# What can be the maximum velocity

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## Homework Statement

A shell flying with velocity v = 500 m/s bursts into three identical fragments so that the kinetic energy of the system increases η = 1.5 times. What maximum velocity can one of the fragments obtain?

## Homework Equations

Conservation of Momentum
Conservation of Energy

## The Attempt at a Solution

$K.E._{initial}=\frac{1}{2} mv^{2}$

Let final velocities be $v_{1}, v_{2} and v_{3}$

$K.E._{final}=\frac{1}{2} \left[ \frac{m}{3} v_{1}^{2}+ \frac{m}{3} v_{2}^{2}+ \frac{m}{3} v_{3}^{2}\right]$

From question
$K.E._{final}= η K.E._{initial}$

Putting the values of K.E. I get a relation
$=3v^{2}η = v_{1}^{2}+ v_{2}^{2}+ v_{3}^{2}$

Also using conservation of momentum

$mv = \frac{m}{3}\left(v_{1}+v_{2}+v_{3}\right)$
which simplifies to
$3v = (v_{1}+v_{2}+v_{3})$

But now I'm stuck here. What to do next to find the maximum velocity?