How Is the Third Piece's Velocity Determined After a Grenade Explosion?

[tornzaer]

Homework Statement

A grenade of mass 1.2 kg is at rest on a smooth frictionless surface when it suddenly explodes into three pieces. A 0.5 kg piece flies off horizontally to the north at 3.0 m/s, a 0.3 kg piece flies off horizontally to the southwest at 4.0 m/s. What is the speed and direction of the third piece?

Homework Equations

P=mv P123=P1+P2+P3 Cosine Law

The Attempt at a Solution

Well, I know the mass and velocity of the first two pieces. I know the mass of the third piece due to the remaining mass from the total and it comes up to 0.4 kg. I know the momentum of the first two pieces. I know I have to draw a diagram and find the momentum of the unknown side, which would be P1+P2. Then, I can solve for v3 by dividing m3 from P1+P2. The problem, however is the direction. Since the second piece flies southwest, I'm assuming the angle is 45 degrees. That leaves me with two unknown angles. I can find this by using the Sine Law, but I don't know which angle to use.

This is where I need help, what is the direction of the third piece? Could someone please show me the diagram and which angle would be it for the third piece. Just as a side note, the velocity of the third piece comes up to a negative number, so does that mean that the direction will be switched? Thank you very much for the help. This is for my exam that's tomorrow.

 

[Delphi51]

The momentum before the explosion is zero - in the east/west direction AND in the north/south direction. Make two headings for the two directions and then write
0 = p1 + p2 + p3 for each. Put in the known quantities, using 4*cos(45) for the east/west part of the velocity of the 2nd piece. I expect you will have two equations with two unknowns - the speed and angle of the 3rd piece. Solve the system of equations to find them.

 

[tornzaer]

Hmmm... Is there a way to solve it by drawing a single triangle? Then I could use the cosine law to find the one side and the sine law from thereon to find the angle of the direction I need. This is how we've done it thus far in class. It's just that I don't know which of the two blank angles is the directional angle. I could use sine law to find either one, just don't know which one.

Thanks.

 

[LowlyPion]

It's much easier to just sum the X-components and Y-components as already suggested.

You end with the numerical values of the x,y of the third piece directly. Then combine them for the |magnitude| and use the tan^-1 to get the angle directly.