91Ó°ÊÓ

Vectors are essential in physics because they help describe quantities that have both magnitude and direction. In the exercise, vectors are used to represent the velocities of the particle at different points in time.

The initial velocity of the particle is noted as a vector pointing east, represented as \((10, 0)\) in vector notation. This means 10 m/s along the eastward direction and 0 m/s northward.

The final velocity points north, represented as \((0, 10)\). Again, this refers to a velocity of 10 m/s northward and 0 m/s eastward. By using vectors, you can effectively manage and manipulate these directions and magnitudes, helping solve problems naturally rich in directionality.

Vector calculations allow us to handle multiple dimensions which are not easy to deal with using scalar numbers alone. They are particularly useful in situations like our particle's, where direction changes are involved.

The concept of change in velocity involves figuring out how the velocity of an object varies over a given period. For this exercise, the change in velocity is determined by subtracting the initial velocity vector from the final velocity vector.

This is done mathematically by finding the difference: \(\Delta \mathbf{v} = \mathbf{v}_f - \mathbf{v}_i\). Here, it's important to keep in mind that direction plays an enormous role. Subtracting the initial velocity \((10, 0)\) from the final velocity \((0, 10)\), we find \(\Delta \mathbf{v} = (-10, 10)\).

This vector \((-10, 10)\) indicates not just a change in speed but also a change in direction from east to north, telling us that the velocity now shifts towards the northwest. The change in velocity, therefore, encompasses both how much and where it changes.

The magnitude of acceleration is a key concept that describes how quickly an object's velocity changes in a particular direction. To find this magnitude, we first need the magnitude of the change in velocity.

Using the Pythagorean theorem, the change in velocity vector \((-10, 10)\) has a magnitude: \[ ||\Delta \mathbf{v}|| = \sqrt{(-10)^2 + (10)^2} = \sqrt{200} = 10\sqrt{2} \text{ m/s} \]

Once we have this change in velocity, we can calculate the acceleration. The formula to find acceleration is \(\mathbf{a} = \frac{\Delta \mathbf{v}}{\Delta t} \). Over the time period \(\Delta t = 5\) seconds, the acceleration is: \[ \mathbf{a} = \frac{10\sqrt{2}}{5} = 2\sqrt{2} \text{ m/s}^2 \]

This indicates the rate at which the particle changes its velocity, effectively highlighting how rapidly it transitions from moving east to north, ultimately moving northwest.