A swimmer is capable of swimming 0.95 m/s in still water.
(a) If she aims her body directly across a 150-m-wide river whose current is 0.65 m/s, how far downstream (from a point opposite her starting point) will she land?
To find out how far downstream the swimmer will land, we need to determine the time it takes for her to cross the river and use this time to calculate the distance traveled downstream.
Let's denote the swimmer's velocity in still water as V_swimmer = 0.95 m/s and the velocity of the river's current as V_current = 0.65 m/s.
The swimmer will be under the influence of both her own velocity and the velocity of the current as she swims across the river. To find the total velocity, we can use the vector addition of these velocities.
The horizontal component of the swimmer's velocity (V_swimmer_horizontal) will remain unaffected by the current since it is perpendicular to the river's flow. It can be calculated using the Pythagorean theorem:
V_swimmer_horizontal = √(V_swimmer^2 - V_current^2)
= √(0.95^2 - 0.65^2)
≈ √(0.9025 - 0.4225)
≈ √0.48
≈ 0.69 m/s
Now we can calculate the time it takes for the swimmer to cross the river using the formula:
Time = Distance / Velocity
Time = 150 m / V_swimmer_horizontal
Time = 150 m / 0.69 m/s
Time ≈ 217.39 s (or seconds)
Lastly, we can calculate the distance traveled downstream using the formula:
Distance downstream = Time * V_current
Distance downstream = 217.39 s * 0.65 m/s
Distance downstream ≈ 141.3 m
Therefore, the swimmer will land approximately 141.3 meters downstream from her starting point.
To find out how far downstream the swimmer will land, we need to consider the combination of the swimmer's velocity and the river's current.
Let's break down the problem:
1. The swimmer's speed in still water is 0.95 m/s. We'll call this the swimmer's "swimming speed" (S).
2. The river has a current of 0.65 m/s. The current is the speed at which the river is flowing (C).
3. The swimmer is aiming directly across the river. This means her motion across the river will be perpendicular to the current.
Now, let's calculate the velocity of the swimmer relative to the ground:
The swimmer's velocity relative to the ground (V) is the vector sum of her swimming speed and the current's speed:
V = S + C
In this case, V = 0.95 m/s + 0.65 m/s = 1.60 m/s
Since the swimmer is aiming directly across the river, her motion perpendicular to the current will be constant. Therefore, we can use the equation:
Distance (d) = Velocity (V) × Time (t)
To find the distance downstream the swimmer will land, we need to determine the time it takes for her to cross the river.
The width of the river is given as 150 m. Therefore, we can rearrange the equation to solve for time:
Time (t) = Distance (d) / Velocity (V)
Substituting the given values:
t = 150 m / 1.60 m/s ≈ 93.75 s
Now that we have the time, we can find out how far downstream the swimmer will land:
Distance downstream (d') = Current (C) × Time (t)
Substituting the values:
d' = 0.65 m/s × 93.75 s ≈ 60.94 m
Therefore, the swimmer will land approximately 60.94 meters downstream from her starting point.
The time it will take to cross the river while aimed straight across will be 150 m/0.95 m/s = 157.9 s
During that time, she will drift
157.9 * 0.65 = 102.6 meters downstream