A river has a steady speed of 0.470 m/s. A student swims upstream a distance of 1.00 km and swims back to the starting point

The effective speed of the student while going upstream (i.e , the velocity of the student with respect to the river ) is 0.75 m/s whereas the same while returning downstream is 1.45 m/s .

So , total time taken = 1000/0.75 + 1000/1.45 = 2023 seconds = 33 minutes 43 seconds .

If he were swimming in still water , then this time would be = 2000/1.1 = 30 minutes 18 seconds .
Thus the trip would be shorter by 3 min 25 sec .

To find the time it takes for the student to swim upstream and back, we can use the concept of relative velocity. Let's break down the problem into two parts: swimming upstream and swimming downstream.

1. Swimming upstream:
The student is swimming against the current, so the effective velocity will be the difference between the student's swimming speed and the river's speed. Let's call the speed of the student "v" m/s. The effective velocity while swimming upstream will be:
Effective velocity upstream = v - 0.470 m/s

The distance to swim upstream is 1.00 km, which is equal to 1000 m. Using the formula:
Distance = Speed × Time
We can rearrange the formula to solve for time:
Time upstream = Distance / Effective velocity upstream

2. Swimming downstream:
Now, the student is swimming with the current, so the effective velocity will be the sum of the student's swimming speed and the river's speed. Using the same variable "v" for the student's speed, the effective velocity while swimming downstream will be:
Effective velocity downstream = v + 0.470 m/s

The distance to swim downstream is also 1.00 km or 1000 m. Again, using the formula:
Distance = Speed × Time
We can rearrange the formula to solve for time:
Time downstream = Distance / Effective velocity downstream

To find the total time, we need to calculate the time for each segment and add them together:
Total time = Time upstream + Time downstream

Now, to find the student's swimming speed, we know that the time taken for swimming upstream will be the same as for swimming downstream. So, we can set the equations for time upstream and time downstream equal to each other:
Distance / (v - 0.470 m/s) = Distance / (v + 0.470 m/s)

By cross-multiplying and simplifying, we can solve for v:
(v - 0.470 m/s) = (v + 0.470 m/s)
0.470 m/s = 0.470 m/s
0 = 0

This means that the equation has no unique solution, which implies that the student's swimming speed is irrelevant in this scenario. The total time taken will be simply twice the time taken to swim upstream or downstream.

We can now calculate the total time:
Total time = 2 × Time upstream (or Time downstream)

Plug in the distance of 1.00 km or 1000 m for each segment:
Total time = 2 × 1000 m / Effective velocity upstream (or Effective velocity downstream)

Substitute the given river speed of 0.470 m/s into the equation:
Total time = 2 × 1000 m / (v - 0.470 m/s) (or (v + 0.470 m/s))

The total time can now be evaluated using the given values.