Question

A body x with a momentum p collides with another identical stationary body y one dimensionally. During the collision y gives an impulse J to the body x. Then, the coefficient of restitution is:

• A. $ \frac{2j}{p}-1 $
• B. $ \frac{j}{p}+1 $
• C. $ \frac{j}{p}-1 $
• D. $ \frac{j}{2p}-1 $

Answer 1

Answer: (B)

Coefficient of restitution $ e=\frac{{{v}_{2}}-{{v}_{1}}}{{{u}_{1}}-{{u}_{2}}} $ Here both the bodies are identical i.e., have the same mass, So, $ e=\frac{m{{v}_{2}}-m{{v}_{1}}}{m{{v}_{1}}-{{\mu }_{2}}} $ $ =\frac{{{P}_{2}}-{{P}_{1}}}{{{p}_{1}}-{{p}_{2}}} $ $ {{p}_{1}}=p $ (Intial momentum of first body $ {{p}_{2}}= $ Initial momentum of second body) = 0 ( $ \because $ Final momentum $ {{p}_{2}}=p+J $ ) ( $ \therefore $ Impulse = change in momentum) $ {{p}_{1}}=0 $ ( $ \because $ When two bodies of equal masses collide elastically then thy exchange then velocities) $ \therefore $ $ e=\frac{p+J}{p} $ $ =1+\frac{J}{p} $