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Standard XII

Physics

COE in Elastic Collision

Mass m 1 hi...

Question

Mass $m_{1}$ hits and sticks with $m_{2}$ while sliding horizontally with velocity $v$ along the common line of centres of the three equal masses ( $m_{1} = m_{2} = m_{3} = m$ ). Initially masses $m_{2}$ and $m_{3}$ are stationary and the spring is unstretched. The maximum compression of the spring is $\sqrt{m / n k} v$ . Find $n$ .

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Solution

After the mass $m 1$ hits $m 2$ , $m 1$ sticks to $m 2$ and will have velocity $v / 2$

Consider the case of maximum compression

At that time the spring will be fully compressed and both the masses will be moving with $e q u a l v e l o c i t y$

As total momentum should be conserved after collision

$2 m v / 2 = (2 m + m) v 1$

$v 1 = 1 / 3 v$

At the time of maximum compression total energy remains conserved after collision

Hence,

$\frac{2 m {v / 2}^{2}}{2} = \frac{k x^{2}}{2} + \frac{3 m {(v / 3)}^{2}}{2}$

$k x^{2} = m v^{2} / 6$

$x = \sqrt{m / 6 k} v$
$∴ n = 6$

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Mass m1 hits and sticks with m2 while sliding horizontally with velocity v along the common line of centres of the three equal masses (m1=m2=m3=m). Initially masses m2 and m3 are stationary and the spring is unstretched. The maximum compression of the spring is √m/nkv. Find n.