91Ó°ÊÓ

Unit-vector notation is a powerful way to express physical quantities, such as velocity or position, in terms of standard directional components.

In a two-dimensional Cartesian coordinate system, vectors can be decomposed into two parts: the horizontal (usually denoted as \(\hat{i}\)) and the vertical (denoted as \(\hat{j}\)).

• \(\hat{i}\) points towards the east.
• \(\hat{j}\) points towards the north.

This notation allows us to easily visualize vector quantities and perform operations like addition, subtraction, and even differentiation.

For instance, in the problem, the velocity of ship B is expressed as \(\vec{v}_B = 31.97\hat{i} + 24.07\hat{j}\) km/h. Here, 31.97 is the eastward component, and 24.07 is the northward component. Such expression simplifies understanding and calculation of vector problems.

The position vector is used to determine the location of one point in relation to another in a coordinate system.

In our scenario, ship A is located 4 km north and 2.5 km east of ship B. We can represent this position using a position vector that details these distances in the north (\(\hat{j}\)) and east (\(\hat{i}\)) directions.

• Initial position: \(\vec{r}_0 = 2.5\hat{i} + 4\hat{j}\) km

This vector tells us how far ship A is from ship B in each directional component. It's a concise way to pinpoint where an object is located in space.

Position vectors are foundational in kinematics because they provide a baseline from which movement or change over time, such as velocity, is calculated.

In problems involving relative motion, the concept of minimum separation is essential for understanding when two moving objects come closest to each other.

Minimum separation tells you the shortest distance between two objects as they move along their trajectories.

To find this, one must first establish a function that describes the separation distance over time and then determine when this separation is at its minimum.

• Function: \( |\vec{r}(t)| = \sqrt{((2.5 - 31.97t)^2 + (4 - 46.07t)^2)} \)
• Calculate minimum by setting derivative to zero.

This process entails finding the derivative, setting it to zero, and solving for time \(t\). For our ships, this moment occurred approximately 49.57 hours after the starting point, indicating when the ships are closest.

Kinematics is the branch of physics that deals with the motion of objects without considering the causes of this motion, like forces. It primarily focuses on descriptive aspects.

• Mastering vectors is key in kinematics to track movement.
• Components like velocity and position are crucial.

In this exercise, kinematics is applied by examining the motion of ships A and B relative to one another.

We calculated the velocity of ship A with respect to ship B to understand how their movements are related. The relative velocity was found to be: \(\vec{v}_{A/B} = -31.97\hat{i} - 46.07\hat{j}\) km/h.

This relative motion helps derive the time-based position function and examines the changing separation distance, concepts central to understanding how objects move concerning each other in kinematic analysis.