According to law of conservation of linear momentum, we get
m→vi+m×0=m→vpf+m→vQf
where →vpf and →vQf are the final velocities of spheres P and Q respectively.
→vi=−→vpf+−−→vQf
(→vi.→vi)=(→vpf+→vQf).(→vpf+→vQf)
=→vpf.→vpf+→vQf.→vQf+2→vpf.→vQf
or v2i=v2pf+v2Qf+2vpfvQfcos θ (1)
According to conservation of kientic energy, we get
12mv2i=12mv2pf+12mv2Qf
v2i=v2pf+v2Qf
Comparing Eqs. (1) and (2), we get cos θ=0⇒θ=90∘