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Let a stationary block explodes into 3 equal pieces .The speeds are also equal which is $v$. If I am asked to determine the angle between the broken pieces that are moving then how to deal with this problem? Because the system must obey the conservation of momentum, can the velocity vectors be arranged by a equilateral triangle?

If the speed or mass varies then what changes of the angles will happen between them? Is there any unique way to solve this type of problem?

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Momentum triangle.

Whether there is a "unique" way to solve this, I'm not sure but assuming that the centre of gravity after the explosion hasn't moved with respect to before the explosion and the other conditions you imposed hold, then a triangular arrangement of the momentum vectors is the only possibility.

For example, with $\alpha = 120 \text { degrees}$, the balance of momenta is on the vertical direction:

$m_1v_1=m_2v_2\cos60 + m_3v_3\cos60$, with $\cos60=0.5$ and $m_1=m_2=m_3$ and $v_1=v_2=v_3$ then this is always true.

It's also true for the projections of momenta on both other axis of symmetry.

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The three momentum vectors of the pieces must form the sides of a triangle so that the total momentum is conserved (all forces are internal to the system, so no momentum is added or removed). The orientation of the triangle will not be unique in space, but the lengths of the sides will uniquely determine the relative angles.

If the masses or speeds vary, that changes the lengths of the vectors. You must still form a triangle.

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