80
2 Particle Dynamics
where u and v with appropriate subscripts refer to the initial and final
velocities.
The negative sign shows that the ball 1 moves toward left after the collision
and hits ball 3. After the second collision with ball 3, ball 2 acquires a velocity
v 2 and moves toward right
v
1 =
2m 3 u 3 + υ 1 (m 1 − m 3 )
m 1 + m 3
=
0 − u 1 (m 1 − 4m)
m + 4m
= 0.36u 1
(3)
But v
1 < v 2 . Therefore ball 1 will not undergo the third collision with ball 2.
Thus in all there will be only two collisions.
2.31 Momentum conservation gives m 1 u 1 + m 2 u 2 = m 2 v 2
(1)
Conservation of kinetic energy in elastic collision gives
1
2
m 1 u
2
1 +
1
2
m 2 u
2
2 =
1
2
m 2 v
2
2
(2)
By the problem
1
2
m 1 u
2
1
=
1
2
m 2 u
2
2
(3)
u 2 = αu 1
(4)
∴ α
2
=
m 1
m 2
(5)
Using (3) in (2)
m 2 u
2
2 =
1
2
m 2 v
2
2
∴ v 2 =
√
2u 2
(6)
Using (4) and (6) in (1)
m 1 u 1 + m 2 αu 1 =
√
2m 2 u 2 =
√
2m 2 αu 1
or
m 1 + m 2 α =
√
2αm 2
Dividing by m 2 and using (5) and rearranging
α
α −
√
2 − 1
= 0
since α = 0, α =
√
2 − 1
∴ α =
u 2
u 1
=
√
2 − 1
∴
u 1
u 2
=
1
√
2 − 1
=
√
2 + 1