Problem 42 - Algebra and Trigonometry

Question

Velocity A migrating salmon heads in the direction \(\mathrm{N} 45^{\circ} \mathrm{E}\) , swimming at 5 \(\mathrm{mi} / \mathrm{h}\) relative to the water. The prevailing ocean currents flow due east at 3 \(\mathrm{mi} / \mathrm{h}\) . Find the true velocity of the fish as a vector.

Step-by-Step Solution

Answer

The true velocity vector of the fish is approximately \(\langle 6.54, 3.54 \rangle \) mi/h.

Step 1: Understand the Problem

We need to find the resultant velocity vector of the salmon, combining its own swimming velocity and the ocean current's velocity.

Step 2: Resolve Salmon Velocity

The salmon's velocity is given as 5 mi/h towards N 45° E. We can interpret this as the salmon moving with an equal component towards the north and east. Hence, the salmon's velocity vector will be: \[ \langle 5\cos(45°), 5\sin(45°) \rangle = \langle \frac{5}{\sqrt{2}}, \frac{5}{\sqrt{2}} \rangle \approx \langle 3.54, 3.54 \rangle. \]

Step 3: Determine Ocean Current Velocity

The ocean current flows due east at 3 mi/h, giving it the vector: \[ \langle 3, 0 \rangle. \]

Step 4: Find the Resultant Velocity

The true velocity of the fish is the sum of the salmon's velocity vector and the ocean current's velocity vector. This is calculated as follows: \[ \langle 3.54, 3.54 \rangle + \langle 3, 0 \rangle = \langle 6.54, 3.54 \rangle. \]

Step 5: Interpret the Resultant Vector

The vector \(\langle 6.54, 3.54 \rangle\) represents the true east and north components of the salmon's velocity. Thus, the fish's true velocity, in mi/h, is approximately 6.54 east and 3.54 north.

Key Concepts

Resultant Velocity

When two or more velocity vectors are combined, the total vector that results from this addition is known as the "resultant velocity." In the case of our migrating salmon, it experiences not just its own swimming speed but also the effect of the ocean currents. Hence, we need to combine these two vectors to find out how fast and in which direction the salmon is truly moving.

• The salmon swims at 5 mi/h in a direction that is North 45° East.
• The ocean current adds a velocity of 3 mi/h directly to the east.

To compute the true velocity, we perform vector addition: add the salmon's swimming vector to the current's vector. The resultant vector gives a true path that the salmon follows, combining the effects of both speeds. It directs the salmon slightly more eastward than north due to the current. Calculating the resultant velocity reveals how these two factors interact, giving us the complete picture of the salmon's movement. This resulting velocity is crucial, as it represents the combined effect of all forces acting upon the fish, measuring its actual path through the water.

Vector Components

Vectors have direction and magnitude, and describing a vector often requires breaking it down into its components. For the salmon's velocity, expressed as N 45° E, it has both northward and eastward components. This helps us think in terms of horizontal and vertical directions, providing a clearer picture of movement in each plane.

• The salmon's northward component arises from its direction, indicating how much of the 5 mi/h is directed north.
• The eastward component explains how much of that same speed goes east.

In mathematical terms, we can resolve this into vector components using trigonometric functions:\[\text{Northward Component} = 5 \times \cos(45^\circ)\]\[\text{Eastward Component} = 5 \times \sin(45^\circ)\]Both components end up being equal, approximately making the salmon's velocity in components \( \langle 3.54, 3.54 \rangle \). These vector components are essential for further operations, like finding resultant velocities or understanding movement patterns across different axes.

Trigonometric Functions

Trigonometric functions play a vital role in resolving vectors into components, which is critical in physics and engineering. They help relate angles to the sides of a right triangle, which is how we determine the components for our vector in question.

• Cosine Function: In our scenario, \( \cos(45^\circ) \) is used to find the component of the salmon's velocity in the northward direction. It calculates how much of the full velocity goes in the vertical or y-axis.
• Sine Function: Similarly, \( \sin(45^\circ) \) helps determine the eastward component, measuring the part of the velocity extending along the horizontal or x-axis.

Using these functions, we divide a vector inclined at an angle into vertical and horizontal pieces. At 45°, both sine and cosine equal \( \frac{1}{\sqrt{2}} \), meaning our vector components turn out identical for this case. This concept is particularly handy when dealing with vectors at an angle, as trigonometry directly links angles with movement in space, offering a precise method to calculate vector components.