m 1 u 1 + m 2 u 2 = m 1 v 1 + m 2 v 2 = 0 m 1 u 1 2 + m 2 u 2 2 = m 1 v 1 2 + m 2 v 2 2 ( m 2 u 2 ) 2 2 m 1 + ( m 2 u 2 ) 2 2 m 2 = ( m 2 v 2 ) 2 2 m 1 + ( m 2 v 2 ) 2 2 m 2 ( m 1 + m 2 ) ( m 2 u 2 ) 2 = ( m 1 + m 2 ) ( m 2 v 2 ) 2 u 2 = − − v 2 ( m 1 u 1 ) 2 2 m 1 + ( m 1 u 1 ) 2 2 m 2 = ( m 1 v 1 ) 2 2 m 1 + ( m 1 v 1 ) 2 2 m 2 ( m 1 + m 2 ) ( m 1 u 1 ) 2 = ( m 1 + m 2 ) ( m 1 v 1 ) 2 u 1 = − − v 1 .