A man swims across the river with a velocity of 2 m/s relative to the flow of water. If the river flows steadily at 4 m/s toward south, what is the velocity of the man with respect to the shore? Solve for both the magnitude and the direction (in degrees).
92
yah dawg
To find the velocity of the man with respect to the shore, we can use vector addition.
Let's define the following vectors:
- Vm: Velocity of the man with respect to the shore
- Vr: Velocity of the river
- Vw: Velocity of the man with respect to the flow of water
Vm = Vw + Vr
Given:
- Vw = 2 m/s
- Vr = 4 m/s (towards the south)
Now, let's find the magnitude of Vm:
Since the man is swimming perpendicular to the flow of water, we can use Pythagoras theorem to find the magnitude of Vm:
|Vm|^2 = |Vw|^2 + |Vr|^2
|Vm|^2 = (2 m/s)^2 + (4 m/s)^2
|Vm|^2 = 4 m^2/s^2 + 16 m^2/s^2
|Vm|^2 = 20 m^2/s^2
Taking the square root of both sides:
|Vm| = sqrt(20 m^2/s^2)
|Vm| ≈ 4.47 m/s
So, the magnitude of the man's velocity with respect to the shore is approximately 4.47 m/s.
Now, let's find the direction of Vm:
Since the river is flowing towards the south, and the man is swimming perpendicular to the flow, the direction of Vm will be at an angle with respect to the shore.
To find this angle, we can use trigonometry. We'll use the inverse tangent function (tan⁻¹) to calculate the angle:
angle = tan⁻¹(Vw / Vr)
angle = tan⁻¹(2 m/s / 4 m/s)
angle = tan⁻¹(0.5)
Using a calculator, we find that angle ≈ 26.57 degrees
So, the velocity of the man with respect to the shore is approximately 4.47 m/s, at an angle of 26.57 degrees.
Vm=velocity of man
Vr=velocity of river
Vmr=velocity of man relative to river
Vmr=Vm+Vr
=2i+4j
=4.47m/s
direction@ an angle of 63.4 degree