A swimmer, capable of swimming at a speed of 1.44 m/s in still water (i.e., the swimmer can swim with a speed of 1.44 m/s relative to the water), starts to swim directly across a 2.21-km-wide river. However, the current is 0.840 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 = Y/X = 1.44/0.840.
A = 55.9o N. of E.
sin55.9 = 1.44/Vr.
Vr = 1.44/sin55.9 = 1.74 m/s. = Resultant velocity.
a. Sin55.9 = 2210/d
d = 2210/sin55.9 = 2670 m To cross.
d = Vr*t = 2670.
1.74*t = 2670.
t = 1534 s. = 25.6 min.
b. D = d*Cos55.9=2670*Cos55.9 = 1497 m.
Well, well, well, looks like our swimmer is in for a wet and wavy ride! Let's dive into these math waters, shall we?
(a) To find out how long it takes for the swimmer to cross the river, we need to consider the swimmer's relative speed to the water. The current is working against our swimmer's progress, so we have to subtract the current's speed from the swimmer's speed.
Relative speed = Swimmer's speed - Current's speed
Relative speed = 1.44 m/s - 0.840 m/s
Relative speed = 0.6 m/s
Now it's time to calculate the time it takes to cross the river. We can use the formula Time = Distance / Speed. In this case, the distance is 2.21 km, but we need to convert it to meters, giving us 2210 m.
Time = Distance / Speed
Time = 2210 m / 0.6 m/s
Time = 3683.33 seconds
So, it takes our swimmer approximately 3683.33 seconds to cross the river. That's one long swim!
(b) Now let's figure out how far downstream our water-loving athlete will end up. To do that, we need to multiply the current speed by the time it took to cross the river.
Distance downstream = Current's speed x Time to cross the river
Distance downstream = 0.840 m/s x 3683.33 seconds
Distance downstream ≈ 3093.33 meters
Hence, our witty swimmer will end up approximately 3093.33 meters downstream. That's quite a journey!
Remember, it's all just math, no swimming pun intended. I bet that swimmer is now a master at going with the flow!
To solve this problem, we can break it down into two components: the swimmer's speed across the river (perpendicular to the current) and the current's speed downstream (parallel to the current).
Step 1: Find the time taken to cross the river.
To find the time taken to cross the river, we use the formula:
Time = Distance / Speed
The distance the swimmer needs to cross is the width of the river, which is 2.21 km or 2210 m. The speed across the river is the swimmer's speed in still water, which is 1.44 m/s.
Time = 2210 m / 1.44 m/s
Time = 1534.72 seconds or 1535 seconds (rounded to the nearest second)
Thus, it will take approximately 1535 seconds for the swimmer to cross the river.
Step 2: Find the distance downstream the swimmer will be.
To find the distance downstream, we can use the formula:
Distance downstream = Current speed x Time
The current speed is given as 0.840 m/s, and the time taken to cross the river is 1535 seconds.
Distance downstream = 0.840 m/s x 1535 s
Distance downstream = 1288.40 meters or 1288 meters (rounded to the nearest meter)
Therefore, the swimmer will be approximately 1288 meters downstream upon reaching the other side of the river.
To answer question (a), we need to find the time it takes for the swimmer to cross the river. We can use the concept of relative velocity to calculate this.
Relative velocity is the vector difference between the velocity of an object and the velocity of another object or medium. In this case, we need to find the relative velocity between the swimmer and the river.
Let's assume the swimmer is moving directly across the river from the starting point to the opposite bank. The river current is flowing perpendicular to the swimmer's motion, so it does not affect the swimmer's speed relative to the water. However, the current will affect the swimmer's motion relative to the ground.
To find the relative velocity between the swimmer and the ground, we can subtract the velocity of the river current from the swimmer's velocity.
Relative velocity = Swimmer's velocity - River current velocity
Given:
Swimmer's velocity (Vw) = 1.44 m/s (speed of the swimmer in still water)
River current velocity (Vc) = 0.840 m/s
Relative velocity (Vr) = Vw - Vc
Now, let's calculate the relative velocity of the swimmer:
Vr = 1.44 m/s - 0.840 m/s
Vr = 0.60 m/s
The swimmer's relative velocity with respect to the ground is 0.60 m/s.
To find the time it takes for the swimmer to cross the river, we need to divide the distance of the river by the swimmer's relative velocity.
Distance of the river = 2.21 km = 2210 m
Time (t) = Distance / Relative velocity
t = 2210 m / 0.60 m/s
Now, let's calculate the time it takes for the swimmer to cross the river:
t = 3683.33 seconds
So, it takes the swimmer approximately 3683.33 seconds (or about 61.39 minutes) to cross the river.
To answer question (b), we need to find the distance the swimmer will be downstream upon reaching the other side of the river. This can be calculated by multiplying the current velocity by the time taken to cross the river.
Distance downstream = River current velocity × Time taken to cross the river
Given:
River current velocity (Vc) = 0.840 m/s
Time taken to cross the river (t) = 3683.33 seconds
Now, let's calculate the distance the swimmer will be downstream:
Distance downstream = 0.840 m/s × 3683.33 seconds
Distance downstream ≈ 3091.67 meters
So, the swimmer will be approximately 3091.67 meters downstream upon reaching the other side of the river.