91Ó°ÊÓ

When understanding the motion of objects in physics, we often break down forces and velocities into vector components. Each vector has two essential parts: a direction and a magnitude. In the provided exercise, after the object explodes into three parts, the momentum of the first two parts are described using vectors: one with momentum

• For the x-direction: \( 2p \mathbf{i} \), indicating momentum along the x-axis.
• For the y-direction: \( p \mathbf{j} \), indicating momentum along the y-axis.

Vectors help represent quantities that have both direction and magnitude. Here, we can treat these parts as being on a grid, with one vector pointing horizontally and the other pointing vertically.
By using vector components, we can easily calculate the resultant effect, considering both direction and magnitude. This becomes crucial in determining the momentum of the third part in the system.

Explosive motion refers to when an object splits apart, resulting in multiple smaller pieces moving away from the original position. This is what happens when the stationary object in space explodes in the exercise.
The law of conservation of momentum plays a vital role here. Even though it might seem like things are moving chaotically, the total momentum among all parts is the same as it was before the explosion.

• In our exercise, initially, the object was at rest, meaning its momentum was zero.
• After the explosion, the parts move in different directions, but the total momentum of these three parts combined must still equate to zero.

This means that as these pieces fly apart, their momenta (plural for momentum) balance each other out to maintain that zero total momentum, keeping true to the conservation law.

Momentum is a quantity of motion an object possesses due to its mass and velocity. In conservation situations like the given problem, you can use that property to predict outcomes, such as motion and velocities post-events, like the given explosion.
In the exercise, calculating the momentum of the third part requires balancing the vector equation \[ 2p \mathbf{i} + p \mathbf{j} + p_3 \mathbf{v} = 0 \].
This involves using the following equations to find unknowns:

• For the x-component: \( 2p + p_{3x} = 0 \) yields \( p_{3x} = -2p \)
• For the y-component: \( p + p_{3y} = 0 \) yields \( p_{3y} = -p \)

With these components, use the Pythagorean theorem to find the magnitude of the third part's momentum, which is \( \sqrt{5}p \).
This captures the geometrical aspect of vectors, showcasing how different directions affect the resultant value. Calculating momentum this way helps to visualize the explosion’s result and ensures all pieces are correctly balanced in motion.