A boat crosses a river and arrives at a point on the opposite bank directly across from its starting position. The boat travels at 4 m/s and the current is 2m/s. If the river is 600 m wide in what direction must the boat steer and how long will it take to cross?
To determine the direction the boat must steer and the time it will take to cross the river, we need to understand the effect of the current on the boat's motion.
Let's break down the motion into two components: the boat's velocity and the current's velocity.
1. The boat's velocity: The boat travels at a constant speed of 4 m/s. This velocity is the boat's speed relative to still water.
2. The current's velocity: The current flows at a speed of 2 m/s. This velocity represents the speed and direction of the river's flow.
Now, let's analyze the boat's motion relative to the ground:
1. Direction to steer: To reach the opposite bank directly across from its starting position, the boat needs to steer slightly upstream. This is because the current will push the boat downstream during its crossing. By steering slightly upstream, the boat can counteract the effect of the current and arrive directly across from the starting point.
2. Time to cross: To calculate the time, we need the component of the boat's velocity in the direction perpendicular to the river. This component is the boat's speed multiplied by the cosine of the angle between the boat's velocity vector and the direction perpendicular to the river.
Let's assume that the angle between the boat's velocity vector and the direction perpendicular to the river is θ.
Using trigonometry, cos(θ) = Adjacent / Hypotenuse
cos(θ) = 4 m/s / (boat's velocity relative to the ground)
Since we want the component of the boat's velocity in the direction perpendicular to the river, we have:
cos(θ) = 4 m/s / (4 m/s)
Taking the inverse cosine, we find: θ = 0 degrees
This means the boat should steer directly across the river, perpendicular to the river's flow.
Now, let's determine the time it takes to cross the river. We can use the formula: time = distance / velocity.
The distance to cross is given as 600 m.
The boat's velocity relative to the ground is the vector sum of its velocity (4 m/s) and the current's velocity (2 m/s), which gives us 6 m/s.
Plugging these values into the formula, we get:
time = 600 m / 6 m/s = 100 s
Therefore, the boat must steer directly across the river and it will take 100 seconds to cross.
Note: This calculation assumes a straight-line path and a constant current speed. In reality, the current might vary or the boat may need to adjust its course to compensate for any lateral drift.