91Ó°ÊÓ

In river crossing problems, understanding vector components is essential. A vector has both magnitude and direction, and can be broken down into horizontal and vertical components. In this exercise, we are dealing with two main velocities: the river flowing south at 2.0 m/s and the motorboat moving east at 4.2 m/s. These represent perpendicular vector components, allowing us to treat them independently in calculations.

• The southward vector (river) influences movement along the north-south axis.
• The eastward vector (boat) affects the east-west movement.

Understanding the independent nature of these components helps in calculating the resultant velocity, which gives us the net effect of both vectors.

The Pythagorean theorem is crucial for calculating the magnitude of the resultant velocity when dealing with orthogonal vector components. The formula states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b):
\[ c = \sqrt{a^2 + b^2} \]
In the context of our problem, the eastward velocity (4.2 m/s) and the southward velocity (2.0 m/s) form a right triangle, where the resultant velocity is the hypotenuse. Plugging in these values:
\[ v_{resultant} = \sqrt{(4.2)^2 + (2.0)^2} \approx 4.65 \text{ m/s} \]
The magnitude of this resultant velocity is the boat's actual speed relative to the earth, factoring in both the river's current and the boat's power.

Trigonometric functions are indispensable in determining the angle of the resultant velocity. Since we have perpendicular vector components, we use the tangent of the angle to find the direction relative to east. The tangent function relates the opposite side to the adjacent side in a right triangle. Here, the components are given as:

• Opposite (southward) = 2.0 m/s
• Adjacent (eastward) = 4.2 m/s

Thus, the angle \(\theta\) can be calculated using:
\[ \tan(\theta) = \frac{2.0}{4.2} \]
\[ \theta = \tan^{-1}\left( \frac{2.0}{4.2} \right) \approx 25.8^\circ \]
This angle, measured from the east direction, illustrates that the boat's path is directed south of east due to the river's influence.

River crossing dynamics involve understanding how a current affects a boat's journey across a body of water. In this situation, the boat aims to cross directly eastward, but the river's southward flow causes a deviation. To solve such a problem, it’s important to consider both the crossing time and any lateral displacement.

• The crossing time is determined by the component of velocity directly across the river, which is 4.2 m/s here. The time to cross the 800-meter-wide river is calculated as:
  \[ \text{time} = \frac{800}{4.2} \approx 190.48 \text{ seconds} \]
• The lateral drift is the downstream movement caused by the river's current, calculated by multiplying the velocity of the current with the time:
  \[ \text{drift} = 2.0 \times 190.48 \approx 380.96 \text{ meters} \]

These calculations reveal how far south the boat lands from its intended destination, demonstrating the complex dynamics between steady movement and natural forces.