We consider the reaction , in which X is taken to be at rest in the laboratory reference frame. The initial kinetic energy in this frame is

We have to show (a) that the initial velocity of the centre of mass is of the system in the laboratory frame is

We have to answer whether this quantity is changed by the reaction. (b) We have to show that the initial kinetic energy, viewed now in a reference attached to the centre of mass of the two particles, is given by

We have to answer whether this quantity is changed by the reaction. (c) In the reaction the kinetic energy of the deuteron, measured in the laboratory frame, is 15.9 MeV. We have to find We may ignore the small relativistic effects.

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(a)

The centre of mass is defined by the equation

Therefore, the velocity of the centre of mass will be

The velocity of the centre of mass is a property of the frame of reference and does not change by the reaction.

(b)

In the centre of mass frame, the velocity of the particle a will be

and that of the particle X will be

Therefore, the kinetic energy in the centre of mass frame will be

If , the will be changed by the reaction.

(c)

In the reaction the kinetic energy of the deuteron, measured in the laboratory frame, is 15.9 MeV.

We have to find

The velocity of the deuteron in the lab frame will be

And, the velocity of the centre of mass will be

The kinetic energy in the centre of mass frame will be