Applying conservation of liner momentum on a firework's explosion

Question

A firework of mass 1 kg is placed on the ground and ignited. The impulse created by the explosion causes it to move vertically upwards with an initial velocity of 50 m/s. After 4 seconds a 2nd explosion takes place in air and the horizontal impulse forces cause it to separate into 4 identical pieces.

2 seconds after the 2nd explosion these 4 pieces travel horizontally in the same plane that the 2nd explosion took place and reach the vertices of a square with a diagonal of 80 meters.

Question: Find the horizontal velocity imparted on a piece after the 2nd explosion.

I tried applying the law of conservation of linear momentum, but I can't get an answer. Can someone explain how I can solve this type of problem?

This is what I attempted:

$$P = m v\\ P_{initial}= P_{final}\\ 1\;\mathrm{kg} \times 50\;\mathrm{ m/s} = 1/4\;\mathrm{kg}\times (V_1 + V_2 + V_3+ V_4)$$

I can't figure out what to do next.

Answer 1

2 seconds after the 2nd explosion these 4 pieces travel horizontally in the same plane that the 2nd explosion took place and reach the vertices of a square of diagonal 80 meters

Question : Find the horizontal velocity imparted on a piece after the 2nd explosion

This text above is all you need. No need for momentum considerations.

• After 2 seconds, the pieces have moved 40 m away from the point of explosion.
• Let me repeat that: It takes 2 seconds to move 40 m for a piece.

And the speed is constant (no acceleration horizontally, since no horizontal forces). There you have the answer.