Vanessa can swim 3.0 m/s in still water. While trying to swim directly across a river from west to east, Vanessa is pulled by a current flowing southward at 2.0 m/s.
A) What is the magnitude of Vanessa’s resultant velocity?
B) If Vanessa wants to end up exactly across stream from where she began, at what angle to the shore must she swim upstream?
X = 3 m/s.
Y = -2 m/s.
a. (Vr)^2 = X^2 + Y^2 = 3^2 + (-2)^2=13
Vr = 3.61 m/s. = Resultant velocity.
b. tan A = Y/X = -2/3 = -0.66666
A = -33.7o = 33.7o S. of E.
Therefore, she must swim 33.7o N. of E.
To determine the magnitude of Vanessa's resultant velocity, we can use vector addition.
A) Let's consider the velocity of Vanessa relative to the ground. Since she is swimming from west to east, her velocity relative to the ground is 3.0 m/s in the eastward direction. However, due to the current flowing southward at 2.0 m/s, her velocity is also affected in the southward direction.
Using the Pythagorean theorem, we can calculate the magnitude of the resultant velocity by finding the hypotenuse of a right triangle formed by Vanessa's eastward velocity and the southward current:
Resultant velocity = √(vertical component^2 + horizontal component^2)
Vertical component (southward): 2.0 m/s
Horizontal component (eastward): 3.0 m/s
Resultant velocity = √(2.0^2 + 3.0^2) = √(4 + 9) = √13 ≈ 3.61 m/s
Therefore, the magnitude of Vanessa's resultant velocity is approximately 3.61 m/s.
B) To end up exactly across the stream from where she began, Vanessa needs to counteract the southward current by swimming slightly upstream. The angle she needs to swim upstream can be found using trigonometry. Let's find the angle, θ:
tan(θ) = vertical component / horizontal component
Since the vertical component is 2.0 m/s (the southward current) and the horizontal component is 3.0 m/s (Vanessa's eastward velocity), we can substitute these values into the equation:
tan(θ) = 2.0 / 3.0
Now we need to find the inverse tangent (arctan) of both sides:
θ = arctan(2.0 / 3.0)
Using a calculator, we can determine that θ is approximately 33.69 degrees.
Therefore, Vanessa needs to swim upstream at an angle of approximately 33.69 degrees to end up exactly across the stream from where she began.