Veronica can swim 3.0 m/s in still water. While trying to swim directly across a river from west to east,?
Veronica is pulled by a current flowing southward at 2.0m/s.
a) What is the magnitude of Vernoicas resultant velocity?
b) If Veronica wants to end up exactly across the stream from where she began. At what angle to the shore must shr swim upstream?
consider the loop in the figure. The area=600m, and it spins with angular velocity =40.0 rad/s in a magnetic field of strnght B=.400 T
To solve this problem, we can use vector addition to find the resultant velocity of Veronica.
a) For the magnitude of Veronica's resultant velocity, we can use the Pythagorean theorem. The vector addition is as follows:
Resultant velocity (V_r) = √((velocity in still water)^2 + (velocity of the current)^2)
= √((3.0 m/s)^2 + (2.0 m/s)^2)
Calculating this expression:
V_r = √(9.0 m^2/s^2 + 4.0 m^2/s^2)
= √(13.0 m^2/s^2)
≈ 3.60 m/s
Therefore, the magnitude of Veronica's resultant velocity is approximately 3.6 m/s.
b) To find the angle at which Veronica needs to swim upstream, we can use trigonometry. We know that the velocity of the current is southward, so the angle will be measured with respect to the north direction (upstream). Let's denote this angle as θ.
Using the formula for tangent (tan):
tan(θ) = (velocity of the current) / (velocity in still water)
= 2.0 m/s / 3.0 m/s
= 0.67
To find the angle, we can take the inverse tangent (arctan) of both sides of the equation:
θ = arctan(0.67)
Calculating this expression:
θ ≈ 33.69°
Therefore, Veronica needs to swim upstream at an angle of approximately 33.69° to end up exactly across the stream from where she began.
To find the magnitude of Veronica's resultant velocity, we can use the concept of vector addition.
a) Magnitude of Resultant Velocity:
1. Draw a diagram to represent the scenario. Draw a horizontal line to represent the river's flow from the west to the east (from left to right). Then draw a vertical line downwards to represent the southward current.
2. Now draw Veronica's velocity vector, pointing towards the east (to the right) with a magnitude of 3.0 m/s.
3. Draw the current velocity vector, pointing directly south (downwards) with a magnitude of 2.0 m/s.
4. To find the resultant velocity, we need to add the two vectors.
- Add the eastward velocity vector (3.0 m/s) to the southward velocity vector (2.0 m/s).
- The resulting vector should connect the starting point of the eastward vector to the ending point of the southward vector.
5. Now, use the Pythagorean theorem to find the magnitude of the resultant velocity vector.
- The magnitude of the resultant velocity (Vres) can be calculated as Vres = √(Vx² + Vy²), where Vx and Vy are the x and y components of the resultant velocity.
- In this case, Vx represents the eastward velocity component (which is 3.0 m/s) and Vy represents the southward velocity component (which is 2.0 m/s).
- Substituting the values: Vres = √((3.0 m/s)² + (2.0 m/s)²)
- Vres = √(9.0 m²/s² + 4.0 m²/s²) = √(13.0 m²/s²) = 3.6 m/s (approximate)
Therefore, the magnitude of Veronica's resultant velocity is approximately 3.6 m/s.
b) Angle to Swim Upstream:
To find the angle at which Veronica must swim upstream, we can use trigonometry.
1. In the diagram, draw a line from the starting point to the ending point of the resultant velocity vector to create a right triangle with the horizontal and vertical axes.
2. The vertical axis represents the southward direction, and the horizontal axis represents the eastward direction.
3. We know the magnitudes of the two sides of the right triangle: the vertical side (southward component) is 2.0 m/s, and the horizontal side (eastward component) is 3.0 m/s.
4. To find the angle, we can use the inverse tangent function. The angle we're looking for, let's call it θ (theta).
- We can use the formula: tan(θ) = (opposite/adjacent), where the opposite side is the southward component, and the adjacent side is the eastward component.
- Rearranging the formula gives: θ = atan(opposite/adjacent)
- Substituting the values: θ = atan(2.0 m/s / 3.0 m/s)
- θ ≈ 33.7° (rounded to one decimal place)
Therefore, Veronica must swim at an angle of approximately 33.7° upstream to end up exactly across the stream from where she began.
b: She has to have a component upstream of 2.0m/s to avoid going down stream.
The angle she goes upstream is
theta=arcSin(2/3)
the velocity across the stream then is
3cosTheta=3cos(arcSin(2/3))
a) if she swims directy across, she does downstream, her velocity is sqrt(2^2+3^2)