v ⃗ 12 = v 1 ⃗ − v 2 ⃗ \vec v_{12}=\vec{v_1}-\vec{v_2} v 12 = v 1 − v 2
v ⃗ 23 = v 2 ⃗ − v 3 ⃗ \vec v_{23}=\vec{v_2}-\vec{v_3} v 23 = v 2 − v 3
v ⃗ 13 = v 1 ⃗ − v 3 ⃗ \vec v_{13}=\vec{v_1}-\vec{v_3} v 13 = v 1 − v 3
m 1 m 2 v ⃗ 12 2 + m 2 m 3 v ⃗ 23 2 + m 1 m 3 v ⃗ 13 2 = m_1m_2\vec v_{12}^2+m_2m_3\vec v_{23}^2+m_1m_3\vec v_{13}^2= m 1 m 2 v 12 2 + m 2 m 3 v 23 2 + m 1 m 3 v 13 2 =
m 1 m 2 ( v 1 ⃗ − v 2 ⃗ ) 2 + m 2 m 3 ( v 2 ⃗ − v 3 ⃗ ) 2 + m 1 m 3 ( v 1 ⃗ − v 3 ⃗ ) 2 = m_1m_2(\vec{v_1}-\vec{v_2})^2+m_2m_3(\vec{v_2}-\vec{v_3})^2+m_1m_3(\vec{v_1}-\vec{v_3})^2= m 1 m 2 ( v 1 − v 2 ) 2 + m 2 m 3 ( v 2 − v 3 ) 2 + m 1 m 3 ( v 1 − v 3 ) 2 =
(After expanding the brackets and grouping the terms ) \text{(After expanding the brackets and grouping the terms )} (After expanding the brackets and grouping the terms )
( m 1 v ⃗ 1 2 + m 2 v ⃗ 2 2 + m 3 v ⃗ 3 2 ) ( m 1 + m 2 + m 3 ) − (m_1\vec v_1^2+m_2\vec v_2^2+m_3\vec v_3^2)(m_1+m_2+m_3)- ( m 1 v 1 2 + m 2 v 2 2 + m 3 v 3 2 ) ( m 1 + m 2 + m 3 ) −
( m 1 v ⃗ 1 + m 2 v ⃗ 2 + m 3 v ⃗ 3 ) 2 ( 1 ) (m_1\vec v_1+m_2\vec v_2+m_3\vec v_3)^2\ (1) ( m 1 v 1 + m 2 v 2 + m 3 v 3 ) 2 ( 1 )
Divide the resulting expression (1) by ( m 1 + m 2 + m 3 ) \text{Divide the resulting expression (1) by }(m_1+m_2+m_3) Divide the resulting expression (1) by ( m 1 + m 2 + m 3 )
We also take into account the fact that: \text{We also take into account the fact that:} We also take into account the fact that:
T = m 1 v ⃗ 1 2 2 + m 2 v ⃗ 2 2 2 + m 3 v ⃗ 3 2 2 T= \frac{m_1\vec v_1^2}{2}+ \frac{m_2\vec v_2^2}{2}+ \frac{m_3\vec v_3^2}{2} T = 2 m 1 v 1 2 + 2 m 2 v 2 2 + 2 m 3 v 3 2
( m 1 v ⃗ 1 2 + m 2 v ⃗ 2 2 + m 3 v ⃗ 3 2 ) = 2 T (m_1\vec v_1^2+m_2\vec v_2^2+m_3\vec v_3^2)= 2T ( m 1 v 1 2 + m 2 v 2 2 + m 3 v 3 2 ) = 2 T
where T − total kinetic energy of the system \text{where }T - \text{total kinetic energy of the system} where T − total kinetic energy of the system
( m 1 v ⃗ 1 + m 2 v ⃗ 2 + m 3 v ⃗ 3 ) 2 (m_1\vec v_1+m_2\vec v_2+m_3\vec v_3)^2 ( m 1 v 1 + m 2 v 2 + m 3 v 3 ) 2
P = m 1 v ⃗ 1 + m 2 v ⃗ 2 + m 3 v ⃗ 3 P = m_1\vec v_1+m_2\vec v_2+m_3\vec v_3 P = m 1 v 1 + m 2 v 2 + m 3 v 3
where P center of gravity impulse \text{where } P\text{ center of gravity impulse} where P center of gravity impulse
T c = 1 2 ( m 1 + m 2 + m 3 ) v c 2 T_c =\frac{1}{2}(m_1+m_2+m_3)v_c^2 T c = 2 1 ( m 1 + m 2 + m 3 ) v c 2
T c − kinetic energy of the center of gravity T_c -\text{kinetic energy of the center of gravity} T c − kinetic energy of the center of gravity
P = ( m 1 + m 2 + m 3 ) v c P =(m_1+m_2+m_3)v_c P = ( m 1 + m 2 + m 3 ) v c
( m 1 v ⃗ 1 + m 2 v ⃗ 2 + m 3 v ⃗ 3 ) 2 = 2 ( m 1 + m 2 + m 3 ) T c (m_1\vec v_1+m_2\vec v_2+m_3\vec v_3)^2= 2(m_1+m_2+m_3)T_c ( m 1 v 1 + m 2 v 2 + m 3 v 3 ) 2 = 2 ( m 1 + m 2 + m 3 ) T c
m 1 m 2 v ⃗ 12 2 + m 2 m 3 v ⃗ 23 2 + m 1 m 3 v ⃗ 13 2 m 1 + m 2 + m 3 = 2 T − 2 T c \frac{m_1m_2\vec v_{12}^2+m_2m_3\vec v_{23}^2+m_1m_3\vec v_{13}^2}{m_1+m_2+m3}=2T-2T_c m 1 + m 2 + m 3 m 1 m 2 v 12 2 + m 2 m 3 v 23 2 + m 1 m 3 v 13 2 = 2 T − 2 T c
by Koenig’s theorem \text {by Koenig's theorem} by Koenig’s theorem
T = T c + T r T = T_c+T_r T = T c + T r
T r − relative kinetic energy of the system T_r -\text{ relative kinetic energy of the system} T r − relative kinetic energy of the system
m 1 m 2 v ⃗ 12 2 + m 2 m 3 v ⃗ 23 2 + m 1 m 3 v ⃗ 13 2 m 1 + m 2 + m 3 = 2 T r \frac{m_1m_2\vec v_{12}^2+m_2m_3\vec v_{23}^2+m_1m_3\vec v_{13}^2}{m_1+m_2+m3}=2T_r m 1 + m 2 + m 3 m 1 m 2 v 12 2 + m 2 m 3 v 23 2 + m 1 m 3 v 13 2 = 2 T r
hence the hypothesis is false \text{hence the hypothesis is false} hence the hypothesis is false
Answer: the proposed proof formula is false \text{Answer: the proposed proof formula is false} Answer: the proposed proof formula is false
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