Final answer:
Using the velocities given and the width of the river, the swimmer will take approximately 151.11 seconds to cross the river, and during this time, the current will carry the swimmer 60.44 meters downstream.
Step-by-step explanation:
The question involves a swimmer who aims to cross a river in a direction perpendicular to the water current. We are asked to calculate how far downstream the swimmer will land, which is a classic problem in vector addition found in many physics textbooks. Applying principles of relative velocity, we calculate the downstream drift caused by the river's current while the swimmer swims across.
To determine the downstream distance, we will use the concept of relative velocities. The swimmer's velocity across the river is perpendicular to the current's velocity. Our strategy involves finding the time it takes for the swimmer to get across the river and then using this time to determine how far the swimmer is carried downstream by the current.
The swimmer's speed in still water is 0.45 m/s, and the river current's speed is 0.40 m/s. Since these two velocities are perpendicular to each other, we can use the Pythagorean theorem to calculate the resultant velocity. However, for this particular problem, we do not actually need to calculate the resultant velocity since the question only asks how far downstream the swimmer will go, not the resultant speed of the swimmer.
To find out how long it will take the swimmer to cross the 68 m-wide river:
- Time to cross = Width of the river / swimmer's speed = 68 m / 0.45 m/s = 151.11 seconds (to two decimal places)
Now, to calculate how far downstream the swimmer will be carried by the current during this time:
- Downstream distance = Current speed * Time to cross = 0.40 m/s * 151.11 s = 60.444 m (to three decimal places)
Therefore, the swimmer will land approximately 60.44 meters downstream from a point opposite her starting point.