# Momentum - explosions

1. Mar 14, 2004

### falcon0311

I'm having trouble solving this:

Vessel at rest explodes. There's 3 pieces. Two pieces, one with twice the mass of the other, fly off perpendicular to one another with the same speed of 31.4 m/s. The third piece has three times the mass of the lightest piece. Find the magnitude and direction of its velocity immediately after the explosion. (Specify direction by giving the angle from the line of travel of the least massive piece.)

I realize the masses can be considered m, 2m, and 3m, and m can be substituted for a mass like 10kg. My initial sketch looks something like this:

-------- m ( 31.4 m/s )
|
|
|
|
2m ( 31.4 m/s )

P = p1 + p2 + p3 = m(31.4m/s) + 2m(31.4m/s) + 3m(v3)

The third momentum vector I realize is not just the bisected angle plus 180. My mind is having trouble coming up with that vector. Any help in solving the problem is much appreciated.

I'm getting 23.4 m/s (approx) at 117 degrees from m.

-Jacob

Last edited: Mar 14, 2004
2. Mar 14, 2004

### HallsofIvy

Staff Emeritus
Trying to do vectors with trigonometry drives me up the wall!

What I would do is this: set up a coofdinate system so that the lightest piece moves along the positive x-axis. Since the second piece moves at right angles, we can take that to be the y-axis. Taking m to be the mass of the lightest piece, its momentum vector is
<34.1m, 0> and the momentum vector of the second piece is <0, (34.1)(2m)>= <0, 64.2m>. Their total momentum is <34.1m, 64.2m>. Since the total momentum of the system is 0, the momentum vector of the third piece is <-34.1m, -64.2m>. Since that third piece has mass 3m, its velocity vector is <-34.1/3, -64.2/3>= <-11.4,-22.7>.

The "length" of that last vector, i.e. the speed of the third piece, is &radic;((11.4)2+ (22.7)2)= 25.5 m/s. Its angle with the positive x-axis (i.e. relative to the motion of the first piece) is arctan(22.7/11.4)= 62.7 degrees except that it is in the third quadrant (both components are negative). The last piece moves at 62.7+ 90= 152.7 degrees measured clockwise from the motion of the first piece or
270-62.7= 207.3 degrees measured counterclockwise.