Problem 41 - Algebra and Trigonometry

Question

Velocity A river flows due south at 3 mi/h. A swimmer attempting to cross the river heads due east swimming at 2 mi/h relative to the water. Find the true velocity of the swimmer as a vector.

Step-by-Step Solution

Answer

The true velocity of the swimmer is \((2, -3)\) mi/h.

Step 1: Identify the Velocities

The velocity of the river is given as 3 mi/h towards the south. This can be represented as a vector \( \mathbf{v}_r = (0, -3) \) because it has no component in the east-west direction (x-axis) but a component of -3 in the north-south direction (y-axis). The swimmer's velocity relative to the river is given as 2 mi/h towards the east, and hence it can be represented as \( \mathbf{v}_s = (2, 0) \).

Step 2: Combine the Velocity Vectors

To find the true velocity of the swimmer, we need to add the swimmer's velocity vector relative to the water with the velocity vector of the river. We use vector addition: \( \mathbf{v}_{ ext{true}} = \mathbf{v}_s + \mathbf{v}_r \).

Step 3: Calculate the Resultant Velocity

Using vector addition, we have \( \mathbf{v}_{ ext{true}} = (2, 0) + (0, -3) = (2, -3) \). This means the true velocity of the swimmer is a vector \( (2, -3) \); 2 mi/h in the eastward direction and 3 mi/h in the southward direction.

Key Concepts

Vector Addition

Vector addition is a crucial concept in understanding how different velocity vectors interact. Imagine vectors as arrows pointing in the direction of the velocity. The river's flow and the swimmer's movement can be depicted as these arrows. Each vector has both a direction and a magnitude, like an arrow pointing south with 3 mi/h speed for the river, and an arrow pointing east with 2 mi/h speed for the swimmer.
By placing these arrows tip to tail, we can visualize how they combine. This principle can be formally expressed as:

• Add the corresponding components: the x-components and the y-components.
• Use the formula: \[ \mathbf{v}_{\text{resultant}} = ( \mathbf{v}_{x1} + \mathbf{v}_{x2}, \mathbf{v}_{y1} + \mathbf{v}_{y2} ) \]

For the swimmer and the river, this results in a single vector that accurately represents both movements. This vector addition gives us the overall motion in two distinct directions at once.

Resultant Velocity

When two or more vectors are combined, the outcome is a resultant vector. In our context, the resultant velocity captures the swimmer's actual path and speed relative to a fixed point, like the riverbank.
This resultant velocity accounts for both the swimmer's eastward push and the river's southward flow. Calculating it involves combining the vectors from each movement:

• First, identify the swimmer's eastward velocity: 2 mi/h
• Second, identify the river's southward velocity: 3 mi/h
• Finally, combine them to find the resultant vector: \[ \mathbf{v}_{\text{true}} = (2, 0) + (0, -3) = (2, -3) \]

The resultant velocity not only tells us the direction of the swimmer's movement but also displays the combined effect of both velocities, showing real-time travel direction and speed.

Components of Vector

Understanding the components of a vector can simplify the process of analyzing motion in two dimensions. Each vector splits into two parts, the horizontal (x-component) and the vertical (y-component), which are essentially projections onto the coordinate axes.
For the given problem:

• The swimmer's velocity vector, moving east, is represented by \( \mathbf{v}_s = (2, 0) \). This indicates 2 units on the x-axis and 0 units on the y-axis.
• The river's velocity vector, flowing south, is depicted as \( \mathbf{v}_r = (0, -3) \). Here, 0 units are on the x-axis, and -3 units are on the y-axis.

Representing each component separately makes complex calculations more accessible, allowing for straightforward vector addition and clearer visualization of the total movement or trajectory in question.