91Ó°ÊÓ

Understanding relative velocity is crucial when analyzing motion in a fluid, such as swimming in a river. It refers to how the speed of one object appears from the perspective of another moving object. In our exercise, the student's swimming speed in still water is different from their speed in a moving river.

When the student swims upstream, we observe their motion relative to the shore. However, the river's current opposes the swimmer, effectively reducing their speed. In physics, we calculate the student's effective speed upstream by subtracting the river's velocity from the swimmer's velocity in still water, which is \(v_{effective, upstream} = v_{swimmer} - v_{river}\).

Conversely, when swimming downstream, the river's current aids the swimmer, increasing their effective speed. It's a classic relative velocity situation: \(v_{effective, downstream} = v_{swimmer} + v_{river}\). This application of relative velocity helps in critical problem-solving involving motion in fluids where two velocities interact.

The concept of effective speed is essential in situations where different velocities combine to affect the actual speed of an object. For the swimming exercise, we calculate the effective speed for both swimming upstream and downstream. This is the swimmer's actual speed relative to the riverbank.

Upstream, the swimmer labors against the current, which results in \(v_{effective, upstream} = 1.20 m/s - 0.500 m/s = 0.700 m/s\). Downstream, the current assists the swimmer, giving an effective speed of \(v_{effective, downstream} = 1.20 m/s + 0.500 m/s = 1.70 m/s\).

The tracking of effective speed in each direction enables us to understand the impact of external factors, like current, on the swimmer's performance and time calculations. This calculation is a perfect demonstration of how environmental conditions modify the capabilities of movers within a medium, such as air or water.

Analyzing time and distance in the context of motion involves a straightforward formula: \(time = \frac{distance}{speed}\). Applying this to our scenario, after calculating the effective speeds, we use them to determine the total time the swimmer takes to cover the distances upstream and downstream.

For the upstream swim, time is found by dividing the distance (1.00 km or 1000 m) by the effective speed upstream (\(time_{upstream} = \frac{1000 m}{0.700 m/s} = 1428.57 s\)). Similarly, for the downstream swim, the calculation is \(time_{downstream} = \frac{1000 m}{1.70 m/s} = 588.24 s\).

To get the total trip time, we simply add the times for both directions, resulting in a comprehensive understanding of the journey's duration. If the water was still, the total time would instead be twice the time taken for one direction at constant speed (\(time_{still water} = 2 \times \frac{1000 m}{1.20 m/s} = 1666.67 s\)), showcasing how currents impact the overall time taken for a swimmer to complete a route.