At what upstream angle must the swimmer aim, if she is to arrive at a point directly across the stream of a 42-m-wide river whose current is 0.50 m/s?
A swimmer is capable of swimming 0.62m/s in still water.
Draw a diagram. If the swimmer swims upstream at an angle of θ, it is clear that
sin θ = 0.50/0.65
Doesn't matter how wide the stream is, does it?
Well, if the swimmer wants to reach the other side of the river, she better aim for somewhere between Zigzag Lane and Crooked Street! But in all seriousness, let's break it down. We have a 42-meter-wide river with a current of 0.50 m/s, and a swimmer who can swim 0.62 m/s in still water. To reach the other side, she needs to compensate for the current's effect. Now, my calculations might be a bit fishy, but if she aims upstream at an angle of π/3 radians, or approximately 60 degrees, she should be able to cross the river with a smile on her face and avoid any fishy business. Good luck to the swimmer, and may the currents be ever in her favor!
To determine the upstream angle the swimmer must aim, we can use the concept of vector addition.
Let's assume that the swimmer's swimming speed relative to the ground is Vg and the speed of the river current is Vc. Given that the swimmer's swimming speed in still water is 0.62 m/s (Vg) and the river current speed is 0.50 m/s (Vc), we can determine the resulting speed (Vs) and direction.
To find the speed of the swimmer relative to the ground (Vs), we can use the Pythagorean theorem:
Vs^2 = Vg^2 + Vc^2
Vs^2 = (0.62 m/s)^2 + (0.50 m/s)^2
Vs^2 = 0.3844 m^2/s^2 + 0.25 m^2/s^2
Vs^2 = 0.6344 m^2/s^2
Vs = √(0.6344 m^2/s^2)
Vs ≈ 0.796 m/s
Now, we can use trigonometry to determine the upstream angle (θ).
tan(θ) = Vc / Vg
tan(θ) = 0.50 m/s / 0.62 m/s
tan(θ) ≈ 0.806
θ ≈ arctan(0.806)
θ ≈ 38.96 degrees
Therefore, the swimmer must aim at an upstream angle of approximately 38.96 degrees to arrive at a point directly across the stream.
To determine the upstream angle the swimmer must aim, we need to consider the velocity of the swimmer relative to the water and the velocity of the river current.
Let's break down the problem step by step:
Step 1: Determine the velocity of the swimmer relative to the water.
The swimmer's swimming speed in still water is given as 0.62 m/s. Since the direction the swimmer is aiming is not specified yet, we'll consider the swimmer swimming at an angle to the current. Let's denote the swimmer's velocity relative to the water as Vswimmer.
Step 2: Determine the velocity of the river current.
The river current is given as 0.50 m/s. Let's denote the velocity of the river current as Vcurrent.
Step 3: Determine the resulting velocity of the swimmer when swimming across the stream (perpendicular to the current).
Since the swimmer is aiming directly across the stream, perpendicular to the current, we can use vector addition to find the resulting velocity (Vresultant). The magnitude of the resulting velocity will be the square root of the sum of the squares of the velocities in the x and y directions.
Vresultant = √(Vswimmer^2 + Vcurrent^2)
Step 4: Determine the angle at which the swimmer must aim.
The angle at which the swimmer should aim can be found using trigonometry. Since we have the magnitudes of the sides opposite and adjacent to the angle, we can use the inverse tangent function (arctan) to find the angle.
Angle = arctan(Vcurrent / Vswimmer)
Step 5: Calculate the answer.
Substitute the given values into the equation:
Angle = arctan(0.50 / 0.62)
Using a calculator, we find:
Angle ≈ 39.6 degrees
Therefore, the swimmer must aim at an upstream angle of approximately 39.6 degrees to end up directly across the stream.