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 clockwise from north) must she aim her canoe? How long will it take her to go directly across the river? Justify and explain your reasoning.
To answer the 2nd part, there's no way she can travel directly across the river. The current is faster than she can paddle, so even if she heads directly upstream, she will be carried south.
So, you ask "In what direction must she aim her canoe?" To do what? She can row due west, and end up 6000 ft downstream, or she can pick any other NW-ish heading and end up a little closer to directly across, but she will still be carried downstream.
To determine the direction the woman must aim her canoe, we can use basic trigonometry. Let's consider the velocity vectors involved:
1. The river's velocity vector is flowing from north to south at 6 mph.
2. The woman's velocity vector is paddling west at 2 mph.
To determine the resulting velocity vector, we can add these two vectors together. Since the vectors are perpendicular to each other (one going north-south and the other going east-west), we can use the Pythagorean theorem to find the magnitude of the resulting vector:
Resultant velocity = √((6 mph)^2 + (2 mph)^2)
= √(36 mph^2 + 4 mph^2)
= √40 mph^2
≈ 6.32 mph
Now, to find the direction of the resultant velocity vector, we can use inverse trigonometric functions. Here, we will use the arctan function, which will give us the angle measured relative to the north direction.
tan(θ) = (2 mph) / (6 mph)
θ = arctan(2/6)
θ ≈ 18.43 degrees
Since we want the direction measured clockwise from north, we subtract this angle from 90 degrees:
Direction = 90 degrees - 18.43 degrees
Direction ≈ 71.57 degrees
Therefore, the woman must aim her canoe approximately 71.57 degrees (measured clockwise from north).
To determine how long it will take the woman to go directly across the river, we can use the concept of relative motion. Since her actual speed is 6.32 mph in the resultant direction, we can calculate the time using the formula:
Time = Distance / Speed
The distance she needs to cross is the width of the river, which is given as 2000 ft. We need to convert this to miles, so we divide it by 5280 (the number of feet in a mile):
Distance = (2000 ft) / (5280 ft/mi)
≈ 0.3788 miles
Time = (0.3788 miles) / (6.32 mph)
≈ 0.06 hours
Therefore, it will take the woman approximately 0.06 hours (or 3.6 minutes) to go directly across the river.
Note: This explanation assumes that the woman's paddling speed remains constant throughout the crossing and that there are no other factors affecting the motion, such as wind or currents.