Question

The captain of a boat wants to travel directly across a river that flows due east. He starts from the south bank of the river and heads toward the north bank. The boat has a speed of \(5.57 \mathrm{~m} / \mathrm{s}\) with respect to the water. The captain steers the boat in the direction \(315^{\circ} .\) How fast is the water flowing? Note that \(90^{\circ}\) is east, \(180^{\circ}\) is south, \(270^{\circ}\) is west, and \(360^{\circ}\) is north.

Short answer

Answer: The speed of the water flow is 3.94 m/s.

Step-by-step solution

Identify the given information

We are given the following information: - Speed of the boat with respect to the water: \(5.57 \mathrm{~m} / \mathrm{s}\) - The boat's direction with respect to the water is \(315^{\circ}\)

Break down the boat's velocity into components

To find the components of the boat's velocity, we can use the following trigonometric relations: \(V_x = V \cdot \cos{\theta}\) \(V_y = V \cdot \sin{\theta}\) Here, \(V_x\) and \(V_y\) are the components of the boat's velocity in the x and y directions, \(V\) is the speed of the boat with respect to the water, and \(\theta\) is the angle. For this problem: \(V = 5.57 \mathrm{~m} / \mathrm{s}\) \(\theta = 315^{\circ}\) Convert the angle to radians: \(\theta = 315^{\circ} \cdot \frac{\pi}{180} = \frac{7\pi}{4}\) radians Now, let's find the components of the boat's velocity: \(V_x = V \cdot \cos{\frac{7\pi}{4}} = 5.57 \cdot \frac{\sqrt{2}}{2}\) \(V_y = V \cdot \sin{\frac{7\pi}{4}} = 5.57 \cdot \frac{-\sqrt{2}}{2}\)

Analyze the components of velocity that relate to the water flow

Since the boat is traveling in the north direction, any eastward component of velocity is due to the water flow. In this case, the \(V_x\) component represents the water flow: Water flow speed = \(V_x = 5.57 \cdot \frac{\sqrt{2}}{2}\) We can now calculate the speed of the water flow: Water flow speed = \(5.57 \cdot \frac{\sqrt{2}}{2} = 3.94 \mathrm{~m} / \mathrm{s}\) So the speed of the water flow is \(3.94 \mathrm{~m} / \mathrm{s}\).

Key concepts

Relative Velocity

When studying motion, it's crucial to consider the concept of relative velocity, which accounts for the fact that motion is observed differently from different reference frames. Imagine you're on a train moving at a constant speed and you start walking down the aisle. Your speed relative to the train is just your walking speed, but to an observer outside the train, your speed is the sum of your walking speed and the train's speed.

Applying this to our river problem, the velocity of the boat as observed from the shore is different from its velocity with respect to the water. The water’s flow adds or subtracts from the boat's velocity depending on the direction of the current. This is why even though the captain sets a direction to go straight across, the boat might end up downstream without the proper adjustment for the river's flow. Calculating the correct heading involves knowing the speed of the river's flow and adjusting the boat's velocity to compensate.

Vector Components

Vectors are mathematical objects used to represent quantities that have both magnitude and direction, such as velocity. When working with vector quantities, it's often useful to break them down into components along the x (east-west) and y (north-south) axes. This process makes it easier to analyze the motion in each direction separately, a method particularly useful when dealing with problems like the motion of a boat in a river.

In our river current problem, we calculate the boat's vector components using trigonometry. The eastward and northward components of the boat's velocity, labeled V_x and V_y respectively, are found by multiplying the boat's speed by the cosine (for the x-component) and the sine (for the y-component) of the angle at which the captain is steering.

Trigonometry in Physics

Trigonometry is not only limited to mathematics but is also a fundamental tool in physics, especially when analyzing forces, motion, and vectors. The trigonometric functions sine, cosine, and tangent are invaluable for resolving a vector into its perpendicular components—essentially, for understanding how motion in one direction affects motion in another.

For our river current problem, we use trigonometry in physics to decompose the boat's velocity into components parallel to the east-west (x-axis) and north-south (y-axis) axes. This step is critical to isolating the effect of the river's current from the boat's engine power. By understanding how the angle of steering translates into horizontal and vertical movement, we can negate the river's influence and guide the boat directly to its intended destination on the opposite bank.

Velocity of a Boat in a River

A boat's motion in a river is a classic problem that requires the understanding of both relative velocity and vector components. The velocity of a boat in a river depends on two main factors: the velocity of the boat relative to the water, and the velocity of the river current itself. To navigate effectively, the captain must steer the boat at an angle that compensates for the current, ensuring that the resultant path of the boat is directly across the river.

This is notably different from navigating in still water, where the boat's velocity would be the same irrespective of the reference frame (assuming no other forces like wind). Here, the velocity vector of the boat must be adjusted so that its component along the direction of the river flow exactly cancels out the river's current, resulting in a net motion directly across the river.