Cantidad de movimiento lineal?

Partimos de los datos :

<img src="https://tex.z-dn.net/?f=%20m_%7Ba%7D%3Dm%20%5C%5C%20%20m_%7Bb%7D%3D3m%20%5C%5C%20%20v_%7B0a%7D%3Di%20%20%20%20%5C%5C%20%20v_%7B0b%7D%3D0%20" />

Llamaremos a las velocidades finales de cada cuerpo :

<img src="https://tex.z-dn.net/?f=%20v_%7Bfa%7D%3D%20f_%7Ba%7D%20%20%20%5C%5C%20%20v_%7Bfb%7D%3D%20f_%7Bb%7D%20%20" />

Sabemos que en la colisión se conserva el vector cantidad de movimiento lineal :

<img src="https://tex.z-dn.net/?f=%5Coverrightarrow%7B%20p%7D_%7B0%7D%3D%5Coverrightarrow%7B%20p%7D_%7Bf%7D%20%5C%5C%20%5C%5C%20m_%7Ba%7D%20%5Coverrightarrow%7Bv%7D_%7Boa%7D%20%2B%20m_%7Bb%7D%20%5Coverrightarrow%7Bv%7D_%7Boa%7D%20%3D%20m_%7Ba%7D%20%5Coverrightarrow%7Bv%7D_%7Bfa%7D%20%2B%20m_%7Bb%7D%20%5Coverrightarrow%7Bv%7D_%7Bfa%7D%20%5C%5C%20%5C%5C%20%28m%29%28i%29%2B%283m%29%280%29%3D%28m%29%28%20f_%7Ba%7D%29%2B%283m%29%28%20f_%7Bb%7D%29%20%5C%5C%20%5C%5C%20%28m%29%28i%29%3D%28m%29%28%20f_%7Ba%7D%29%2B%283m%29%28%20f_%7Bb%7D%29%20%5C%5C%20%5C%5C%20%28%20%5Cdiagup%5C%21%5C%21%5C%21%5C%21%5C%21m%29%28i%29%3D%28%5Cdiagup%5C%21%5C%21%5C%21%5C%21%5C%21m%29%28%20f_%7Ba%7D%29%2B%283%5Cdiagup%5C%21%5C%21%5C%21%5C%21%5C%21m%29%28%20f_%7Bb%7D%29%20%5C%5C%20%5C%5C%20i%3D%20f_%7Ba%7D%2B3%20f_%7Bb%7D%20%5C%5C%20%20%5C%5C%20%20f_%7Ba%7D%3Di-3f_%7Bb%7D%20%20%20%5C%20%5C%20%5C%20%281%29" />

Esa será la ecuación (1).

Por otro lado, si el choque es elástico significa que la energía cinética del sistema se conserva :

<img src="https://tex.z-dn.net/?f=E_%7Bko%7D%3D%20E_%7Bkf%7D%20%5C%5C%20%20%5C%5C%20%20%5Cdfrac%7B1%7D%7B2%7D%20m_%7Ba%7D%20v_%7Boa%7D%5E%7B2%7D%20%20%2B%20%5Cdfrac%7B1%7D%7B2%7D%20m_%7Bb%7D%20v_%7Bob%7D%5E%7B2%7D%3D%5Cdfrac%7B1%7D%7B2%7D%20m_%7Ba%7D%20v_%7Bfa%7D%5E%7B2%7D%2B%5Cdfrac%7B1%7D%7B2%7D%20m_%7Bb%7D%20v_%7Bfb%7D%5E%7B2%7D%20%5C%5C%20%20%5C%5C%20%20%5Cdiagup%5C%21%5C%21%5C%21%5C%21%5C%21%5C%21%5C%21%5Cdfrac%7B1%7D%7B2%7D%20m%20%5Ccdot%20i%5E%7B2%7D%3D%5Cdiagup%5C%21%5C%21%5C%21%5C%21%5C%21%5C%21%5C%21%5Cdfrac%7B1%7D%7B2%7D%20m%20%5Ccdot%20%20f_%7Ba%7D%20%5E%7B2%7D%2B%5Cdiagup%5C%21%5C%21%5C%21%5C%21%5C%21%5C%21%5C%21%5Cdfrac%7B3%7D%7B2%7D%20m%20%5Ccdot%20%20f_%7Bb%7D%20%5E%7B2%7D%20%5C%5C%20%20%5C%5C%20%20i%5E%7B2%7D%3D%20f_%7Ba%7D%20%5E%7B2%7D%2B%203%20f_%7Bb%7D%5E%7B2%7D%20%20%20%5C%20%20%5C%20%20%5C%20%282%29" />

Reemplazamos la ecuación (1) en la (2) :

<img src="https://tex.z-dn.net/?f=%20i%5E%7B2%7D%3D%28i-3f_%7Bb%7D%29%5E%7B2%7D%2B3%20f_%7Bb%7D%5E%7B2%7D%20%5C%5C%20%20%5C%5C%20%20%5Cdiagup%5C%21%5C%21%5C%21%5C%21%5C%21i%5E%7B2%7D%3D%20%5Cdiagup%5C%21%5C%21%5C%21%5C%21%5C%21i%5E%7B2%7D-6i%20f_%7Bb%7D%2B12f_%7Bb%7D%20%5E%7B2%7D%20%5C%5C%20%20%5C%5C12f_%7Bb%7D%20%5E%7B2%7D-6i%20f_%7Bb%7D%3D0%20%5C%5C%20%20%5C%5C%206f_%7Bb%7D%20%5E%7B2%7D-i%20f_%7Bb%7D%3D0%20%5C%5Cf_%7Bb%7D%286f_%7Bb%7D-i%29%3D0~%5CLongrightarrow%20%5C%20%5Cleft%5Clangle%5Cbegin%7Barray%7D%7Bc%7D%20%20f_%7Bb%7D%20%3D0%20%5C%5C%20%20%5C%5C%206%20f_%7Bb%7D%20-i%3D0%5CLongrightarrow%20%20%5C%20%5Cboxed%7Bf_b%3Di%2F6%7D%5Cend%7Barray%7D%5Cright%20%5C%5C%20%20%5C%5C%20%20%20%20%20%20%20%20%20%20" />

Reemplazamos este hecho en la ecuación (1) :

<img src="https://tex.z-dn.net/?f=%20f_%7Ba%7D%3Di-3%5Cleft%28%5C%20%5Cdfrac%7Bi%7D%7B6%7D%5Cright%29%20%5C%5C%20%20%5C%5C%20%20f_%7Ba%7D%3Di-%20%5Cdfrac%7Bi%7D%7B2%7D%20%5C%5C%20%20%5C%5C%20%20%5Ctherefore%20%20%5Cboxed%7Bf_%7Ba%7D%3D%20i%2F2%7D%20%20%20%20" />

Espero haberte ayudado, un saludo : ).