Problem 55 - Precalculus Mathematics for Calculus

Question

Velocity A river flows due south at 3 mi/h. A swimmer attempting to cross the river heads due east swimming at \(2 \mathrm{mi} / \mathrm{h}\) relative to the water. Find the true velocity of the swimmer as a vector. GRAPH CANT COPY

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to find the true velocity of the swimmer, considering both the river's current and the swimmer's own velocity. This involves vector addition.

Step 2: Define the Vectors

Let the velocity of the river be represented by the vector \(\vec{R} = 3\,\text{mi/h south}\). The swimmer's velocity relative to the water is \(\vec{S} = 2\,\text{mi/h east}\).

Step 3: Break Down Vectors into Components

Express \(\vec{R}\) as a component vector: \(\vec{R} = \langle 0, -3 \rangle\). Express \(\vec{S}\) as a component vector: \(\vec{S} = \langle 2, 0 \rangle\).

Step 4: Add the Vectors

The true velocity of the swimmer \(\vec{V}\) is the sum of the vectors \(\vec{R}\) and \(\vec{S}\). Compute \(\vec{V} = \vec{R} + \vec{S} = \langle 0, -3 \rangle + \langle 2, 0 \rangle = \langle 2, -3 \rangle\).

Step 5: Interpret the Result

The vector \(\langle 2, -3 \rangle\) represents the true velocity of the swimmer: 2 mi/h east and 3 mi/h south.

Key Concepts

Velocity Vector

Imagine you're the swimmer trying to cross that river. Your swimming speed and direction form what we call a "velocity vector." This velocity vector is crucial because it combines both speed and the direction you're moving in. In this exercise, the river creates its own velocity vector, flowing south at 3 mi/h. Meanwhile, the swimmer strives to swim east at 2 mi/h relative to the water. To find out where the swimmer actually ends up, both these vectors need to be taken into account together. By combining these velocities, you find the swimmer's true path across the river.

Vector Components

To make calculations easier, vectors can be split into what we call "components." Think of a vector as having two important characteristics: how far it goes in one direction and how far it goes in another—usually horizontal and vertical directions. For the river's vector, pointing south at 3 mi/h, it can be broken down into a component vector of

• In the horizontal direction: 0 mi/h (since it’s totally vertical)
• In the vertical direction: -3 mi/h (going south)

Similarly, the swimmer's eastward vector is:

• Horizontal: 2 mi/h
• Vertical: 0 mi/h

By understanding these components, we can more easily calculate the overall effect of these vectors when combined.

Vector Representation

Vectors can be visualized on a graph, which helps us see how different directions and speeds interact. In mathematical terms, we often represent vectors as pairs of numbers in angled brackets, like \(\langle 2, -3 \rangle\), where the first number refers to the horizontal component and the second to the vertical component. This notation provides a clear picture of how far something travels in each direction. By representing the swimmer's and the river's velocities this way, we can precisely calculate the real movement as the sum of the vectors. So, for the swimmer, the true velocity vector \(\langle 2, -3 \rangle\) tells us they are truly moving 2 mi/h east and 3 mi/h south, simultaneously.

Relative Velocity

Relative velocity considers how fast one object, like a swimmer, moves compared to another motion, such as the river's flow. Even if a swimmer aims to go east at 2 mi/h, the river pulling south at 3 mi/h changes the resulting path. This difference between the swimmer’s intended direction and the actual outcome is captured in the concept of relative velocity.

• The swimmer's speed relative to still water is adjusted by the river’s speed.
• The final path, or resulting velocity, is a combination of both movements.

Understanding relative velocity helps explain why the swimmer doesn't go straight east but moves southeast instead. It is through vector addition that we can accurately predict this path.