Question

The captain of a boat wants to travel directly across a river that flows due east with a speed of \(1.00 \mathrm{~m} / \mathrm{s} .\) He starts from the south bank of the river and heads toward the north bank. The boat has a speed of \(6.10 \mathrm{~m} / \mathrm{s}\) with respect to the water. In what direction (in degrees) should the captain steer the boat? Note that \(90^{\circ}\) is east, \(180^{\circ}\) is south, \(270^{\circ}\) is west, and \(360^{\circ}\) is north.

Short answer

Answer: The captain should steer the boat at an angle of approximately 260.58°, which is in the northwest direction.

Step-by-step solution

Identifying given vectors and the desired vector

Identify the given vectors: boat's velocity vector with respect to the water (let's call it \(\vec{v}_{bw}\)), water's velocity vector (let's call it \(\vec{v}_{w}\)), and the desired velocity vector of the boat with respect to the ground (let's call it \(\vec{v}_{bg}\)). The desired vector is from south bank to north bank which is in the northward direction.

Vector addition

We need to find the vector sum of \(\vec{v}_{bw}\) and \(\vec{v}_{w}\) to obtain the desired vector, \(\vec{v}_{bg}\). Let the angle of the boat be θ. The relation between the three velocities is: \[\vec{v}_{bg} = \vec{v}_{bw} + \vec{v}_{w}\]

Using components to find the angle

The water's velocity vector \(\vec{v}_{w}\) has only eastward component which is \(1.00 \mathrm{~m} / \mathrm{s}\). Break \(\vec{v}_{bw}\) into its components in the eastward and northward directions. The northward component will be \(6.10 \sin(θ) \mathrm{~m} / \mathrm{s}\) and eastward component will be \(6.10 \cos(θ) \mathrm{~m} / \mathrm{s}\). The boat's desired velocity vector components will be: \[\vec{v}_{bg}^{north} = 6.10 \sin(\theta) \mathrm{~m} / \mathrm{s}, \vec{v}_{bg}^{east} = 6.10\cos(\theta) + 1.00 \mathrm{~m} / \mathrm{s}\]

Make use of the fact that boat needs to go directly north

The boat's desired velocity vector has no eastward component as it should go directly north. So, we equate the eastward component to zero, which gives: \[6.10 \cos(\theta) + 1.00 = 0\]

Solve for angle θ

Rearrange the equation derived in Step 4 and solve for the angle θ: \[\theta = \arccos(-\frac{1.00}{6.10})\] Compute the value of θ: \[\theta \approx 99.42^{\circ}\] As we assumed the counterclockwise angle from east to be positive, the angle measured clockwise from east will be \(360^{\circ} - θ = 360^{\circ} - 99.42^{\circ} \approx 260.58^{\circ}\). Thus, the captain should steer the boat at an angle of approximately \(260.58^{\circ}\), which is in the northwest direction.

Key concepts

Vector Addition

Vector addition is a cornerstone of physics, particularly when analyzing motion and forces. To visualize vector addition, imagine arrows pointing in different directions where the length represents the magnitude, and the direction represents the way the vector is pointing. In the river crossing problem, the captain must account for two vectors: the boat's velocity relative to the water and the water's velocity relative to the ground.

To reach directly across the river, the captain must 'add' these vectors so that their combined effect leads to a straight path to the opposite bank. Mathematically, vector addition involves combining the corresponding components of each vector. If one vector has an eastward component and another has a northward component, their sum will be the diagonal of the rectangle formed by putting these two vectors at right angles to each other, following the rules of the parallelogram law of vector addition.

For the river crossing, this means calculating how the river's flow affects the boat's desired straight path across the river. The captain must steer the boat at an angle to counteract the river's flow, and vector addition helps determine this angle precisely.

Vector Components

Vector components are crucial to solving multidimensional problems, like navigating a river's current. Any vector can be broken down into its components along the axes of a coordinate system, usually horizontal (x-axis) and vertical (y-axis) components. When dealing with vectors in two dimensions, these components are essential for understanding the vector's overall effect.

In our river crossing example, we consider the boat's velocity vector in relation to the water and break it into components parallel to the river flow (eastward) and perpendicular to it (northward). The northward component of this vector is important because it represents the boat's speed directly across the river, unaffected by the current. By analyzing the boat's speed in terms of its components, it becomes easier to calculate the exact heading the captain needs to steer.

Calculating Vector Components

To calculate the components of the boat's velocity, we use trigonometric functions—specifically, sine for the northward component and cosine for the eastward component. Since the aim is to reach directly to the opposite bank, the eastward component must be neutralized by the current's effect. This aspect demonstrates the power of decomposing vectors: it simplifies complex motion into manageable calculations.

Relative Velocity

Relative velocity is the concept of measuring the velocity of an object, such as our boat, relative to another object’s frame of reference, which in this scenario is the river bank or the ground. The key to understanding this problem is realizing the effect of the water current on the boat's course. The ground or the riverbank serves as our frame of reference for measuring how the boat's velocity relative to the water combines with the river's current.

The captain wants to travel directly across the river, which means the boat's velocity relative to the bank should have no eastward or westward component. This is where the concept of relative velocity becomes essential. The water’s eastward velocity adds to the boat’s velocity when measuring from the riverbank’s frame of reference, potentially pushing the boat downstream.

To counter this, the captain must steer the boat upstream at an angle that eliminates the eastward component created by the current. The steering angle is calculated using the boat's speed, the current’s speed, and trigonometry, ensuring the only velocity from the captain's efforts is northward. Thus, the concept of relative velocity enables us to understand how different motions affect each other from a chosen reference point.