91Ó°ÊÓ

Understanding the idea of breaking down a vector into components is fundamental in physics. When you have a vector, like the camel's initial 75 km journey at 37° north of due east, you're dealing with a direction and a magnitude. In order to simplify motion analysis, we use vector components. These components are like slices of a pie — splitting the vector into more manageable horizontal (east-west) and vertical (north-south) parts.

To find these components, we apply trigonometric functions: the cosine function helps find the horizontal (east) component, and the sine function helps find the vertical (north) component.

• The eastward component: \[ 75 imes \cos(37^\circ) \approx 59.88 \text{ km} \]
• The northward component: \[ 75 \times \sin(37^\circ) \approx 45.14 new.\text{ km} \]

These components tell us how far the camel goes in each direction.

The concept of average speed is simply about how much distance the camel covers over a certain amount of time, without concerning itself with direction. It's calculated by taking the total distance traveled and dividing it by the total time taken.

In our exercise, the camel travels a path of 75 km on its first leg and 65 km on its second, totaling 140 km. The total time spent moving, including the rest, is 90 hours:
\[\text{Average Speed} = \frac{75+65}{90} = 1.56 \ \text{km/h}\]
This calculation emphasizes speed, not the actual direction the camel takes.

Average velocity is slightly different from average speed. While speed covers how much ground is covered, average velocity concerns itself with the change in position from start to end, plus direction. For our camel at rest, it took a total of 90 hours to travel from its start to its resting position.

The displacement, the straight-line measurement from point \(A\) to the point of rest, is the total displacement we calculated using vector components:\[\sqrt{(59.88)^2 + (-19.86)^2} \approx 62.27 \ \text{km}\]
The average velocity is then:\[\text{Average Velocity} = \frac{62.27}{90} \approx 0.692 \ \text{km/h}\]
This also includes direction, approximately 18.26° south of east.

Trigonometry serves as a vital tool in physics for analyzing directions and distances. It is extensively used to resolve directions into simpler components. In our scenario with the camel, trigonometry assists in determining the exact path and displacement of the camel's journey.

When the camel moves 75 km at 37° north of east, we used the cosine and sine functions to resolve this movement into horizontal and vertical components, pinpointing how much of this distance is due east and how much shifted north. Further trigonometric calculations like tangent help calculate the angle of resultant displacement:\[\theta = \tan^{-1}\left(\frac{-19.86}{59.88}\right) \approx -18.26^\circ\]
The angle tells us about the direction relative to the east-west line — revealing how far south of east the camel ends up. Such components and angles simplify the analysis of the camel's overall path.