The correct option is A (e+1)N−12N−1u
Step 1: Find the velocity after first collision.
After the first collision, velocity of first ball becomes u.
Using conservation of momentum
mu=mv1+mv2
u=v1+v2 …(i)
e=−(v1−v2)u
e=−(u−v2−v2)u
eu=2v2−u
v2=(e+1)u2
Step 2: Find the velocity after second collision.
Conservation of momentum
mv2=mv′2+mv3
v2=v′2+v3
v′2=v2−v3
e=−(v′2−v3)v2
ev2=−(v2−v3−v3)
ev2+v2=2v3
v3=v2(e+1)2
v3=(e+1)2 (e+1)2u
v3=(e+1)222u
Step 3: Find the velocity after Nth collision.
v3=(e+1)222u
Similarly,
v4=(e+1)323u
vn=(e+1)N−12N−1u