A vessel at rest at the origin of an xy coordinate system explodes into three pi

Question

A vessel at rest at the origin of an xy coordinate system explodes into three pieces. Just after the explosion, one piece, of mass m, moves with velocity v in the negative x direction, and a second piece, also of mass m, moves with the same velocity in the negative y direction. The third piece has mass 3m. Just after the explosion, (a) what is the magnitude of the velocity of the third piece in terms of the given variables? (b) What is the direction of its motion (as an angle relative to the NEGATIVE x axis)?

Explanation / Answer

Using conservation of momentum, the x-component of the 3rd peice can be found by
0 = -V*m + 0*m + 3m*Vx ==> V = 3Vx ==> Vx = V/3.
Likewise, for the y component, we get
0 = 0*m - V*m + 3m*Vy ==> Vy = V/3.
Therefore the magnitude of the velocity is
|V| = sqrt(2)*(V/3) m/s
and the direction is

Vy = Vx
theta = arctan(Vy/Vx) = 45 degrees.

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