Work and Energy in Different Reference Frames.pdf
The main result I’ll focus on is the work and energy in different inertial frame
Newton’s second law for inertial and noninertial (nonrotating) reference frames:
$$ m_j\frac{d\bold v'_j}{dt}=\bold F_j-m_j\bold A(t) $$
Where $\bold F_j$ is the net real force on the $j$th particle, $\bold v_j'=\bold v-\bold V(t)$, $\bold V(t)=\bold V_0+\int_0^tA(t')dt'$, $\bold F_{fict}=-m_j\bold A(t)$, where $\bold V_0$ is the initial value of $\bold V$. If we dot each side with $d\bold r'_j$ on each side we obtain the work-energy theorem: $dK'=dW'$ and if we expand on the terms we get the general result for inertial frames frame $\Sigma$ and $\Sigma'$ that moves a velocity $\bold V$ with respect to $\Sigma$ for the $jth$ particle:
$$ dK'=dK_j + dZ_j\\ dW'= dW_j + dZ_j\\ ~\\ dZ_j=-\bold V(t)\cdot\bold F_jdt=-\bold V\cdot d\bold P_j $$
Summing over $j$ we obtain:
$$ dK'=dK-\bold V\cdot d\bold P\\ dW'=dK-\bold V\cdot d\bold P $$
where $dW$$'$ and $dK'$ indicates the work and kinetic energy in the $\Sigma'$ frame and $dW$ and $dK$ the work and KE in the $\Sigma$ frame
Elastic collsions in addition to momentum conseravtion, conserve kinetic energy: $T_i=T_f$
In the case that two particles have equal speeds, $v_0$, and opposite velocities then the final velocities $v_m,v_M$ are given by
$$ v_m=\frac{3M-m}{M+m}v_0\\ v_M=\frac{M-3m}{M+m}v_0 $$
In general if the initial speeds are of different magnitudes, the relative difference between initial velocities is equal and opposite to the relative difference to final velocities
$$ \bold v_1-\bold v_2=-(\bold v_1'-\bold v_2') $$
If a mass moves in a circular orbit in the field of an attractive central force with potentials of $U=k r^n$, then $T=\frac{nU}{2}$
The Virial Theorem is much more general than this, but this is one important special case.