A steel ball of mass $2 m$ suffers one dimensional elastic collision with a row of three steel balls each of mass $m$ . If ball of mass $2 m$ has velocity $v$ initially and the three balls with small seperation were at rest. Then, after all possible collisions.

Ball

1

and ball of mass

2 m

would remain at rest

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Ball

2

moves with velocity

\frac{4 v_{0}}{9}

towards right

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Ball

3

moves towards right with velocity

\frac{4 v_{0}}{3}

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Ball of mass

2 m

moves towards right with velocity one fourth that of ball

1

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Solution

The correct option is D Ball of mass $2 m$ moves towards right with velocity one fourth that of ball $1$ .

When two masses collide elastically, velocities after collision is
$v_{1} = \frac{m_{1} - m_{2}}{m_{1} + m_{2}} u_{1} + \frac{2 m_{2}}{m_{1} + m_{2}} u_{2}$
$v_{2} = \frac{m_{2} - m_{1}}{m_{1} + m_{2}} u_{2} + \frac{2 m_{1}}{m_{1} + m_{2}} u_{1}$
When $2 m$ collides with ball $1,$
$m_{1} = 2 m m_{2} = m u_{1} = v_{0} u_{2} = 0$
$v_{1} = \frac{v_{0}}{3} & v_{2} = \frac{4 v_{0}}{3}$

when ball $1$ , collides elastically with ball $2,$ which inturn collides with $3$ and velocities gets interchanged as shown below.

Now $2 m$ again strikes with ball $1,$
which in turn with ball $2,$ then

$m_{1} = 2 m m_{2} = m u_{1} = \frac{v_{0}}{3} u_{2} = 0$
$v_{1} = \frac{v_{0}}{9} & v_{2} = \frac{4 v_{0}}{9}$
when ball $1$ , collides elastically with ball $2,$ velocity gets interchanged as shown below.

Now $2 m$ again strikes with ball $1,$
which in turn with ball $2,$ then

$m_{1} = 2 m m_{2} = m u_{1} = \frac{v_{0}}{9} u_{2} = 0$
$v_{1} = \frac{v_{0}}{27} & v_{2} = \frac{4 v_{0}}{27}$

No collision is possible after this, hence final velocities can be shown as,

Hence $b, c, d$ are correct options.

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A steel ball of mass 2m suffers one dimensional elastic collision with a row of three steel balls each of mass m. If ball of mass 2m has velocity v initially and the three balls with small seperation were at rest. Then, after all possible collisions.

A steel ball of mass $2 m$ suffers one dimensional elastic collision with a row of three steel balls each of mass $m$ . If ball of mass $2 m$ has velocity $v$ initially and the three balls with small seperation were at rest. Then, after all possible collisions.