Problem 55 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The swimmer's true velocity is \( 2i - 3j \) mph.

Step 1: Understand the Problem

To solve this problem, we need to determine the swimmer's actual velocity when accounting for the river's flow. This involves decomposing and combining two velocity vectors: the river's and the swimmer's own velocity relative to the water.

Step 2: Define the Vectors

The river's velocity vector can be defined as \( \overrightarrow{v}_{\text{river}} = 0i - 3j \), since it flows south at 3 mph. The swimmer's velocity relative to the water can be defined as \( \overrightarrow{v}_{\text{swimmer}} = 2i + 0j \), as he heads east at 2 mph.

Step 3: Combine the Vectors

The true velocity of the swimmer, \( \overrightarrow{v}_{\text{true}} \), is found by adding the river’s and the swimmer's velocity vectors: \( \overrightarrow{v}_{\text{true}} = \overrightarrow{v}_{\text{swimmer}} + \overrightarrow{v}_{\text{river}} = 2i - 3j \).

Step 4: Interpret the Result

The true velocity vector \( 2i - 3j \) indicates that the swimmer is moving 2 mph east and 3 mph south. This vector represents the swimmer's real-world direction and speed, taking into account both his own swimming and the river's flow.

Key Concepts

Velocity Vectors

Velocity vectors are a fundamental concept in physics and mathematics when it comes to describing the movement of objects. A vector is essentially a quantity that has both magnitude and direction. In the context of velocities, it describes how fast something is moving and in which direction. Imagine standing by the riverbank and watching a swimmer. You see the river pushing south, and the swimmer swimming east. The actual path the swimmer takes is a combination of both influences. This is what velocity vectors help us determine—how these individual motions combine.

• The magnitude tells you how fast: For example, the swimmer swims at 2 mph.
• The direction tells you where: For the swimmer, this is east. The river adds a southward component at 3 mph.

Using velocity vectors, we can precisely figure out the swimmer's true motion in the water, considering both his effort and the river's flow.

Decomposition of Vectors

To truly grasp how different forces affect an object's motion, we decompose a vector into its components. This process involves breaking down a vector into simpler parts that are aligned with specific directions. In our swimming scenario, both the swimmer's stroke and the river's current have their influences, which we express as vectors.Each vector can be split into "i" and "j" components:

• "i" represents the east-west direction. The swimmer's effort of 2 mph becomes the vector component \( \overrightarrow{i} \), or specifically \( 2i \).
• "j" typically represents the north-south direction. The river's current affects this component, resulting in \( -3j \) as it flows south.

This decomposition allows us to visualize and mathematically manipulate the vectors to find the net effect. By simply adding these components, \( 2i - 3j \), we derive the total impact on the swimmer's motion, giving us their actual velocity vector.

Pythagorean Theorem

The Pythagorean theorem comes into play when we need to calculate the magnitude of a resultant vector formed by two perpendicular components. Once we decompose our vector into east-west and north-south components, these can be seen as the sides of a right triangle.In the case of our swimmer:

• The eastward component is 2 mph.
• The southward component is 3 mph.

We apply the Pythagorean theorem to determine the magnitude of the resultant vector, which represents the swimmer's actual speed across the river:\[\text{Magnitude} = \sqrt{(2^2) + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}\]This result tells us how fast the swimmer’s overall motion is, irrespective of direction. Using the theorem, we accurately condense the complex interplay of various forces into a single, understandable measure of speed. This way, we can see how both the swimmer's efforts and the river's push culminate in the swimmer's real travel rate.