Problem 93 - Fundamentals of Physics

Question

\(A\) 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

Answer

(a) 63.06 km; (b) 18.28° south of east; (c) 0.7007 km/h; (d) 18.28° south of east; (e) 1.556 km/h; (f) 1.0035 km/h; (g) due east.

Step 1: Calculate the Initial Displacement Vector Components

First, find the components of the camel's initial displacement of 75 km at 37° north of due east. Use trigonometry:- Eastward (x-axis) component: \[75 \times \cos(37°) \]- Northward (y-axis) component: \[75 \times \sin(37°) \]Calculating these, we get:- Eastward component: \[75 \times 0.7986 = 59.895\] km- Northward component: \[75 \times 0.6018 = 45.135\] km.

Step 2: Calculate the Second Displacement Vector Components

Next, calculate the components of the camel's second displacement. The camel walks 65 km due south. The components are: - Southward (y-axis) component: -65 km (since south is negative direction on y-axis) - Eastward (x-axis) component remains 0 km.

Step 3: Determine the Total Displacement Components

Add the components from the first and second displacements to determine the total displacement with respect to the starting point A:- Total eastward component: \[59.895 + 0 = 59.895\] km- Total northward component: \[45.135 - 65 = -19.865\] km.

Step 4: Calculate the Magnitude of the Camel's Displacement

Use the Pythagorean theorem to find the magnitude of the total displacement:\[ \sqrt{(59.895)^2 + (-19.865)^2} \approx \sqrt{3584.19 + 394.62} = \sqrt{3978.81} \approx 63.06 \] km.

Step 5: Determine the Direction of the Camel's Displacement

Calculate the angle from east using the arctangent function:\[ \theta = \arctan\left( \frac{-19.865}{59.895} \right) \approx -18.28° \]This angle south of the east corresponds to 18.28° south of east.

Step 6: Calculate the Total Time Moved

The total time includes walking time and rest period:\[ 50 \text{ h} + 35 \text{ h} + 5 \text{ h} = 90 \text{ h}. \]

Step 7: Calculate Average Velocity

Average velocity is the total displacement divided by the total time:\[ \text{Average velocity} = \frac{63.06 \text{ km}}{90 \text{ h}} \approx 0.7007 \text{ km/h}. \]The direction is the same as the overall displacement: 18.28° south of the east.

Step 8: Calculate Average Speed

The total distance traveled by the camel is the sum of all segments:\[ 75 \text{ km} + 65 \text{ km} = 140 \text{ km}. \]Average speed is total distance over total time:\[ \text{Average speed} = \frac{140 \text{ km}}{90 \text{ h}} \approx 1.556 \text{ km/h}. \]

Step 9: Determine Remaining Time to Reach B

The remaining time for the camel to reach oasis B is:\[ 120 \text{ h} - 90 \text{ h} = 30 \text{ h}. \]

Step 10: Calculate the Displacement Required to Reach B

The remaining displacement to reach oasis B eastward is:\[ 90 \text{ km} - 59.895 \text{ km} = 30.105 \text{ km}. \]The y-component needs to be zero as the oasis B is directly east.

Step 11: Calculate Required Average Velocity After Rest

The camel needs to travel the remaining 30.105 km east in 30 hours:\[ \text{Average velocity} = \frac{30.105 \text{ km}}{30 \text{ h}} = 1.0035 \text{ km/h}. \]Direction: due east.

Key Concepts

Trigonometry in Physics

Trigonometry plays a crucial role in physics problems that involve angles and vector components. In our exercise, the camel moves primarily in two directions: first at an angle north of east, and then due south. The use of trigonometry helps break down these movements into component parts that can be managed independently.

When dealing with angles, the trigonometric functions cosine and sine become your trusted allies. By calculating the cosine of the angle, we determine the component of the movement along the x-axis (eastward). Similarly, the sine of the angle provides the component along the y-axis (northward or southward depending on the direction).

• Eastward component: Calculated using cosine, representing horizontal movement.
• Northward component: Calculated using sine, representing vertical movement.

Using these components makes it easier to determine the total displacement of the camel as it provides a detailed understanding of its movement in a two-dimensional plane.

Pythagorean theorem

The Pythagorean theorem is a fundamental concept in physics, especially when dealing with vectors. This theorem helps us find the magnitude of a resultant vector when we know its perpendicular components, making it indispensable when calculating the camel's displacement.

In our problem, the camel's movement can be visualized as forming a right-angled triangle, with its eastward and north-south components as the legs. The Pythagorean theorem states that the square of the hypotenuse (which is the magnitude of the displacement) is equal to the sum of the squares of the other two sides.

• Formula: \[ \text{Magnitude of displacement} = \sqrt{(\text{eastward component})^2 + (\text{north-south component})^2} \]

This simple yet powerful formula allows us to calculate the camel's total displacement from the starting point, ensuring that we capture both its directional changes in one clean calculation.

Average speed and velocity

Average speed and velocity are critical concepts to understand when assessing an object's motion. While they are related, they are not the same, and it's essential to distinguish between them in physics.

• Average speed: This is the total distance traveled divided by the total time taken, providing a scalar quantity that disregards direction.
• Average velocity: In contrast, this is a vector quantity calculated as the total displacement divided by the total time. It incorporates direction and gives a more nuanced view of motion.

In our camel exercise, we calculated both the average speed and average velocity. The average speed considers the whole journey's length, while the average velocity focuses on the camel's starting and ending points, encapsulating the direction and efficiency of its movement.

Vectors and components

Vectors and their components are foundational in physics, as they allow us to describe quantities that have both magnitude and direction. In the camel's journey, each leg of the path is considered a vector.

• Vectors are broken into components to handle them efficiently. Typically, vectors in problems like this are divided into x (horizontal) and y (vertical) components.
• We determine these components using trigonometric functions, turning complex, angled journeys into straightforward calculations along each axis.

By understanding vectors in terms of their components, we simplify the camel's travels into manageable pieces, making it easier to analyze, calculate, and comprehend its displacement and direction over the journey.

Problem-solving in physics

Problem-solving in physics involves a methodical approach to dissecting complex situations into simpler parts. The camel problem showcases the importance of structured planning and execution.

• First, identify known variables and what you need to find, using given values like distances, angles, and timings.
• Utilize physics principles, breaking down vectors into components, using trigonometrical identities, and applying appropriate formulas like the Pythagorean theorem.
• Perform calculations step-by-step, checking each result before moving to the next to ensure accuracy.

The structured method not only provides clarity but also cultivates confidence as you progress through each calculation, ensuring you arrive at correct and meaningful results. By fostering such an approach, you can solve a wide array of physics problems effectively.