It is given that collision between the protons is elastic. We know that for such collisions both momentum and kinetic energy are conserved.
The total system kinetic energy before the collision equals to the total system kinetic energy after the collision.
Also, in a perfectly elastic collision coefficient of restitution is
Let two protons be designated as
Momentum of system before collision
Momentum of system after collision
By Law of conservation of momentum we have
Since it is proton-proton collision, mass of both is same. As such the above expression reduces in terms of velocities
From Law of Conservation of Energy we get
As mass of both protons is identical kinetic energy is shared equally between the two after the collision.
Inserting in(2) we get
Inserting given value