Problem 58 - Precalculus Mathematics for Calculus

Question

Velocity A migrating salmon heads in the direction 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 is approximately \(6.54 \hat{i} + 3.54 \hat{j}\).

Step 1: Understand the problem

We need to find the true velocity of the salmon, taking into account both its swimming direction and the ocean current. The salmon swims at a velocity of 5 mi/h in the direction N 45° E, while the current flows due east at 3 mi/h.

Step 2: Express salmon's velocity as a vector

The direction N 45° E can be interpreted as 45 degrees east of north. The velocity vector for the salmon can be given by breaking it into its north and east components: \[V_s = 5 \cos 45^{\circ} \hat{i} + 5 \sin 45^{\circ} \hat{j}\]Simplifying, we get:\[V_s = \frac{5}{\sqrt{2}} \hat{i} + \frac{5}{\sqrt{2}} \hat{j}\]This results in:\[V_s = \frac{5\sqrt{2}}{2} \hat{i} + \frac{5\sqrt{2}}{2} \hat{j}\]where \(\hat{i}\) represents the east direction and \(\hat{j}\) the north direction.

Step 3: Express current's velocity as a vector

The ocean current flows due east at 3 mi/h. This can be represented as a vector where the north component is zero:\[V_c = 3 \hat{i} + 0 \hat{j}\]

Step 4: Add the vectors to find the true velocity

To find the true velocity of the salmon, we add the salmon's velocity vector to the current's velocity vector:\[V_{total} = V_s + V_c\]\[V_{total} = \left( \frac{5\sqrt{2}}{2} + 3 \right) \hat{i} + \frac{5\sqrt{2}}{2} \hat{j}\]

Step 5: Calculate and simplify the components

Calculate the components:\[ \frac{5\sqrt{2}}{2} \approx 3.54 \]So the east component is:\[ 3.54 + 3 = 6.54 \] and the north component remains:\[ 3.54 \]Thus, the total velocity vector is approximately:\[ V_{total} \approx 6.54 \hat{i} + 3.54 \hat{j} \]

Step 6: State the final true velocity vector

The true velocity vector of the salmon is approximately:\[ V_{total} \approx 6.54 \hat{i} + 3.54 \hat{j} \]where \(\hat{i}\) is the unit vector in the east direction and \(\hat{j}\) is the unit vector in the north direction.

Key Concepts

Vector Addition

Vector addition is an important concept in physics and mathematics that involves combining different vectors to form a single resultant vector. Imagine you are trying to figure out the salmon's true velocity as it swims in the ocean with a prevailing current. This means two velocity vectors are affecting the salmon - its swimming speed in the water and the speed of the ocean currents.
To add vectors, you should:

• Break down each vector into its components, typically along the east (horizontal) and north (vertical) directions.
• Add the corresponding components from each vector together to get the resultant vector.

By adding the vectors for swimming and current, you determine the fish's actual path and speed. This process allows you to accurately calculate the true velocity using vector algebra, ultimately leading to better understanding of how different forces combine in directional motion.

Velocity

Velocity is a vector quantity, which means it has both magnitude and direction. It is an essential part of understanding how objects move through space. In the case of the salmon moving through water, its velocity depends on its swimming speed and direction relative to the ocean current.
To calculate velocity:

• Determine the magnitude, which is how fast the object is moving (in mi/h in this case).
• Establish the direction, given initially as N 45° E for the salmon.

Combining these two elements helps you establish what is known as the velocity vector. Without considering both magnitude and direction, you don't have a complete picture of the object's movement path. By analyzing velocity, you gain a comprehensive view of how various factors influence motion.

Components of Vectors

Understanding components of vectors is crucial when dealing with problems involving motion in two dimensions. Vectors like the salmon's swimming path need to be broken down into smaller, manageable parts to simplify calculations.
The components:

• Make use of trigonometric functions to resolve the vector into perpendicular directions, often marked as east (î) and north (ĵ).
• In the given scenario, the salmon's velocity of 5 mi/h north-east is split into east ( î) and north (ĵ) parts, using cos and sin functions of 45°.

These vector components facilitate the addition by enabling you to simply add the north and east components of two vectors separately. By understanding components, it's much easier to visualize and calculate how different vectors interact, leading to solutions for problems such as the salmon's true velocity.