Problem 67 - Physics for Scientists and Engineers with Modern Physics

Question

(1I) 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 -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) 45.84 m downstream. (b) 91.67 seconds.

Step 1: Determine the time to cross the river

The swimmer's speed perpendicularly to the riverbank is 0.60 m/s. The river is 55 m wide. To find how long it takes to cross, use the formula for time: \[ t = \frac{\text{distance}}{\text{speed}} = \frac{55 \, \text{m}}{0.60 \, \text{m/s}} \]. Calculating this gives: \[ t = \frac{55}{0.60} \approx 91.67 \, \text{s} \].

Step 2: Calculate downstream distance

While the swimmer moves across the river, the current pushes her downstream with a speed of 0.50 m/s. To find out how far downstream she lands, use the relation:\[ \text{Distance}_{\text{downstream}} = \text{speed}_{\text{current}} \times \text{time} \]. Using Step 1's result, calculate:\[ \text{Distance}_{\text{downstream}} = 0.50 \, \text{m/s} \times 91.67 \, \text{s} \approx 45.835 \, \text{m} \].

Key Concepts

Relative Velocity

Relative velocity is an important concept in understanding the motion of objects in different frames of reference. In the context of river crossing, it refers to the effective velocity of the swimmer relative to the banks of the river. This velocity is a combination of the swimmer's velocity in still water and the velocity of the river current.

To visualize this, imagine the swimmer aiming directly across the river. In still water, she swims at a speed of 0.60 m/s. The river also has its own current moving sideways at 0.50 m/s. These two vectors combine to create the swimmer's actual or relative velocity in relation to the riverbank.

To understand relative velocity more intuitively, it may help to consider a general formula: when two velocities are at right angles (as is often the case with river crossings), use the Pythagorean theorem to determine any resultant velocities. In this case, the swimmer's effective path across the river is diagonal, composed of her own swimming and the current's push downstream. Understanding how vectors interact in this way is crucial for tackling similar physics problems.

River Crossing

Crossing a river presents a unique challenge due to the presence of the current. It involves navigating both across and downstream.

• When crossing, swimmers must factor in the width of the river and their ability to go straight across, irrespective of the current's influence.
• The current impacts where the swimmer actually lands, carrying them further downstream.

In this example, imagine if the swimmer wanted to land directly opposite the starting point on the other bank. To counteract the current pushing her downstream, she would have to angle upstream—akin to aiming for a spot upriver. It's a balance between fighting the current and maintaining her intended path across. Therefore, river crossing scenarios in physics often force us to think about the interaction of different forces and speeds, and how they compose the resultant motion of an object.

Time Calculation

Calculating the time required for the swimmer to cross the river requires a straightforward application of the formula for time, given constant speed. The formula is simply:\[ t = \frac{\text{distance}}{\text{speed}} \]In this instance, the distance is the width of the river (55 m), and the speed is the swimmer's speed perpendicular to the river (0.60 m/s).

• By substituting the known values into the formula, we find that it takes her approximately 91.67 seconds to cross the river.
• This calculation assumes the current only affects lateral movement, not speed across.

This understanding is crucial as the calculated time informs subsequent downstream calculations, by establishing how long the swimmer is subjected to the current's influence. Accurately calculating the time helps in solving for other factors affected by the time taken to traverse the river.

Distance Calculation

Distance calculation is critical in determining how far downstream the swimmer will end up due to the river's current. As the swimmer crosses, the river current pushes her consistently sideways, and we measure this effect using the following relation:\[ \text{Distance}_{\text{downstream}} = \text{speed}_{\text{current}} \times \text{time} \]Here, the given speed of the river is 0.50 m/s. With the time calculated as 91.67 seconds, the swimmer's downstream drift is:

• Approximately 45.835 meters.
• This drift means even though she aimed directly across the river, the swimmer would land significantly further downstream from her starting point.

Comprehending how distance is calculated with continuous lateral influence by a current provides a foundational principle in physics, illustrating how the environment affects an object's trajectory and correcting for external factors becomes vital in navigation and motion prediction.