A river is 2000 ft wide and flowing at 6 mph from north to south. A woman in a canoe starts on the eastern shore and heads west at her normal paddling speed of 2 mph. In what direction (measured from North) will she actually be traveling. How far downstream from a point directly across the river will she land?

Question

A river is 2000 ft wide and flowing at 6 mph from north to south. A woman in a canoe starts on the eastern shore and heads west at her normal paddling speed of 2 mph. In what direction (measured from North) will she actually be traveling. How far downstream from a point directly across the river will she land?

Answer 1

she is traveling on an angle west of south

... the tangent of the angle is ... 2/6

the river is flowing 3 times as fast as the woman paddles
... she will end up three times the width of the river , downstream

Answer 2

All angles are measured CW from +y-axis.

a. Vr = 2mi/h[270] + 6mi/h[180o],
Vr = (2*sin270+6*sin180) + (2*cos270+6*cos180)I,
Vr = (-2+0) + (0-6)I = -2 - 6i = 6.32[18.4o] W of S. = 6.32mi/h[198.4o] CW.
Direction = 198.4 degrees CW.

b. Tan18.4 = 2000/Y,
Y = 6,012 Ft. downstream.

Answer 3

To determine the direction in which the woman will be traveling, we need to understand the concept of vector addition. The woman's velocity has two components: one due to her normal paddling speed to the west and the other due to the river's flow to the south.

First, let's find the vector representing the woman's normal paddling speed. Since she is paddling west at a speed of 2 mph, this can be represented as a vector with a magnitude of 2 mph pointing directly west.

Next, let's find the vector representing the river's flow. The river is flowing from north to south at a speed of 6 mph. Since the woman is crossing the river from east to west, the river's flow can be represented as a vector with a magnitude of 6 mph pointing directly south.

To find the resulting vector representing the woman's actual direction, we need to add these two vectors. To do this, we can use the Pythagorean theorem.

The woman's actual velocity vector can be represented as the hypotenuse of a right triangle, with the legs representing her paddling speed and the river's flow. The magnitude of the resulting vector can be found using the equation:

Resultant_velocity = sqrt((paddling_speed^2) + (river_flow^2))

Substituting the values, we get:

Resultant_velocity = sqrt((2^2) + (6^2))
= sqrt(4 + 36)
= sqrt(40)
= 2 * sqrt(10)

So, the magnitude of the woman's resultant velocity is 2 * sqrt(10) mph.

To find the angle at which she will be traveling from the north, we can use inverse trigonometry. The angle (θ) can be calculated using the equation:

θ = arctan(paddling_speed / river_flow)

Substituting the values:

θ = arctan(2 / 6)
= arctan(1 / 3)

Using a calculator or trigonometry tables, we find that:

θ ≈ 18.43 degrees

Therefore, the woman will be traveling at an angle of approximately 18.43 degrees west of north.

To find how far downstream she will land, we need to calculate the time it takes to cross the river and then multiply it by the river's flow rate.

The time taken to cross the river can be calculated using the equation:

Time = Distance / Speed

The distance across the river is given as 2000 ft. Since the woman's paddling speed is 2 mph, we need to convert her speed to ft/min to match the units of distance:

Paddling speed = 2 mph = (2 * 5280) ft / (60 * 60) min = 58.67 ft/min

Substituting the values:

Time = 2000 ft / 58.67 ft/min
≈ 34.06 min

Therefore, it will take approximately 34.06 minutes for the woman to cross the river.

Multiplying the time by the river's flow rate of 6 mph, we get:

Distance downstream = 6 mph * (34.06 min / 60 min/hr)
≈ 3.42 miles

So, the woman will land approximately 3.42 miles downstream from a point directly across the river.