Problem 25 - Fundamentals of Physics

Question

Oasis \(B\) is \(25 \mathrm{~km}\) due east of oasis \(A\). Starting from oasis \(A\), a camel walks \(24 \mathrm{~km}\) in a direction \(15^{\circ}\) south of east and then walks \(8.0 \mathrm{~km}\) due north. How far is the camel then from oasis \(B\) ?

Step-by-Step Solution

Answer

The camel is 2.54 km away from Oasis B.

Step 1: Understand the problem

We need to determine the camel's final position relative to Oasis B after it walks in two different segments. We'll calculate its final position using vector addition.

Step 2: Decompose the first walk into components

The camel initially walks 24 km at a direction of 15° south of east. We convert this path into eastward and southward components using trigonometry: - Eastward component: \[ 24 \cos(15°) = 23.19 \text{ km} \]- Southward component: \[ 24 \sin(15°) = 6.21 \text{ km} \]

Step 3: Evaluate the second component of walk

Next, the camel walks 8 km due north. This only affects the north-south axis and simplifies to moving 8 km north.

Step 4: Calculate net displacement north-south

The camel initially moves 6.21 km south and then 8 km north:\[8 - 6.21 = 1.79 \text{ km north}\]

Step 5: Calculate net displacement east-west

The camel moves 23.19 km east from the first leg. There's no other east-west movement to add.

Step 6: Calculate camel's final position relative to Oasis B

Since Oasis B is 25 km east of Oasis A, we calculate the distance between the camel's final position and Oasis B:- East-west difference: \[ 25 - 23.19 = 1.81 \text{ km west} \]- The north-south position is already 1.79 km north.

Step 7: Use Pythagoras to find the total distance from Oasis B

The distance from the camel's final position to Oasis B can be found using the Pythagorean theorem:\[\sqrt{(1.81)^2 + (1.79)^2} \approx \sqrt{3.2761 + 3.2041} \approx \sqrt{6.4802} \approx 2.54 \text{ km}\]

Step 8: Conclusion

The camel is approximately 2.54 km away from Oasis B after completing its path.

Key Concepts

Trigonometry Fun with Angles and Directions

Trigonometry helps us understand movements and directions in a geometric way. It's especially handy for problems where angles and vector components are involved, like with our wandering camel. When the camel walks 24 km in a direction of 15° south of east, we need to break down this journey into understandable parts.
We use trigonometric functions for this:

• **Cosine Function**: Useful for finding the eastward (horizontal) component. It calculates how far along the "east" line the camel walked.
• **Sine Function**: Helps find the southward (vertical) component, determining how much the camel deviated from directly heading east.

By calculating these components, we understand the camel's actual path more clearly. From trigonometry, we see the eastward part is 24 times cos(15°) and the southward is 24 times sin(15°). Breaking down movements like this is a key step in vector addition.

The Pythagorean Theorem in Action

The Pythagorean Theorem is a powerful tool for calculating distances in right-angled triangles. It's handy when we need to find out just how far one point is from another, which is what we do to find how far our camel ends up from the oasis.
The core formula of the theorem is:\[a^2 + b^2 = c^2\]Here, "a" and "b" are the legs (small sides) of the triangle, and "c" is the hypotenuse (long side).
When the camel's path created a right triangle with 1.81 km west and 1.79 km north, we plugged these into the theorem. We solve it to find the straight-line distance from the camel's position to Oasis B. This helps us understand not just the direction or path taken but also the effective distance from one point to another.

Understanding Displacement Calculation

Displacement is a measurement of change in position. It's not just about the total distance walked but includes the direction, making it a vector. For our camel, even though it walked some kilometers around, its final displacement tells us exactly how far and in what direction the camel is from Oasis B after its journey.
Here's how you can think about it: - **Displacement vs. Distance**: Distance measures the total path length, while displacement is the direct line between start and end points. - **Net Calculations**: By using vector components, we can find the net movements—separately for east-west and north-south. - **Combining Components**: Once we have the net movements, combining them gives the overall displacement. In our solution, by combining the east-west and north-south distances using the Pythagorean theorem, we discover the camel's displacement relative to Oasis B is about 2.54 km. This tells us precisely and clearly how far off the camel is from its destination, no matter the winding path it took.