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When a vessel explodes, it affects the motion of its fragments. According to the conservation of momentum principle, the total momentum before and after the explosion must remain the same, as long as there are no external forces acting on the system. In this case, the vessel starts off at rest, indicating its initial momentum is zero.

Now, let's think about what happens after the explosion. The vessel breaks into three pieces, each moving in different directions. Because momentum is conserved, the total momentum of these pieces combined must also equal zero. This principle helps us figure out missing data, like the speed or direction of one of the pieces.

When the pieces move, they create momentum in different directions. Hence, analyzing the individual momentum components along the x-axis and y-axis helps us understand how the entire scenario breaks down. Remember that each momentum vector includes a mass and a velocity component.

Velocity is a vector quantity, meaning it has both magnitude and direction. To solve problems involving vectors, it can be useful to decompose them into their components. In cases like explosions where parts move in different perpendicular directions, calculating the x and y components is critical.

For a piece with a certain velocity, its x and y components describe its motion along those axes. With vectors, you use trigonometry to break apart and then analyze these components. In our example, two pieces move solely along the x and y axes, which makes the math straightforward. The third piece's velocity has to be understood from the conservation law applied to these axes.

We identified one piece had a velocity component at \(-30 \, \mathrm{m/s}\) on the x-axis, while another had \(-30 \, \mathrm{m/s}\) on the y-axis. For the third piece to satisfy the momentum conservation, its motion adds up the resulting effect of the other pieces to zero.

The magnitude of a velocity vector indicates the speed of an object in its path. To find this, use the Pythagorean theorem. It applies when you have two perpendicular velocity components—as seen in this case with the x and y components of the third piece.

The formula \[ v_3 = \sqrt{v_{3x}^2 + v_{3y}^2} \] gives the overall speed. For our third piece, substituting the previously found velocities \( v_{3x} = 10 \, \mathrm{m/s}\) and \( v_{3y} = 10 \, \mathrm{m/s}\) leads us to a magnitude of roughly \( 14.14 \, \mathrm{m/s}\).

This magnitude offers a single number representing the energy transfer through motion from the explosion. It's how fast the third piece is moving onward once it has left the explosion site fully.

Aside from speed, knowing the direction of an object's movement is crucial. For that, we use angles to provide insight into exactly where an item is heading.

In physics, the direction angle tells how far off from a reference axis (usually the x-axis) the vector is pointing. Typically, you calculate this using the tangent function, \[ \theta = \tan^{-1} \left( \frac{v_{3y}}{v_{3x}} \right) \].

This inverse tangent function helps us find \( \theta \), ensuring we understand the angle relative to the axes. With velocity components \( v_{3x} = 10 \, \mathrm{m/s}\) and \( v_{3y} = 10 \, \mathrm{m/s}\), the angle came out to be \( 45^\circ \).

The positive angle here indicates the direction lies between the positive x- and y-axes, moving diagonally away from the explosion point and continuing through that designated path.