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A sphere P of mass m and velocity vi und...

A sphere P of mass m and velocity `v_i` undergoes an oblique and perfectly elastic collision with an identical sphere Q initially at rest. The angle `theta` between the velocities of the spheres after the collision shall be

A

0

B

`45^@`

C

`90^@`

D

`180^@`

Text Solution

Verified by Experts

The correct Answer is:
C
According to law of conservation of linear momentum , we get
`mvecv_i+mxx0=mvecv_(Pf)=mvecv_(Qf)`
where `vecv_(Pf) and vecv_(Qf)` are the final velocities of spheres P and Q after collision respectively.
`vecv_i=vecv_(Pf)+vev_(Qf)`
`(vecv_i.vecv_i)=(vecv_(Pf)+vecv_(Qf)).(vecv_(Pf)+vecv_(Qf))`
`vecv_(Pf). vecv_(Pf)+vecv_(Qf). vecv_(Qf)+2vecv_(Pf). vecv_(Qf)`
or `v_i^2=v_(Pf)^2+v_(Qf)^2 + 2v_(Pf) v_(Qf) cos theta`...(i)
According to conservation of kinetic energy , we get `1/2mv_i^2 =1/2mv_(Pf)^2 + 1/2mv_(Qf)^2 implies v_i^2=v_(Pf)^2 +v_(Qf)^2`...(ii)
Comparing (i) and (ii), we get
`2v_(Pf)v_(qf)cos theta=0 implies cos theta =0 or theta=90^@`
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