We consider the reaction , in which X is taken to be at rest in the laboratory reference frame. In an endothermic reaction , the interacting particles and must have a kinetic energy, measured in the centre-of-mass frame, of at least if the reaction is to “go”. We have to show using the result

that the threshold energy for the particle , measured in the laboratory frame, is

We have to answer whether it is reasonable that should be greater than .

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We note that

In the centre-of-mass frame the sum of the momentum of the particles and is zero. So for the endothermic reaction to “go” through the total energy in the centre-of-mass frame has to be such that

.

Or

The required minimum kinetic energy of the incident particle will therefore be

It is reasonable that should be greater than , because in the centre-mass-frame in addition to particles and their centre-of-mass is also moving with respect to the laboratory frame and carries kinetic energy.