Please help me understand this equation :
Total kinetic energy of two particle system = K.E. of system w.r.t. centre of mass + K.E. of centre of mass
Here's the derivation from my book :
The author starts off by deriving the individual momenta of the two particles w.r.t. the centre of mass, and later uses the formula given below to find their kinetic energies in the centre of mass frame. (Is that the correct way to put it?)
$$K = {p^2}/2m$$
However, when they add the two energies (please see the pictures), they get a value which seems to be half of the correct value (1/2 + 1/2 = 1 and not 1/2). This is one of my doubts.
Next, they find the kinetic energy of the centre of mass in the ground frame. This is pretty straightforward :
$$K_{com (ground frame)} = (1/2)({m_1} + {m_2}){(V_{com (ground frame)})}^2$$
Now comes the confusing part (my second doubt) :
The kinetic energy of a system of particles ($K$) is equal to the kinetic energy of the system relative to centre of mass ($K'$) plus kinetic energy of the centre of mass.
$K_{Total}$ = K.E. of system w.r.t. centre of mass + KE of centre of mass
(Again, please refer the pictures.)
They are almost treating kinetic energies like vectors and adding them up! How is that correct?
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I found a related question, but didn't understand the answer provided. :(
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A few solved examples related to this are provided in my textbook. Here's one of them :
Why have they used such a complicated formula to calculate the change in kinetic energy? Couldn't they have used a much simpler formula :
$$K = {1/2}mv^2 + {1/2}MV^2$$
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Any help is appreciated. 👍