A vessel at rest explodes breaking into three pieces. Two pieces, having equal mass fly off perpendicular to one another with the same speed of 30 m/s. The third piece had three times the mass of each other piece. What are the magnitude and direction of its velocity immediately after the explosion?
What would be the correct way to set up the problem? I think to find the direction I would do the arc tan of (x/y). Be what would be the x and y? How would I find the magnitude of the third piece? THanks for helping! :D
The correct way is to break all momentums into vectors, in the two dimensional plane.
To solve this problem, we can consider the conservation of momentum. According to the law of conservation of momentum, the total momentum before the explosion is equal to the total momentum after the explosion.
Let's denote the velocity of the first piece as v1, the velocity of the second piece as v2, and the velocity of the third piece as v3. We are given that v1 and v2 have the same magnitude of 30 m/s.
Since there are no external forces acting on the system, the total momentum before the explosion is zero, as the vessel is at rest. Therefore, the sum of the momenta of the three pieces after the explosion must also be zero.
To set up the problem, we need to consider the components of momentum in the x and y directions separately. Let's assume that the x-axis is horizontal and the y-axis is vertical.
Let's assign the positive x-direction to the piece 1 and the positive y-direction to the piece 2.
We have:
m1 * v1x + m2 * v2x + m3 * v3x = 0 (in the x-direction)
m1 * v1y + m2 * v2y + m3 * v3y = 0 (in the y-direction)
Since v1 and v2 have the same magnitude, their x and y components will also have the same magnitude.
Let's assume m1 = m2 = m, and m3 = 3m (as given in the problem).
In the x-direction:
m * v1x + m * v2x + 3m * v3x = 0
Since v1x and v2x are in opposite directions, their sum is zero. Therefore:
3m * v3x = 0
Solving for v3x, we get:
v3x = 0
In the y-direction:
m * v1y + m * v2y + 3m * v3y = 0
Since v1y and v2y are perpendicular and have the same magnitude, their sum is zero. Therefore:
3m * v3y = 0
Solving for v3y, we also get:
v3y = 0
So, the velocity of the third piece after the explosion is v3 = v3x i + v3y j = 0 i + 0 j = 0.
The magnitude of the velocity of the third piece is |v3| = sqrt(v3x^2 + v3y^2) = sqrt(0^2 + 0^2) = 0. Therefore, the magnitude of the third piece's velocity is 0 m/s.
To find the direction, you are correct in using the arctan(y/x) formula. But since both x and y components of the third piece's velocity are zero, the arctan(y/x) cannot be determined. We can conclude that the third piece has no velocity in any specific direction; its velocity is zero.
Therefore, the magnitude of the third piece's velocity is 0 m/s, and its direction is undefined.