Question

A grenade having mass 10 kg flying horizontally with a velocity of 10m/s explodes into 2 fragments. The larger fragment has the velocity of 25 m/s and smaller fragment has the velocity of 12.5 m/s. What are the masses of the fragments.

Answer 1

Let the larger mass be x kg.
So the smaller mass= 10-x kg
Linear momentum of the initial system = 100 kgm/s
Linear Momentum of final system = 25x+ 12.5(10-x) = 12.5x+125kgm/s
Since no external force is applied, so the linear momentum remains constant.
So, 100= 12.5+25x
87.5 = 25x
x = 3.2

So larger mass = 10-3.2 kg
= 6.8 kg
But I think the larger mass will have smaller velocity, that is what I am getting.

Answer 2

 I guess you can use conservation of momentum in this case. Consider that the explosion gases expand in all directions, so no momentum is added to the system from the explosion. 

Momentum before explosion = (10 kg)(10 m/s) = 100 kg*m/s 
Let x = mass of larger piece, and y = mass of smaller piece. 

Momentum after = (x)(25 m/s) - (y)(12.5 m/s) = 100 kg*m/s {momentum is conserved} they are subtracted because in opposite directions. 

The other is conservation of mass {here we are assuming that only a small portion of mass of explosives turn to gas} 

x + y = 10 kg 
We have two equations and two unknowns. 
25x - 12.5y = 100 kg {I divided through by 1 m/s} and x + y = 10 kg 

Solve with your favorite method and find that x = 6 kg, y = 4 kg 

So if my assumptions hold true, then the larger 25 m/s piece is 6 kg, and the smaller piece is 4 kg (traveling backward at 12.5 m/s)