A swimmer is capable of swimming 1.00 m/s in still water.
If she aims her body directly across a 150-m-wide river whose current is 0.80 m/s, how far downstream (from a point opposite her starting point) will she land?
To solve this problem, we can use the concept of relative velocity. Here are the steps to determine how far downstream the swimmer will land:
1. Determine the swimmer's speed relative to the ground:
- The swimmer's speed in still water is given as 1.00 m/s.
- Since the river current is 0.80 m/s, the swimmer's speed relative to the ground will be the vector sum of her speed in still water and the current.
- The speed relative to the ground can be calculated using the formula: speed relative to the ground = √((speed in still water)^2 + (current speed)^2).
- Plugging in the values, we get: speed relative to the ground = √((1.00 m/s)^2 + (0.80 m/s)^2).
2. Calculate the time it takes for the swimmer to cross the river:
- The width of the river is given as 150 m.
- The time taken to cross the river can be calculated using the formula: time = distance / speed.
- Plugging in the values, we get: time = 150 m / (sqrt((1.00 m/s)^2 + (0.80 m/s)^2)).
3. Determine the distance downstream:
- The swimmer's speed relative to the ground will also be the rate at which they will be carried downstream.
- The distance downstream can be calculated by finding the product of the swimmer's speed relative to the ground and the time taken to cross the river.
- Plugging in the values, we get: distance downstream = (sqrt((1.00 m/s)^2 + (0.80 m/s)^2)) * (time).
Now, you can perform the calculations to find the distance downstream where the swimmer will land.
To find out how far downstream the swimmer will land, we need to determine the effect of the river's current on her motion.
Let's break down the problem into two components: the swimmer's forward velocity across the river and the river's current velocity.
1. The swimmer's forward velocity across the river:
- The swimmer's velocity in still water is given as 1.00 m/s. This is her speed relative to the riverbanks.
- Since she aims her body directly across the river, her velocity components can be represented as follows: The forward velocity across the river is equal to her speed in still water, which is 1.00 m/s.
2. The river's current velocity:
- The river's current is given as 0.80 m/s. This represents the velocity of the river's flow.
- The current is acting perpendicular to the swimmer's intended path.
To determine the net velocity of the swimmer relative to the riverbanks, we need to use vector addition. The net velocity (v_net) is the resultant vector formed by adding the swimmer's forward velocity (v_swimmer) and the river's current velocity (v_current).
Using the Pythagorean theorem, we can find the magnitude of the net velocity:
v_net = √(v_swimmer^2 + v_current^2)
v_net = √(1.00^2 + 0.80^2)
v_net = √(1.00 + 0.64)
v_net = √1.64
v_net ≈ 1.28 m/s
Now that we have the magnitude of the net velocity, we can calculate the distance the swimmer will drift downstream. This can be done using the formula:
Distance downstream = (Net velocity) × (Time taken)
In this case, the time taken can be determined by considering the river's width:
Time taken = (River width) / (Net velocity)
Time taken = 150 m / 1.28 m/s
Time taken ≈ 117.19 s
Finally, we can find the distance downstream using:
Distance downstream = (Current velocity) × (Time taken)
Distance downstream = 0.80 m/s × 117.19 s
Distance downstream ≈ 93.75 m
Therefore, the swimmer will land approximately 93.75 meters downstream from her starting point, across the 150-meter-wide river.
I see two similar triangles, her distance downstream as x m
x/.8 = 150/1
x = 120
Her time to cross will be T = 150/0.8 seconds. During that time, she will drift downstream by 0.80 T meters.
Do the numbers.