Problem 67 - Physics for Scientists and Engineers with Modern Physics

Question

A swimmer is capable of swimming \(0.60 \mathrm{~m} / \mathrm{s}\) in still water. \((a)\) If she aims her body directly across a \(55-\mathrm{m}\) -wide river whose current is \(0.50 \mathrm{~m} / \mathrm{s}\), how far downstream (from a point opposite her starting point) will she land? (b) How long will it take her to reach the other side?

Step-by-Step Solution

Answer

(a) She lands 45.84 meters downstream. (b) It takes her 91.67 seconds to cross.

Step 1: Understanding the Problem

We have a swimmer who swims straight across a river. We need to determine how far downstream she will land due to the river's current and how long it will take to cross the river.

Step 2: Analyzing the Problem Vertically

The width of the river determines the vertical distance the swimmer must cross. She swims with a velocity of \(0.60 \mathrm{~m/s}\) perpendicularly to the current. The width of the river is 55 meters. The time to cross can be calculated by dividing the river width by her swimming speed.

Step 3: Calculating the Crossing Time

The time \(t\) to cross can be calculated as:\[ t = \frac{\text{width of river}}{\text{speed of swimmer in still water}} = \frac{55 \text{ m}}{0.60 \text{ m/s}} \]Calculating this gives:\[ t = 91.67 \text{ seconds} \]

Step 4: Analyzing Downstream Displacement

While the swimmer is crossing the river, the current moves her downstream. The downstream velocity is \(0.50 \text{ m/s} \). To find how far downstream she lands, we multiply the time to cross by the speed of the current.

Step 5: Calculating Downstream Distance

The downstream distance \(d\) is given by:\[ d = \text{velocity of current} \times t = 0.50 \text{ m/s} \times 91.67 \text{ s} \]Calculating this gives:\[ d = 45.84 \text{ meters} \]

Key Concepts

River Crossing

When a swimmer attempts to cross a river by swimming straight across, their trajectory doesn’t follow a straight line due to the river's current. The motion combines the swimmer's own swimming speed with the river's current. When we analyze such a river crossing scenario, two main components come into play:

• The width of the river, which is the perpendicular distance the swimmer needs to cover.
• The lateral movement caused by the river current, which pushes the swimmer downstream.

In our example, the swimmer aims at swimming perpendicular to the current. Her crossing is influenced by the river's current, which moves her downstream from the direct landing spot. By understanding these components, we can calculate both her actual trajectory across the river and where she will finally land on the opposite bank.

Velocity Analysis

Analyzing motion in problems involving multiple velocities requires breaking down motions into components. This is especially true in projectile motion or in our swimmer's case while crossing a river. The swimmer’s speed and the river current speed both contribute to her actual path.
The swimmer's velocity is directed perpendicularly across the river. This is her speed determined by how fast she can swim in still water, here it's given as \(0.60 \text{ m/s}\).

• This forms the vertical component of her velocity.

The river current, flowing parallel to the banks, represents the horizontal velocity component pushing the swimmer downstream at \(0.50 \text{ m/s}\).

• By examining the two velocities separately, we understand the resultant path.

To find out how far downstream the swimmer ends up, we use the resultant effect of the horizontal velocity over the crossing time.

Time of Flight

Calculating the time required for the swimmer to reach the opposite bank is crucial for understanding her entire crossing journey. In this scenario, often referred to as time of flight, the river's width and the swimmer's pace in still water decide the crossing time.
By dividing the river's width by the swimmer's speed, we obtain the time taken to traverse the distance perpendicularly. For our swimmer:\[ t = \frac{55 \text{ m}}{0.60 \text{ m/s}} = 91.67 \text{ seconds}\]This calculated time also helps determine how much influence the current has during her crossing. While she is swimming across, it's over this time period that the current shifts her position downstream. This highlights the concept of time of flight in two-dimensional motion, essential in understanding the complete trajectory of her crossing.