91Ó°ÊÓ

In physics, vector addition helps us find a resultant vector when multiple vectors are acting together. Imagine vectors as arrows. They have both direction and magnitude. To find the overall effect or resultant vector, we need to "add" these arrows. In our river crossing problem, the boat is subject to two main vectors:

• The current's velocity of 2.0 m/s flowing south.
• The boat's own velocity of 4.2 m/s heading east.

Instead of just adding them like numbers, we treat them geometrically. We form a right triangle where these two velocities are the legs and the resultant velocity is the hypotenuse. This tribunal approach gives us a simple yet effective way to visualize how different forces—or vectors—interact in the real world.

The Pythagorean theorem is crucial when dealing with perpendicular vectors. It allows us to calculate the magnitude of the resultant vector. This theorem 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). Mathematically, it's written as: \[ c = \sqrt{a^2 + b^2} \]In our scenario, the boat's eastward speed is 4.2 m/s, and the river's southward speed is 2.0 m/s. These values serve as a and b, respectively. Plugging into the formula, we find:\[ v_{earth} = \sqrt{(2.0)^2 + (4.2)^2} \approx 4.65 \, \text{m/s} \]This resultant speed is the boat's velocity relative to the earth, considering both the river's flow and the boat's motor.

Calculating the angle of direction helps us understand where the resultant velocity is pointing. When two vectors are acting, they won't always point straight in one direction. They create an angle, showing the mixed direction of forces. To find this angle, we use tangent, a trigonometric function. Knowing the opposite (southward) and adjacent (eastward) sides of our triangle:\[ \theta = \arctan\left(\frac{\text{opposite}}{\text{adjacent}}\right) \]In this river crossing, it looks like:\[ \theta = \arctan\left(\frac{2.0}{4.2}\right) \approx 25.5^\circ \]So, our angle indicates a path that is about 25.5 degrees south of east. This means the boat moves eastward, but also slightly towards the south due to the river's current.

When crossing a river with a current, understanding both the vector addition and the time it takes is fundamental. Here, even if the motorboat travels directly across the river (eastward), the flow will carry it downstream (southward). Hence, it doesn't end up directly across but displaced somewhat south.To find how long it takes to cross, we consider the distance perpendicular to the current—here, the river's width of 800 meters—and divide by the component of the velocity pointing in that direction (4.2 m/s east). Thus:\[ t = \frac{800}{4.2} \approx 190.5 \, \text{seconds} \]Additionally, the boat is pushed southward during this time by the current. The southward displacement, calculated using the velocity along the current and time, is:\[ d = 2.0 \times 190.5 \approx 381 \, \text{m} \]This value tells us how far downstream the boat ends up from its initial point once it reaches the opposite bank.