91Ó°ÊÓ

When the woman rows across the river, she has to account for the current to ensure that she reaches her intended destination. This concept is known as relative velocity. In simple terms, relative velocity is the velocity of an object considering the velocities of other influencing factors, such as the current of the river in this case.

Imagine the woman rowing directly across at her full speed of 6.4 km/h without considering the current. The river's current, moving at 3.2 km/h, would push her downstream. Thus, to achieve her desired destination directly across the river, she needs to incorporate the current's velocity into her planning. This involves adjusting the angle at which she rows, influencing her relative velocity components in two directions:

• Across the river (perpendicular to the current).
• Along with the river (in the direction of the current).

By considering the current, she ensures that her path results in her desired endpoint across the river. Understanding relative velocity simplifies determining the right angle and speed needed to navigate accurately.

The angle at which the rower crosses the river is vital for compensating for the current. If she rows straight across, the current would carry her downstream. Instead, she must row at an angle to negate this effect. The key is finding the correct angle, denoted by \(\theta\), to ensure she ends up directly opposite her starting point.

To find this angle, consider her speed in still water (6.4 km/h). When adjusting for the current, the downstream velocity component must cancel out. If the downstream speed needs to be zero, according to the problem:

\[6.4 \sin(\theta) + 3.2 = 0\]

Solving this equation gives her the correct angle \(\theta = 210^\circ\). This angle ensures that the upstream rowing component perfectly counters the river's current, allowing her to reach her target directly across the river.

Time to cross the river depends on the effective speed across the river. This effective speed is determined using the angle of crossing. Given the boat can row at 6.4 km/h in still water and her calculated angle \(\theta\), her effective speed is defined by:

\[v_{\text{effective}} = 6.4 \cos(210^\circ) = -5.536 \text{ km/h}\]

Despite the negative sign indicating direction, we care about the speed's magnitude. The width of the river is given as 6.4 km. Thus, the time to cross is:

\[t = \frac{6.4}{5.536} \approx 1.156 \text{ hours}\]

This calculation shows how quickly she can traverse the river when accounting for her angle of crossing. Understanding effective speed is crucial for determining crossing time accurately.

The river's current significantly affects her motion and travel time. When considering the current, it's essential to acknowledge its two main impacts:

• Pushing downstream: The current at 3.2 km/h pulls her away from a straight crossing path.
• Impact on time: Traveling upstream or downstream affects her total travel time.

**Crossing Without Correction:**
Without correcting for the current by setting an angle, she would need to row directly across at her full speed of 6.4 km/h. In this case, her shortest crossing time without current influence is just:
\[t = \frac{6.4}{6.4} = 1 \text{ hour}\]

**Round Trips Involving Current:**
In scenarios where she rows downstream and back or upstream and back, her travel time changes significantly. For instance:

• Downstream: Faster due to additional current speed.
• Upstream: Slower as the current resists her motion.

Recognizing the river current's role is fundamental for planning her journey effectively. This understanding ensures she not only reaches her destination but does so in the desired time frame.