A swimmer, capable of swimming at a speed of 1.81 m/s in still water (i.e., the swimmer can swim with a speed of 1.81 m/s relative to the water), starts to swim directly across a 2.29-km-wide river. However, the current is 1.31 m/s, and it carries the swimmer downstream. (a) How long does it take the swimmer to cross the river? (b) How far downstream will the swimmer be upon reaching the other side of the river?
Tan A = 1.31/1.81 = 0.72376.
A = 35.9o
Tan 35.9 = d/2.290.
d1 = 2.290*Tan35.9 = 1.658 km downstream.
sin35.9 = 1.31/V2.
V2 = 1.31/sin35.9 = 2.23 m/s
d2 = sqrt(2.29^2+1.66^2) = 2.83 km =
Distance across with wind.
d2 = V2*t = 2830 m.
2.23*t = 2830.
t = 1269 s. = 21 Min. to cross.
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To solve this problem, we need to break it down into components and apply the concept of relative velocity.
Let's start by finding the time it takes for the swimmer to cross the river (part a).
The swimmer's velocity relative to the water is 1.81 m/s. Since the current is flowing downstream, it adds to the swimmer's velocity, making the effective velocity across the river greater than the swimmer's velocity relative to the ground.
To find the effective velocity of the swimmer across the river, we can use the concept of relative velocity:
Effective velocity = swimmer's velocity relative to the water + current's velocity
Effective velocity = 1.81 m/s + 1.31 m/s
Effective velocity = 3.12 m/s
Now, to find the time it takes for the swimmer to cross the river, we can use the formula:
Time = distance / velocity
Distance across the river = 2.29 km = 2290 m
Velocity = effective velocity = 3.12 m/s
Time = 2290 m / 3.12 m/s
Time ≈ 733.97 seconds (rounded to two decimal places)
Therefore, it takes approximately 733.97 seconds for the swimmer to cross the river (part a).
Now, let's move on to part b, which asks how far downstream the swimmer will be upon reaching the other side of the river.
To find the distance downstream, we can use the formula:
Distance downstream = current's velocity × time taken to cross the river
Current's velocity = 1.31 m/s
Time taken to cross the river = 733.97 seconds (rounded from part a)
Distance downstream = 1.31 m/s × 733.97 seconds
Distance downstream ≈ 961.98 meters (rounded to two decimal places)
Therefore, the swimmer will be approximately 961.98 meters downstream when reaching the other side of the river (part b).