Problem 65 - 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

(a) Velocity vector: \((7.42, 22.80)\) (b) Water speed: 7.42 mi/h, true boat speed: 22.80 mi/h

Step 1: Understanding the Direction

The boat is heading \(\mathrm{N} 72^{\circ} \mathrm{E}\). This means it is 72 degrees east from north. We need to convert this direction into vector component form.

Step 2: Find the Vector Components of the Boat's Velocity

The speed of the boat relative to the water is 24 mi/h. The boat's velocity vector can be split into the following components using trigonometry:- The northward component (y-axis) is: \[ v_y = 24 \cos(72^{\circ}) \]- The eastward component (x-axis) is:\[ v_x = 24 \sin(72^{\circ}) \]Calculating these gives:\( v_y = 24 \cos(72^{\circ}) \approx 7.42 \mathrm{mi/h} \)\( v_x = 24 \sin(72^{\circ}) \approx 22.80 \mathrm{mi/h} \) So, the velocity vector of the boat relative to the water is approximately \(7.42, \,22.80\).

Step 3: Understand the Influence of Water Current

The water is flowing directly south, which means it affects only the northward component of the boat's velocity. The boat is observed to move east, meaning the northward component of the resultant velocity is zero.

Step 4: Calculate the Water's Speed

Since the northward component of the boat's true velocity is zero, the southward (water) component must negate the boat's northward component. Thus, the water's speed is equal to the northward component of the boat's velocity:\[ s = 7.42 \mathrm{mi/h} \]

Step 5: Find the True Speed of the Boat

The eastward component of the boat's velocity remains unaffected by the water.Therefore, the true speed of the boat is the same as the eastward component:\[ v_{true} = 22.80 \mathrm{mi/h} \]

Key Concepts

Trigonometry Applications

Trigonometry plays a crucial role in breaking down the velocity of a boat into vector components. In this exercise, we work with the boat's heading direction, given as \( \mathrm{N} 72^{\circ} \mathrm{E} \). To convert this into a vector form, trigonometry helps us figure out the northward and eastward components.

These components are derived using

• The cosine function for the northward component: \( v_y = 24 \cos(72^{\circ}) \).
• The sine function for the eastward component: \( v_x = 24 \sin(72^{\circ}) \).

Applying these functions takes the magnitude (here, the boat's speed of 24 mi/h) and breaks it down into the horizontal and vertical components using angles. This method is fundamental in trigonometric applications to determine vectors, especially when dealing with navigation and physics problems.

Velocity Vector

A velocity vector represents the direction and speed of an object in motion through a specific path. In the context of our exercise, the boat's velocity relative to water is a vector represented by its components: northward and eastward.

This concept harbors two main elements:

• Magnitude: The speed of the boat relative to the water, which is 24 mi/h in this scenario.
• Direction: Specified as \( \mathrm{N} 72^{\circ} \mathrm{E} \), indicating how it moves through the water.

The velocity vector is then given as \( (7.42, \, 22.80) \), where 7.42 is the northward component, and 22.80 is the eastward component. Understanding these components and how they interact underlines the essence of navigating via vectors.

Relative Velocity

Relative velocity is the velocity of an object as observed from another reference frame. Here, it reflects the motion of the boat relative to the water's flow.

When the water is moving south and the boat is observed to travel directly east, we compare the boat's movement concerning still water and moving water:

• The boat's velocity (relative to still water) has components both northward and eastward.
• The water's velocity affects only the northward component, negating it, because the current flows south.

This results in a net eastward true velocity of the boat of 22.80 mi/h, showing how components adjust when interacting with other moving surfaces.

Directional Angles

Directional angles are key to determining the path of movement concerning cardinal points and degrees. In our problem, working with angles helps express the initial direction of the boat (\( \mathrm{N} 72^{\circ} \mathrm{E} \)) and assists in converting these into meaningful velocity components.

• This angle gives a split between heading north and east by 72 degrees from the north.
• Affects how the velocity is perceived and calculated in terms of components.

Generic directional angle interpretation helps in constructing the baseline for motion studies, guiding both navigation in this exercise and broader applications in physics and engineering.