91Ó°ÊÓ

Momentum is a fundamental concept in physics. It represents the quantity of motion an object has and is given by the formula \( p = mv \), where \( p \) stands for momentum, \( m \) for mass, and \( v \) for velocity. In the given exercise, we observe two objects with their respective mass and velocity values. The first object has a mass of 3.00 kg moving at 5.00 m/s in the \( i \) direction, resulting in a momentum of \( 15.00 \, \text{i} \, \text{kgm/s} \). The second object, with a mass of 2.00 kg, is moving at -3.00 m/s in the \( j \) direction, giving it a momentum of \( -6.00 \, \text{j} \, \text{kgm/s} \). The direction in which the object moves is crucial here, as momentum is a vector quantity. Thus, the direction must be accounted for, which brings us to vector addition.

When dealing with vectors like momentum, direction matters just as much as magnitude. Consider each component separately: the i (horizontal) and j (vertical) directions. The exercise shows how the momenta of the two objects are combined.
The first object has a momentum of \( 15.00 \, \text{i} \, \text{kgm/s} \), while the second has \( -6.00 \, \text{j} \, \text{kgm/s} \). To find the total momentum, we "add" these vectors:

• Horizontal (i-component): \( 15.00 \, \text{i} \)
• Vertical (j-component): \( -6.00 \, \text{j} \)

Both components form the total momentum vector \( 15.00 \, \text{i} \, \text{kgm/s} - 6.00 \, \text{j} \, \text{kgm/s} \). Vector addition allows us to understand how these distinct directional components combine to form a new resultant vector for the entire system. This resultant vector represents the conserved momentum before the collision and is crucial for determining the final velocity after the collision.

Collisions illustrate the principle of conservation of momentum in an engaging way. When two objects collide and stick together, like the ones in the example, they form a composite object.
The law of conservation of momentum tells us that the total momentum before the collision is the same as after, given no external forces act on it.
Let's examine the collision:

• Total initial momentum: \( 15.00 \, \text{i} \, \text{kgm/s} - 6.00 \, \text{j} \, \text{kgm/s} \)
• The mass after sticking: sum of the two masses \( 3.00 + 2.00 = 5.00 \, \text{kg} \)

To find the final velocity of the combined object, divide the total momentum by the total mass. Component-wise calculation yields \( 3.00 \, \text{i} \, \text{m/s} - 1.20 \, \text{j} \, \text{m/s} \). This calculation not only adheres to the principles of physics but also shows how vector addition aids in determining the new direction and speed of a compound system post-collision.