A boat is to be used to cross a river of width w that is flowing with a speed v. The boat travels with a constant speed 2v relative to water.
At what angle should the boat be directed in order to arrive at a point directly across from its starting point?
(draw a diagram showing the three relative velocity vectors.)
To determine the angle at which the boat should be directed in order to arrive directly across from its starting point, we need to consider the velocity vectors involved. Let's break it down step by step:
1. Draw a diagram:
- Start by drawing a horizontal line to represent the direction the river is flowing (from left to right).
- Label this line as "v", which represents the velocity vector of the river.
- Now, draw another line starting from the left side of the river at an angle, representing the direction at which the boat is traveling. Label it as "2v", representing the velocity vector of the boat relative to the water.
- Finally, draw a vertical line starting from the end of the "2v" vector to represent the resulting velocity vector of the boat relative to the ground. This is the vector we need to determine the angle for.
2. Analyze the vectors:
- The velocity vector of the boat relative to the ground is the vector sum of the velocity vector of the boat relative to the water (2v) and the velocity vector of the river (v).
- To arrive directly across from its starting point, the vertical component of the resulting velocity vector (which is perpendicular to the river) should be zero.
3. Set up the equation:
- Using the components of the vectors, we can set up the equation: 2v*sinθ = v*cosθ, where θ represents the angle between the boat's direction and the horizontal axis.
4. Solve the equation:
- Divide both sides of the equation by v: 2*sinθ = cosθ.
- Rearrange the equation: tanθ = 2.
- Take the inverse tangent (arctan) of both sides: θ = arctan(2).
- Use a calculator or reference table to find the value of arctan(2).
The result will be the angle at which the boat should be directed in order to arrive directly across from its starting point.