Problem 49 - Algebra and Trigonometry

Question

Velocity of a Boat A boat heads in the direction \(\mathrm{N} 72^{\circ} \mathrm{E}\) The speed of the boat relative to the water is 24 \(\mathrm{mi} / \mathrm{h}\) . The water is flowing directly south. It is observed that the true direction of the boat is directly east. (a) Express the velocity of the boat relative to the water as a vector in component form. (b) Find the speed of the water and the true speed of the boat.

Step-by-Step Solution

Answer

Velocity: \( \langle 22.824, 7.416 \rangle \); Water speed: 7.416 mi/h; Boat speed: 22.824 mi/h.

Step 1: Understand the Vector Representation

To express the velocity of the boat relative to the water in component form, we'll split the given velocity into north and east components. Since the boat's direction is given as \( N 72^\circ E \), we'll interpret this as an angle \( 72^\circ \) to the east of north. The speed of the boat relative to the water is 24 mi/h.

Step 2: Calculate North and East Components

The north component of the boat's velocity is \( 24 \cos(72^\circ) \) and the east component is \( 24 \sin(72^\circ) \). This gives the velocity vector of the boat relative to the water as \( \langle 24 \sin(72^\circ), 24 \cos(72^\circ) \rangle \).

Step 3: Substitute and Calculate

Calculate the trigonometric functions: \( \cos(72^\circ) \approx 0.309 \) and \( \sin(72^\circ) \approx 0.951 \). This gives the vector as \( \langle 24 \times 0.951, 24 \times 0.309 \rangle \), simplifying to \( \langle 22.824, 7.416 \rangle \).

Step 4: Express Velocity of Water as Vector

Since the water flows directly south, its velocity vector is \( \langle 0, -v \rangle \) where \( v \) is the speed of the water.

Step 5: Determine True Velocity Vector of the Boat

The true direction of the boat is directly east, hence its velocity vector relative to the ground is \( \langle u, 0 \rangle \), where \( u \) is the boat's true speed. This comes from combining the velocity of the boat relative to the water and the water's velocity.

Step 6: Set Up and Solve Equations

The components of the velocity vectors must add: \( 22.824 = u \) and \( 7.416 + (-v) = 0 \). Solving the second equation gives \( v = 7.416 \).

Step 7: Find True Speed of the Boat

The east component directly gives the true speed of the boat as \( u = 22.824 \).

Step 8: Conclusion

The velocity of the boat relative to the water in component form is \( \langle 22.824, 7.416 \rangle \). The speed of the water is 7.416 mi/h to the south and the true speed of the boat is 22.824 mi/h eastward.

Key Concepts

Understanding Velocity Vectors

Velocity vectors are essential in understanding motion in two dimensions. A vector has both magnitude and direction. When we talk about the velocity of a boat, we're considering how fast and in which direction it's moving in relation to another object, in this case, water. To solve problems involving velocity vectors:

• Understand that velocity is a vector quantity with two components: horizontal (x-axis) and vertical (y-axis).
• Visualize the situation: Here, the boat moves at an angle relative to the north, and the water moves southwards.
• True velocity results from the combination of velocity relative to water and water flow itself.

Vectors help determine both the true path and speed of the object in question. In our exercise, the boat's true velocity ends up pointing due east, demonstrating how combined vector components result in specific directions.

Applying Trigonometric Functions

Trigonometric functions come in handy when breaking vectors into components, especially when the direction of movement is given as an angle. For the boat, its initial direction is \( N 72^\circ E \), which means the direction is 72 degrees eastward from North. Here's how trigonometry helps:

• Cosine function: Used to find the adjacent side of the right triangle, which represents the northward component. For our boat, it is \( 24 \cos(72^\circ) \).
• Sine function: Used for the opposite side, finding the eastward component: \( 24 \sin(72^\circ) \).

By calculating these components, we achieve the velocity vector \( \langle 22.824, 7.416 \rangle \). Trigonometric functions convert the angle information into usable vector components, directly affecting the calculation of the accurate speed and direction.

Breaking Down Vector Components

Vector components are the building blocks for understanding the precise movement in the chosen coordinate system, often represented as \( \langle x, y \rangle \). Here's a breakdown of components for our exercise:

• **East Component (x-axis):** Calculated using the sine of the direction angle \( 72^\circ \). The east component shows how much the velocity contributes to horizontal movement.
• **North Component (y-axis):** Determined using cosine, representing the boat's northward velocity part.

In our example, the water flows south with vector \( \langle 0, -v \rangle \). The true eastward movement combines boat and water, creating a known "resultant vector," directly east. Solving gives the true speed east \( u = 22.824 \) mi/h, and water speed south \( v = 7.416 \) mi/h. Understanding vector components allows us to deconstruct the boat’s actual navigation from given conditions, revealing individual influences of each motion element.