Question #233159

If m1,m2 and m3 be the masses of three particles v12, v23 and v13 be their relative velocities, prove that the total kinetic energy (E) of the system about the centre of mass is given by

m1m2v212+ m2m3v223+ m1m3v213 / m1+ m2 + m3

1
Expert's answer
2021-09-04T15:19:31-0400

v12=v1v2\vec v_{12}=\vec{v_1}-\vec{v_2}

v23=v2v3\vec v_{23}=\vec{v_2}-\vec{v_3}

v13=v1v3\vec v_{13}=\vec{v_1}-\vec{v_3}


m1m2v122+m2m3v232+m1m3v132=m_1m_2\vec v_{12}^2+m_2m_3\vec v_{23}^2+m_1m_3\vec v_{13}^2=

m1m2(v1v2)2+m2m3(v2v3)2+m1m3(v1v3)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=

(After expanding the brackets and grouping the terms )\text{(After expanding the brackets and grouping the terms )}

(m1v12+m2v22+m3v32)(m1+m2+m3)(m_1\vec v_1^2+m_2\vec v_2^2+m_3\vec v_3^2)(m_1+m_2+m_3)-

(m1v1+m2v2+m3v3)2 (1)(m_1\vec v_1+m_2\vec v_2+m_3\vec v_3)^2\ (1)

Divide the resulting expression (1) by (m1+m2+m3)\text{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:}

T=m1v122+m2v222+m3v322T= \frac{m_1\vec v_1^2}{2}+ \frac{m_2\vec v_2^2}{2}+ \frac{m_3\vec v_3^2}{2}

(m1v12+m2v22+m3v32)=2T(m_1\vec v_1^2+m_2\vec v_2^2+m_3\vec v_3^2)= 2T

where Ttotal kinetic energy of the system\text{where }T - \text{total kinetic energy of the system}


(m1v1+m2v2+m3v3)2(m_1\vec v_1+m_2\vec v_2+m_3\vec v_3)^2

P=m1v1+m2v2+m3v3P = m_1\vec v_1+m_2\vec v_2+m_3\vec v_3

where P center of gravity impulse\text{where } P\text{ center of gravity impulse}

Tc=12(m1+m2+m3)vc2T_c =\frac{1}{2}(m_1+m_2+m_3)v_c^2

Tckinetic energy of the center of gravityT_c -\text{kinetic energy of the center of gravity}

P=(m1+m2+m3)vcP =(m_1+m_2+m_3)v_c

(m1v1+m2v2+m3v3)2=2(m1+m2+m3)Tc(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


m1m2v122+m2m3v232+m1m3v132m1+m2+m3=2T2Tc\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

by Koenig’s theorem\text {by Koenig's theorem}

T=Tc+TrT = T_c+T_r

Tr relative kinetic energy of the systemT_r -\text{ relative kinetic energy of the system}


m1m2v122+m2m3v232+m1m3v132m1+m2+m3=2Tr\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

hence the hypothesis is false\text{hence the hypothesis is false}


Answer: the proposed proof formula is false\text{Answer: the proposed proof formula is false}



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