Mechanics doubt

All statements are true, but I don’t understand how to prove statement 1.

Initial KE = 1/2(m*u^2)

Final KE = PE at compression = ?

Can someone help me in finding PE at compression (in terms of u and θ).

They have asked at maximum compression, so take both the balls velocity to be same and then apply various equation, i think you might then get it… ( i don’t have any theoretical reason for the velocity thing, but we use the same thing in spring block collisions)…

When PE will be maximum, at that instant the relative velocities of the balls in the direction of collision (AB) will be zero.
conservation of momentum along that direction will give there velocities as (ucos(theta))/2 .
now you can use conservation of energy to find PE(max) at that instant

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I see, but why did you take the velocity component as ucosθ/2

Could you show your working for that?

before collision the velocity of first ball is ucos(theta) in AB direction…during collision when there will be maximum compression both will have equal velocities (relative velocity zero)(as we do for spring block collisions)
By conservation of momentum(along AB) we’ll get the velocity at that instant

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Thanks, I understand now.