91Ó°ÊÓ

In physics, the velocity of an object is often not in a straight line, hence, it's essential to break it into components. For a boat moving through water, like in our original exercise, this means dividing the velocity into directions that are easier to work with. Typically, these directions are north, south, east, and west.
To find these components, we use trigonometry, particularly the sine and cosine functions. For the boat moving at a velocity of 8 m/s, aimed 30° west of north, we determine:

• The northward component: Using the cosine function, \( v_{b,n} = 8.0 \cdot \cos(30^{\circ}) \approx 6.93 \ ext{m/s} \).
• The westward component: With the sine function, \( v_{b,w} = 8.0 \cdot \sin(30^{\circ}) = 4\ ext{m/s} \).

By resolving these components, we can predict how the boat travels northward and westward, relative to the water's current.

Trigonometry plays a significant role in physics, particularly when analyzing non-linear motion like projectile or simultaneous target problems. It allows us to break down motion into perpendicular components, using three main functions: sine, cosine, and tangent.
In the given scenario, the angle provided (30°) assists in determining how much of the boat's speed is directed north and how much is west.

• Sine (\( \sin \)) delivers the component opposite the angle, representing the boat's westward movement, achieved via \( v_{b,w} = 8.0 \cdot \sin(30^{\circ}) \).
• Cosine (\( \cos \)) yields the adjacent side of the angle triangle, illustrating the northward velocity, found by \( v_{b,n} = 8.0 \cdot \cos(30^{\circ}) \).

This breakdown is indispensable, as it simplifies complex motion problems, translating angles into measurable directions, which can then be processed using simple vector addition or subtraction to uncover combined effect or resultant velocities.

Relative velocity is essential to understand real-world scenarios where objects move differently from their surroundings. It depicts the speed of one object as observed from another moving object.
In our exercise, the boat's relative velocity to the ground differs from the velocity relative to the water. The river's current impacts the boat's path, demanding an adjustment in velocity components. The river moves east at 2.5 m/s, so even if the boat is set just at 30° from a northward path, real-world effects skew it westward, requiring calculations like:

• Northward velocity remains as \( v_{n} = 6.93 \ ext{m/s} \).
• Eastward affects yield \( v_{e} = 2.5 - 4 = -1.5 \ ext{m/s} \), indicating a resultant westward movement.

By assessing these relative flows, you can accurately determine how long it takes the boat to bridge a given span. Moreover, it gets easier to compute resultant velocity using Pythagorean theory, allowing us to give 6.93 m/s with a direction of 77.5° north of west, forming the finalized path observed from the riverbank.