Momentum & Energy: A Firecracker Explodes

[FredGirl13]

Homework Statement

A firecracker of mass 18g explodes into three pieces of equal mass. The first piece flies off at a speed 24m/s. The second piece flies at 18m/s at an angle 110 Degrees from the direction of the first piece. Find the speed and direction of the third piece.

Homework Equations

P=m∆v Ek=0.5mv^2 W=F∆dcos∅ W=∆Ek And I'm not really sure...

The Attempt at a Solution

Now I know that the momentums have to add up to zero. So mv1+mv2+mv3=0.

p1=(0.006)(24)=0.144kg m/s
x1=-0.1353 y1=-0.049251

p2=(0.006)(18)=0.108kg m/s
x2=0.101484 y2=0.0369384

0.144+0.108+0.006v3=0
v3=42m/s

I don't know if that's right but now I don't know how to find the direction?

V1y=-8.2085m/s V1x=-22.5526m/s

V2y=6.1564m/s V2x=16.914m/s

This is just for a study sheet, I don't hand it in. But the test is tomorrow so I really need help!

 

[haruspex]

It's a really good idea to work entirely with symbolic variables, only plugging in numbers at the final step. In the present case, you would discover the masses all cancel, so you don't need the 18g at all. More generally, it makes it easier to spot mistakes, to track down mistakes (by plugging in whatever convenient numbers you like at different stages), to avoid units conversion errors, to check dimensional consistency, and for others to follow the logic. E.g., I can't check your resolution of velocities into components without reverse engineering your calculation to figure out which axes you chose.
You will need to assign unknowns for the speed and direction for the third piece, then obtain your two momentum equations. Please post what you get.

 

[FredGirl13]

Umm, I tried this question yesterday, and looking at it now I don't actually know what I used to split those into components. And I can't seem to find a way from scratch :/

 

[haruspex]

Pick the trajectory of one of the first two pieces as 'angle zero' and specify all other angles from that base line. Anticlockwise, say.

 

[FredGirl13]

I did that, I made a x/y coordinate plane with velocity vector 1 as my zero but then I tried to make a right angle triangle and couldn't figure out any of the other angles in the triangle other than the 90 degree one

 

[haruspex]

Why a right-angled triangle?
You know the second piece goes at 110 degrees to the first, so you can draw that in. For the third, you know roughly which way it goes (it cannot be within 70 degrees of either of the other two), so draw that in and set its angle from vector 1 to be theta.
Now you should be able to resolve in two directions and write out the equations.

 

[FredGirl13]

I still don't get it. You're not really helping.

 

[haruspex]

There's a line missing from my previous post. Strange.
You've drawn X and Y axes. Let's say the first piece goes along the +ve X axis from the origin at 24m/s. Draw a line and arrow for that, labelling it with the speed. We'll measure angles anticlockwise from there. The second piece goes at 110 degrees at 18m/s. Draw a line, arrow and label for that. The third piece will go off somewhere in the third quadrant, probably, so draw a line and arrow for that. Mark in the angle as theta (to the positive x axis) and label it with the unknown speed V.
Now apply conservation of momentum in the x and y directions. Since all the masses are the same, you can omit those and just work with the speeds and angles.
Figure out the components of each speed in the x and y directions.
Please post all your working as far as you get, but not the diagram for now. We can come back to that if necessary.