A rowboat crosses a river with a velocity of 4.23 mi/h at an angle 62.5o north of west relative to the water. The river is 0.543 mi wide and carries an eastward current of 0.73 mi/h. How far upstream is the boat when it reaches the opposite shore?
To solve this problem, we need to break down the velocities into their horizontal and vertical components.
Let's start with the velocity of the boat. The given velocity is 4.23 mi/h at an angle of 62.5° north of west. To calculate the horizontal and vertical components of this velocity, we can use trigonometry.
Horizontal Component of Boat's Velocity:
Vx = Vboat * cos(θ)
Vx = 4.23 mi/h * cos(62.5°)
Vx = 4.23 mi/h * 0.4545
Vx = 1.924 mi/h
Vertical Component of Boat's Velocity:
Vy = Vboat * sin(θ)
Vy = 4.23 mi/h * sin(62.5°)
Vy = 4.23 mi/h * 0.891
Vy = 3.773 mi/h
Next, let's consider the velocity of the river. The given current is moving eastward at 0.73 mi/h. Since the boat is crossing the river, we need to consider this current's effect on the boat's motion.
Horizontal Component of River's Velocity: This component is in the same direction as the boat's velocity, so we add them.
Vrx = Vcurrent = 0.73 mi/h
Vertical Component of River's Velocity: This component does not affect the boat's motion in the y-direction since the boat is moving perpendicular to it.
Vry = 0 mi/h
Now we can calculate the net velocity of the boat relative to the shore by considering the horizontal and vertical components.
Net Velocity in the x-direction:
Vnet,x = Vx + Vrx
Vnet,x = 1.924 mi/h + 0.73 mi/h
Vnet,x = 2.654 mi/h
Net Velocity in the y-direction:
Vnet,y = Vy + Vry
Vnet,y = 3.773 mi/h + 0 mi/h
Vnet,y = 3.773 mi/h
To find the time it takes for the boat to cross the river, we can use the distance and the net velocity in the x-direction.
Time = Distance / Velocity
Time = 0.543 mi / 2.654 mi/h
Time ≈ 0.2046 hours
Finally, we can find the distance the boat drifts upstream during this time, considering the net velocity in the y-direction.
Distance Upstream = Vnet,y * Time
Distance Upstream = 3.773 mi/h * 0.2046 hours
Distance Upstream ≈ 0.772 mi
Therefore, the boat is approximately 0.772 miles upstream when it reaches the opposite shore.