91Ó°ÊÓ

Oasis \(A\) is \(90 \mathrm{~km}\) due west of oasis \(B .\) A desert camel leaves \(A\) and takes \(50 \mathrm{~h}\) to walk \(75 \mathrm{~km}\) at \(37^{\circ}\) north of due east. Next it takes \(35 \mathrm{~h}\) to walk \(65 \mathrm{~km}\) due south. Then it rests for \(5.0 \mathrm{~h}\). What are the (a) magnitude and (b) direction of the camel's displacement relative to \(A\) at the resting point? From the time the camel leaves \(A\) until the end of the rest period, what are the (c) magnitude and (d) direction of its average velocity and (e) its average speed? The camel's last drink was at \(A ;\) it must be at \(B\) no more than \(120 \mathrm{~h}\) later for its next drink. If it is to reach \(B\) just in time, what must be the (f) magnitude and (g) direction of its average velocity after the rest period?

Step by step solution

01

Determine the Camel's Displacement After First Walk

Convert the orientation angle into components. The camel walks 75 km at an angle of \(37^\circ\) north of due east.- Eastward component \( (x_1): x_1 = 75 \cos(37^\circ) \)- Northward component \( (y_1): y_1 = 75 \sin(37^\circ) \)

02

Determine the Camel's Displacement After Second Walk

The camel then walks 65 km due south.- If the south is considered negative in the y-direction, the southward component is simply \( y_2 = -65 \) km.

03

Calculate Total Displacement

The sum of east-west components:- Total eastward displacement, \( x = x_1 \)The sum of north-south components:- Total northward displacement, \( y = y_1 + y_2 \)Determine the resultant displacement using Pythagoras' Theorem: \[ d = \sqrt{x^2 + y^2} \]Calculate the direction of the displacement \( \theta \) using:\[ \theta = \arctan\left(\frac{y}{x}\right) \]

04

Calculate Average Velocity and Speed During Trip

Determine the total time: travel time plus rest time, \( t = 50 + 35 + 5 = 90 \) hours.- Average velocity magnitude: \( v_{avg} = \frac{d}{t} \)- Average speed is given by: \( s_{avg} = \frac{d_{total}}{t} \), where \(d_{total} = 75 + 65 \) km.Direction of average velocity remains the same as that of the displacement.

05

Determine Remaining Velocity to Oasis B

First, calculate the remaining distance to B from the rest location in the East-West direction.- Remaining distance, \( x_{remaining} = 90 - x \).Then, calculate the velocity needed to cover this distance in the remaining time:- Time remaining, \( t_{remaining} = 120 - 90 = 30 \) hours.- Required velocity magnitude, \( v_{req} = \frac{x_{remaining}}{t_{remaining}} \).Direction of the velocity will be due east.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Components

Understanding vector components is the key to solving many physics problems involving direction and magnitude. When a vector, like a displacement vector, has a direction and a magnitude, it can be broken down into components. These components are essentially projections onto the axes of a coordinate system.

For instance, when a camel walks 75 km at an angle of 37° north of due east, as in the exercise, we can break this down into eastward and northward components. To do this:

• The eastward component (x) is calculated using the cosine function: \(x = 75 \cos(37°)\).
• The northward component (y) uses the sine function: \(y = 75 \sin(37°)\).

By calculating these components, you can analyze the problem more easily without getting bogged down by diagonal distances.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept often used in physics to find the magnitude of a resultant vector when you have its components. According to this theorem, if you have a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In our example, after determining the emirate hustling. camel's eastward and northward displacements, we use Pythagorean Theorem to find out the overall displacement from its starting point.

• If the eastward displacement is considered as \(x\) and the northward as \(y\), then the resultant displacement \(d\) can be calculated as: \[ d = \sqrt{x^2 + y^2} \]

Consequently, this gives us a clear picture of the camel's actual position relative to its starting point after both legs of the journey.

Average Velocity

Average velocity gives you an idea of how fast an object is moving and in which direction, averaged over a period of time. Unlike average speed, which only considers distance, average velocity takes both magnitude and direction of displacement into account.

To find the average velocity of the camel from the start until it rests:

• First, determine the total displacement \(d\).
• Divide this by the total time taken, \(t = 90 \text{ hours}\).
• The formula for average velocity \(v_{avg}\) is: \[ v_{avg} = \frac{d}{t} \]

This will tell you the average rate and direction at which the camel displaced itself from the original oasis, taking into account all travel segments and rest periods.

Directional Angles

Directional angles help us understand the orientation of a vector concerning a reference direction, usually part of a compass like due east or north. It's common to use trigonometric functions to calculate these angles once you have vector components.

For example, once the camel's total sense of displacement is known from its respective components, the direction angle \(\theta\) can be found as:

• \(\theta = \arctan\left(\frac{y}{x}\right)\)

This angle forms a crucial part of describing any vector fully, as it not only gives us the scalar magnitude of the vector but also its specific direction in a two-dimensional plane.