A swimmer can swim in still water at a speed of 9.42 m/s. He intends to swim directly across a river that has a downstream current of 3.04 m/s.
hope he makes it. If it's the Mississippi, he might not.
To determine the swimmer's actual speed and direction relative to the riverbank, we need to break down the swimmer's speed into its horizontal and vertical components.
Let's call the swimmer's speed in still water "v_swimmer" and the speed of the downstream current "v_current."
The swimmer's actual speed relative to the riverbank (v_swimmer_actual) can be found using the Pythagorean theorem:
v_swimmer_actual = √((v_swimmer)^2 + (v_current)^2)
Plugging in the given values:
v_swimmer_actual = √((9.42 m/s)^2 + (3.04 m/s)^2)
v_swimmer_actual = √(88.5364 m^2/s^2 + 9.2416 m^2/s^2)
v_swimmer_actual = √(97.778 m^2/s^2)
v_swimmer_actual ≈ 9.89 m/s
Therefore, the swimmer's actual speed relative to the riverbank is approximately 9.89 m/s.
To determine the swimmer's direction relative to the riverbank, we need to use trigonometry. We can find the angle (θ) that the swimmer makes with the riverbank using the inverse tangent function:
θ = arctan(v_current / v_swimmer)
Plugging in the given values:
θ = arctan(3.04 m/s / 9.42 m/s)
θ = arctan(0.323)
Using a calculator, we find that θ is approximately 17.32 degrees.
Therefore, the swimmer needs to swim at an actual speed of approximately 9.89 m/s and at an angle of approximately 17.32 degrees relative to the riverbank to cross the river.