The reference frame, in which the centre of inertia of a given system of particles is at rest, translates with a velocity V relative to an inertial reference frame K. The mass of the system of particles equals m, and the total energy of the system in the frame of the centre of inertia is equal to `overset~E`. Find the total energy E of this system of particles in the reference frame K.

Explanation

To find the relationship between the values of the mechanical energy of a system in the K and C reference frames, let us begin with the kinetic energy T of the system. The velocity of the i-th particle in the K frame may be represented as `vecv_1=overset~vecv_i+vecv_c`. Now we can write
`T=sum1/2m_iv_i^2=sum 1/2m_i(overset~vecv_1+vecv_C)*(overset~vecv_i+vecv_C)`
`=sum1/2m_ioverset~v_i^2+vecv_Csum m_ioverset~vecv_1+sum 1/2m_iv_C^2`
Since in the C frame. `summ_ioverset~vecv_i=0`, the previous expression takes the form
`T=overset~T+1/2mv_C^2=overset~T+1/2mV^2` (since according to the problem `v_C=V`) (1)
Since the internal potential energy U of a system depends only on its configuration, the magnitude U is the same in all reference frames. Adding U to the left and right hand sides of Eq. (1), we obtain the sought relationship
`E=overset~E+1/2mV^2`