When two bodies of masses m and M respectively are connected by a light string. masses m & M are moving with velocities v & u in opposite direction respectively. maximum compression of spring is

Dear Arshad,

As the masses m and M connected by a light spring are moving towards each other, the spring keeps on compressing as long as they move towards each other. Hence the compression keeps on increasing as long as the  masses move towards each other.

This means that, when the two masses come to rest the spring no longer undergoes any further compression hence it reached its maximum compression when both the masses come to rest.

using conservation of energy, ( as there isn't any non conservative force ), initially the energy of the masses plus spring system is,

E initial  = 1/2 mv² + 1/2Mu²

Finally the Kinetic energies are zero and they are converted to spring potential energy

E final = 1/2 k_Xmax² where X_max is the maximum compression in the spring

E initial = E final from conservation of energy

from this equation we can find maximum compression of the spring, if variables m,M,v,u and k are given.

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