A rowboat crosses a river with a velocity of 3.42 mi/h at an angle 62.5° north of west relative to the water. The river is 0.805 mi wide and carries an eastward current of 1.25 mi/h. How far upstream is the boat when it reaches the opposite shore?
north speed = 3.42 sin 62.5 = 3.03
time = .805/3.03 = .265 hours
west speed = 3.42 cos 62.5 - 1.25
= .329 mi/hr
distance west = .329*.265 = .0872 miles
By the way to cross a river fast, forget where you end up downstream and row or swim directly across in direction, not trying to get the right component upstream to end up across from where you started. This is important if swimming and tired. Do not fight the current.
To determine how far upstream the boat is when it reaches the opposite shore, we need to break down the given information and solve the problem step by step.
1. Draw a diagram: Draw a diagram representing the situation. Label the river's width as 0.805 mi and the boat's velocity as 3.42 mi/h at a 62.5° angle north of west.
2. Resolve the boat's velocity: Divide the boat's velocity into horizontal and vertical components. The vertical component is given by: v_vertical = v * sin(θ), where v is the velocity (3.42 mi/h) and θ is the angle (62.5°). Calculate the value for v_vertical.
v_vertical = 3.42 mi/h * sin(62.5°)
= 2.994 mi/h
3. Calculate the time to cross the river: Since the boat crosses the river, we need to determine how long it takes to cross. The time can be calculated as t = d / v_horizontal, where d is the river's width (0.805 mi) and v_horizontal is the horizontal velocity of the boat. Calculate the value for t.
t = 0.805 mi / (3.42 mi/h)
≈ 0.235 h (rounded to three decimal places)
4. Determine the distance traveled upstream: The distance traveled upstream can be found using the formula: d_upstream = v_vertical * t. Calculate the value for d_upstream by multiplying the vertical velocity (2.994 mi/h) by the time taken to cross (0.235 h).
d_upstream = 2.994 mi/h * 0.235 h
≈ 0.705 mi (rounded to three decimal places)
Therefore, the boat is approximately 0.705 miles upstream when it reaches the opposite shore.