cinética:
![{\displaystyle K_{i}=K_{f}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d06de8ff385a917a5d83f351fcf77cb50f37c2d)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c8119555c179ecf3e892bcaa53629afd83fb874)
Reacrupando os termos
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bc42d1ba4c59e31f93b80d587fca562e03693ce)
![{\displaystyle m_{1}(v_{1,i}^{2}-v_{1,f}^{2})=m_{2}v_{2,f}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/637cdafb448aece5e8d78e6f9c2034de7a709497)
![{\displaystyle m_{1}(v_{1,i}-v_{1,f})(v_{1,i}+v_{1,f})=m_{2}v_{2,f}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80b516af4ab34d46e02019581befd0a738b088c1)
Usando a equação oriunda da conservação do momentum na equação anterior:
![{\displaystyle m_{1}(v_{1,i}-v_{1,f})(v_{1,i}+v_{1,f})=m_{2}v_{2,f}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80b516af4ab34d46e02019581befd0a738b088c1)
![{\displaystyle {\color {red}m_{2}{v}_{2,i}}(v_{1,i}+v_{1,f})=m_{2}v_{2,f}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e31bfad0b2089e9461eb7f6bcae7a29baa957a4)
![{\displaystyle {\color {red}{\cancel {m_{2}}}{v}_{2,i}}(v_{1,i}+v_{1,f})={\cancel {m_{2}}}v_{2,f}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae997781a653f125cbb024a33f904e7c717d9ce0)
![{\displaystyle {\color {red}{\cancel {{v}_{2,f}}}}(v_{1,i}+v_{1,f})=v_{2,f}^{\cancel {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aeba8a27c16cb44f2d7bc544be2014828d771221)
![{\displaystyle (v_{1,i}+v_{1,f})=v_{2,f}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/889e7faea7d5237e2d18444b9bca0512cb950bc8)
Substituindo na equação do momentum:
![{\displaystyle m_{1}{v}_{1,i}=m_{1}{v}_{1,f}+m_{2}{\color {red}{v}_{2,f}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/141c2c04a4cbdc6698a57f7cb5fcf54373667a22)
![{\displaystyle m_{1}{v}_{1,i}=m_{1}{v}_{1,f}+m_{2}{\color {red}(v_{1,i}+v_{1,f})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4be254a104896d68f782c5b0b43394a099928fd8)
Reagrupando de maneira a escrever a velocidade final em termos da velocidade
inicial:
![{\displaystyle m_{1}{v}_{1,i}-m_{2}v_{1,i}=m_{1}{v}_{1,f}+m_{2}{v_{1,f}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e3658c75662744a482b16ced3b9ee9df06ac2bf)
logo:
![{\displaystyle {v}_{1,f}={\frac {{v}_{1,i}(m_{1}-m_{2})}{m_{1}+m_{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e771a784026cdd47527cd628d0db580df349150)
Substituindo na equação que relaciona as velocidades teremos:
![{\displaystyle (v_{1,i}+{\color {red}v_{1,f}})=v_{2,f}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a17aa53bcb49c4a0580d763e216ee55e26498e)
![{\displaystyle (v_{1,i}+{\color {red}{\frac {{v}_{1,i}(m_{1}-m_{2})}{m_{1}+m_{2}}}})=v_{2,f}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00d7eaa9582e5d6ed094931aa0e374854cfe48ad)
Reescrevendo:
![{\displaystyle {\frac {(v_{1,i}({m_{1}+m_{2}})+{v}_{1,i}(m_{1}-m_{2})}{m_{1}+m_{2}}}=v_{2,f}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ec46b20bc5aec145e683e053dd8007dd6901ae)
![{\displaystyle v_{2,f}={\frac {2m_{1}{v}_{1,i}}{m_{1}+m_{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2dfd421e2bd52501893f6dde2851e279659ba0d)
Resumindo:
Antes da colisão:
após a colisão:
![{\displaystyle {v}_{1,f}={\frac {{v}_{1,i}(m_{1}-m_{2})}{m_{1}+m_{2}}}v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5b804fefeb223043ff77efc2fb01204e9636738)
![{\displaystyle v_{2,f}={\frac {2m_{1}{v}_{1,i}}{m_{1}+m_{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2dfd421e2bd52501893f6dde2851e279659ba0d)
![{\displaystyle m_{1}=m_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ac4f8db979aad35e260ec4878a1cb1456b85de4)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43b02c9e330d0c7f12cc8c57fbf217d96880778a)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77caf2606c1e104d03219ad249f6ddae5465cf67)
![{\displaystyle m_{1}<<m_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a9b6aa488bb3c2840408ff6e77c03fad5a8ca6a)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecf849b24256e9dc0ceb900371ca36a83bc6795f)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7f6207d0e64381bd130afb0c92496b770a5f0a7)
![{\displaystyle m_{1}>>m_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb90d7dd41dfacaa9f1b8ec03b716511cfae43e0)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bece5d3d61f3dff8f45bba6fddec4659613699e)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f9617d65239af91ab0cbc045bb37218b1e4d742)
Se usarmos um referencial que se move com a velocidade inicial
da partícula 2 (partícula alvo) caimos no mesmo caso anterior (alvo em repouso)
![{\displaystyle V_{1,i}=v_{1,i}-v_{2,i};V_{1,f}=v_{1,f}-v_{2,i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd5e3acb5ba69e202a774b9b1c7104b547d0794)
![{\displaystyle V_{2,i}=v_{2,i}-v_{2,i}=0;V_{2,f}=v_{2,f}-v_{2,i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00659972e211a22401b6f0b05ddb076f3d4bcab5)
Usando os resultados anteriores:
![{\displaystyle {V}_{1,f}={\frac {{V}_{1,i}(m_{1}-m_{2})}{m_{1}+m_{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfc84eff3b14082990babf898c723cccfedfc7cb)
![{\displaystyle V_{2,f}={\frac {2m_{1}{V}_{1,i}}{m_{1}+m_{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5edecfebdc23ec4e5fcee3d71e278cbecc0240b)
Lembrando que:
![{\displaystyle V_{1,f}=v_{1,f}-v_{2,i}\Longrightarrow v_{1,f}=v_{2,i}+V_{1,f}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f50e14bd3532af3bf4f599ed650fe2a00d02ff7)
![{\displaystyle V_{2,f}=v_{2,f}-v_{2,i}\Longrightarrow v_{2,f}=v_{2,i}+V_{2,f}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7ad039f0ebed25709d72c496f8b643a90dee283)
![{\displaystyle V_{1,i}=v_{1,i}-v_{2,i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/354671afd80a895bcba67a0f4a8ce82d0b1c96c6)
Logo:
![{\displaystyle v_{1,f}=v_{2,i}+{\frac {(v_{1,i}-v_{2,i})(m_{1}-m_{2})}{m_{1}+m_{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89aae6330041c28e22ce48658cdde26a990c841e)
![{\displaystyle v_{2,f}=v_{2,i}+{\frac {2m_{1}(v_{1,i}-v_{2,i})}{m_{1}+m_{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46caa90c336e931a33ec29d586489ccc4e380b70)