A river 99.1 m wide flows toward the south at 31.5 m/min. A girl on the west bank wishes to reach the east bank in the least possible time. She can swim 99.1 m in still water in 1.80 min.
What is her velocity relative to land?
magnitude
In what direction must she swim if she wishes to travel straight across the river?________upstream
To find the girl's velocity relative to the land, we need to find the resultant velocity of her swim and the river's flow.
First, let's find the girl's swimming velocity in still water. She can swim 99.1 m in 1.80 min.
Velocity (v) = Distance (d) / Time (t)
v = 99.1 m / 1.80 min
Next, let's find the river's flow velocity. The river is flowing toward the south at 31.5 m/min.
Now, to find the girl's velocity relative to the land (Vr), we use vector addition.
If the girl swims perpendicular to the river's flow, her velocity relative to the land (Vr) will be the vector sum of her swimming velocity (ves) and the river's flow velocity (ver).
Vr = ves + ver
If we consider the direction of the river's flow as positive south and the direction of the girl's swim as positive west, we can use the Pythagorean theorem to find the magnitude of her velocity relative to land (Vr) and trigonometry to find the angle (θ) at which she must swim.
Magnitude of Vr:
Vr = sqrt((ves)^2 + (ver)^2)
Direction of Vr (θ):
θ = tan^(-1) (ver / ves)
By calculating the magnitude and direction, we can determine the girl's velocity relative to the land.
Do you want to calculate the magnitude and direction?
To find the girl's velocity relative to land, we need to use the concept of vector addition.
Let GW be the velocity of the girl in still water (99.1 m/1.80 min) and VR be the velocity of the river (31.5 m/min) towards the south.
The girl's velocity relative to land (VL) can be determined by finding the vector sum of GW and VR. Since the river flows southward, the direction will be opposite to the river's velocity.
To find VL, we can use the Pythagorean theorem:
VL^2 = GW^2 + VR^2
Substituting the given values:
VL^2 = (99.1 m/1.80 min)^2 + (31.5 m/min)^2
Simplifying the equation:
VL^2 = (551.39 m^2/min^2) + (992.25 m^2/min^2)
VL^2 = 1543.64 m^2/min^2
Taking the square root of both sides:
VL ≈ 39.3 m/min
Therefore, the girl's velocity relative to land is approximately 39.3 m/min.
To travel straight across the river, the girl must swim upstream at a velocity equal to the velocity of the river. Since the river is flowing towards the south, the girl must swim northward, opposite to the direction of the current.