A river has a steady speed of 0.640 m/s. A student swims upstream a distance of 1.00 km and swims back to the starting point.
(a) If the student can swim at a speed of 1.15 m/s in still water, how long does the trip take?
(b) How much time is required in still water for the same length swim?
(c) Intuitively, why does the swim take longer when there is a current?
time = distance/speed, so
(a) 1000/(1.15+.640) + 1000/(1.15-.640) = 2519.44 s ≈ 42 minutes
see what you can do with the other parts
(a) Well, let's calculate the time it takes for the student to swim upstream first. The student's effective speed against the current would be 1.15 m/s - 0.640 m/s = 0.510 m/s. Using the formula time = distance / speed, we can find the time it takes to swim upstream: time = 1000 m / 0.510 m/s = 1960.78 seconds.
Now, let's calculate the time it takes to swim downstream. The student's effective speed with the current would be 1.15 m/s + 0.640 m/s = 1.790 m/s. Again using the formula time = distance / speed, the time it takes to swim downstream would be: time = 1000 m / 1.790 m/s = 558.66 seconds.
To find the total time for the trip, we need to add the time for swimming upstream and the time for swimming downstream: total time = 1960.78 seconds + 558.66 seconds = 2519.44 seconds.
Therefore, the trip takes approximately 2519.44 seconds.
(b) If there were no current, the student's speed would simply be 1.15 m/s. Using the formula time = distance / speed, the time it would take for the same length swim in still water would be: time = 1000 m / 1.15 m/s = 869.57 seconds.
So, without the current, it would take approximately 869.57 seconds.
(c) Well, intuitively speaking, the swim takes longer when there is a current because the current opposes the swimmer's motion, making it harder and slower to swim upstream. It's like trying to walk on a treadmill going the opposite direction – you have to work harder and it takes more time to reach your destination. It's just nature's sneaky way of adding a little challenge to our aquatic adventures!
To find the answers to the questions, we can use the formula:
Time = Distance/Speed
(a) To find the time it takes for the trip, we first need to calculate the total distance traveled. The student swims upstream for a distance of 1.00 km, and then swims back to the starting point, covering the same distance again. Therefore, the total distance traveled is 2.00 km.
To find the time, we can use the formula:
Time = Distance/Speed
Time = 2.00 km / (1.15 m/s - 0.640 m/s)
Note: We subtract the river's speed because swimming upstream is against the current, thus reducing the effective speed.
Converting km to meters:
1.00 km = 1000 meters
Time = 2000 meters / (1.15 m/s - 0.640 m/s)
(b) To find the time required for the same length swim in still water (without any current), we can simply use the formula:
Time = Distance/Speed
Time = 2.00 km / 1.15 m/s
Converting km to meters:
1.00 km = 1000 meters
Time = 2000 meters / 1.15 m/s
(c) Intuitively, the swim takes longer when there is a current because the student has to swim against the current while going upstream. This reduces the effective speed, making it harder for the student to cover the same distance in the same amount of time compared to swimming in still water. On the way back downstream, the current helps the student, resulting in a faster effective speed and a shorter time for the same distance.
To solve this problem, we can use the concept of relative velocity. Let's break down the problem into three parts:
(a) Swimming upstream: When swimming upstream, the student swims against the current. The effective speed will be the difference between the student's swimming speed and the river's speed.
The effective speed while swimming upstream is: 1.15 m/s - 0.640 m/s = 0.51 m/s
The distance to swim upstream is 1.00 km which is equal to 1000 m.
Using the formula time = distance/speed, we can calculate the time taken:
Time taken to swim upstream = Distance/Speed = 1000 m / 0.51 m/s ≈ 1960.78 seconds.
(b) Swimming downstream: When swimming downstream, the student swims with the current. The effective speed will be the sum of the student's swimming speed and the river's speed.
The effective speed while swimming downstream is: 1.15 m/s + 0.640 m/s = 1.79 m/s
The distance to swim downstream is the same, 1.00 km or 1000 m.
Using the formula time = distance/speed, we can calculate the time taken:
Time taken to swim downstream = Distance/Speed = 1000 m / 1.79 m/s ≈ 558.66 seconds.
(c) The swim takes longer when there is a current because when swimming upstream, the student has to swim against the current, which reduces their effective speed. On the other hand, when swimming downstream, the student benefits from the current, which increases their effective speed. Hence, the time taken to swim against the current is longer compared to swimming in still water, while the time taken to swim with the current is shorter.