Problem 58 - 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

True velocity of salmon: approximately \( (6.54, 3.54) \) mi/h.

Step 1: Define Directions

First, we need to understand the directions given in the problem. The vector for the salmon's velocity is directed north-east (N 45° E), meaning it is 45 degrees towards east from the north direction.

Step 2: Set Up the Salmon's Velocity Vector

The salmon's velocity is 5 mi/h at an angle of 45 degrees from the north. We can use trigonometry to find its components. \( v_{salmon} = (5 \cos(45°), 5 \sin(45°)) \) which simplifies to \( v_{salmon} = \left( \frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2} \right) \approx (3.54, 3.54) \) mi/h.

Step 3: Define the Current's Velocity Vector

The ocean current flows due east at 3 mi/h, so its vector is \( v_{current} = (3, 0) \).

Step 4: Add the Vectors

The true velocity of the fish is the vector sum of the salmon's velocity and the current's velocity. Add the vectors: \( v_{true} = v_{salmon} + v_{current} = (3.54, 3.54) + (3, 0) = (6.54, 3.54) \).

Step 5: Conclusion

The true velocity of the salmon, taking into account the ocean current, is a vector with components approximately \( (6.54, 3.54) \) mi/h, indicating its velocity eastward and northward.

Key Concepts

Understanding Trigonometry in Vector Decomposition

Trigonometry is a powerful tool for breaking down vectors into components. When dealing with directions, such as a salmon swimming at an angle, we can use trigonometric functions to simplify the problem. Consider a vector represented by its magnitude and direction on a coordinate plane. Decomposing it involves determining how much of the vector points in each direction, usually the x (horizontal) and y (vertical) axes.

For example, a salmon moving at 45 degrees north-east with a speed of 5 mi/h uses trigonometry to find how far it goes in each direction. The cosine function helps us find the horizontal component, while the sine function helps with the vertical component:

• Horizontal (x-axis) component: \( 5 \cos(45^{\circ}) \)
• Vertical (y-axis) component: \( 5 \sin(45^{\circ}) \)

These calculations yield the vector's decomposition: \( \left( \frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2} \right) \) or approximately \( (3.54, 3.54) \). Understanding these components is crucial because they help determine the true path of an object, factoring in both directional movement and speed.

Analyzing Velocity Vectors

Velocity vectors are vital in physics as they represent the speed and direction of moving objects. They are typically expressed in components, such as the horizontal and vertical velocities, which allow for detailed analysis and calculations. When you visualize a velocity vector, imagine an arrow where the length corresponds to speed and the angle indicates direction.

In our salmon example, we have two key vectors: the salmon's own swimming velocity and the water current. Each influences the fish's actual path. For the salmon:

• Speed: 5 mi/h
• Direction: 45° north-east

For the ocean current:

• Speed: 3 mi/h
• Direction: due east

By adding these vectors, you calculate the resultant vector that shows the salmon's true velocity relative to the earth. This process, called vector addition, involves combining the individual components to get a final vector that reflects both influences.

Impact of Ocean Currents on Movement

Ocean currents are powerful forces that can significantly alter the course of objects moving through water. They are like conveyor belts that carry water and everything in it over great distances. When an object, such as a migrating salmon, moves through these currents, its path can change based on the current's direction and speed.

Let's say a current moves due east at 3 mi/h. Any object traveling within this water will also experience a push of this velocity in the same direction. For our salmon, this means its north-east trajectory will shift eastward.

• The salmon's own vector is \( (3.54, 3.54) \), representing a north-east path.
• The eastward current vector is \( (3, 0) \), adding directly to the horizonal component.

The combined effect, calculated as \( (6.54, 3.54) \), shows the salmon's true speed and course. Factoring in ocean currents is essential in navigation and planning to predict accurate paths for ships, sea animals, and drifting objects.