A river has a steady speed of 0.276 m/s. A student swims upstream a distance of 1.4 km and swims back to the starting point. If the student can swim at a speed of 1.15 m/s in still water, how long (in minutes) does the trip take?
How do I solve this problem? Thank You
speed upstream = 1.15 - .276 = .874 m/s
time upstream = 1.4 * 10^3/.874 = 1601 seconds
speed downstream - 1.15 + .276 = 1.426
time downstream = 1.4*10^3/1.426 = 982 seconds
total time = 1601+982 = 2583 seconds
2583/60 = 43 minutes
Why didn't you add vectorially
To solve this problem, we need to use the concept of relative velocity and the formula \( \text{{speed}} = \frac{{\text{{distance}}}}{{\text{{time}}}} \).
Let's break down the problem:
1. The river has a steady speed of 0.276 m/s. This means that the current of the river is flowing downstream at a speed of 0.276 m/s.
2. The student swims upstream a distance of 1.4 km. Since the student is swimming against the current, the effective speed will be the difference between the student's swimming speed and the river's speed. Therefore, the student's effective speed while swimming upstream will be \(1.15 \, \text{m/s} - 0.276 \, \text{m/s} = 0.874 \, \text{m/s}\).
3. Now, we need to calculate the time the student takes to swim upstream. We can use the formula \( \text{{speed}} = \frac{{\text{{distance}}}}{{\text{{time}}}} \) to find the time. Rearranging the formula, we have \( \text{{time}} = \frac{{\text{{distance}}}}{{\text{{speed}}}} \). Converting the distance to meters, we have \( \text{{distance}} = 1.4 \, \text{km} \times 1000 \, \text{m/km} = 1400 \, \text{m} \). Plugging in the values, we get \( \text{{time}}_{\text{{upstream}}} = \frac{{1400 \, \text{m}}}{{0.874 \, \text{m/s}}} \).
4. Next, the student swims back to the starting point, downstream this time. When the student is swimming downstream, the effective speed is the sum of the student's swimming speed and the river's speed. Therefore, the student's effective speed while swimming downstream will be \(1.15 \, \text{m/s} + 0.276 \, \text{m/s} = 1.426 \, \text{m/s}\).
5. We can use the same formula \( \text{{speed}} = \frac{{\text{{distance}}}}{{\text{{time}}}} \) to find the time taken to swim downstream. Rearranging the formula, we have \( \text{{time}} = \frac{{\text{{distance}}}}{{\text{{speed}}}} \). Plugging in the values, we get \( \text{{time}}_{\text{{downstream}}} = \frac{{1400 \, \text{m}}}{{1.426 \, \text{m/s}}} \).
6. Finally, to find the total time for the trip, we add the time taken to swim upstream and the time taken to swim downstream: \( \text{{total time}} = \text{{time}}_{\text{{upstream}}} + \text{{time}}_{\text{{downstream}}} \).
Now you can substitute the values into the equations and calculate the total time taken, in minutes.