Homework Help: What can be the maximum velocity

1. The problem statement, all variables and given/known data
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?

2. Relevant equations
Conservation of Momentum
Conservation of Energy

3. The attempt at a solution
[itex]K.E._{initial}=\frac{1}{2} mv^{2}[/itex]

Let final velocities be [itex]v_{1}, v_{2} and v_{3}[/itex]

[itex]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] [/itex]

From question
[itex] K.E._{final}= η K.E._{initial} [/itex]

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

Also using conservation of momentum

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

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

 

Thanks. I got my answer correct.