Answer
The speed of the third piece is $170~m/s$
Work Step by Step
Just before the firecracker explodes, the speed is zero, and so the initial momentum is zero. By conservation of momentum, the final momentum is also zero.
Let $M$ be the mass of each of the three pieces. We can find the magnitude of the vector sum of the momentum of the two pieces moving with a speed of $120~m/s$ at right angles to each other.
$\sqrt{[(120~m/s)~M]^2+[(120~m/s)~M]^2} = (170~m/s)~M$
For the vector sum of the momentum of all three pieces to equal zero, the third piece must of a momentum of $(170~m/s)~M$ in the opposite direction. We can find the speed of the third piece:
$Mv = (170~m/s)~M$
$v = 170~m/s$
The speed of the third piece is $170~m/s$.
