# Find the trajectory of a boat that moves in a river

• david22
In summary, the problem involves a boat starting from point P on the bank of a river, with width D, and moving towards point Q on the opposite side of the river at a constant velocity VB and a constant velocity VR with respect to the river. The boat's trajectory can be determined by the instantaneous distance r from Q and the angle θ between r and the line PQ, which can be expressed as r=Dsecθ/(secθ+tanθ)VB/VR. The boat's position and velocity can be treated as vectors, with the velocity towards Q and the velocity with respect to the shore being important in solving the problem.
david22

## Homework Statement

A boat part from the point P of the bank of a river, with width D, and that flows with a constant velocity VR, and moves with a constant velocity VB, directed towars a point Q, located on the other side of the river directly in front of P. If r is the instantaneous distance of the boat respect to Q and θ is the instantaneous angle between r and PQ, show that the trajectory of the boat is determined by:

r=Dsecθ/(secθ+tanθ)VB/VR

## The Attempt at a Solution

I am completely lost in this problem; I to put the instantaneous distance of the boat r as a vector and a took Q as my origin r = cos(90-θ)i+sen(90-θ)j but then i don't know what else to do; I would appreciate your help a lot

david22 said:

## Homework Statement

A boat part from the point P of the bank of a river, with width D, and that flows with a constant velocity VR, and moves with a constant velocity VB, directed towars a point Q, located on the other side of the river directly in front of P. If r is the instantaneous distance of the boat respect to Q and θ is the instantaneous angle between r and PQ, show that the trajectory of the boat is determined by:

r=Dsecθ/(secθ+tanθ)VB/VR

## The Attempt at a Solution

I am completely lost in this problem; I to put the instantaneous distance of the boat r as a vector and a took Q as my origin r = cos(90-θ)i+sen(90-θ)j but then i don't know what else to do; I would appreciate your help a lot

Have you copied the problem text correctly? "A boat part from the point P of the bank of a river, with width D, and that flows with a constant velocity VR, and moves with a constant velocity VB, directed towars a point Q, located on the other side of the river directly in front of P." If the boat moves with constant velocity it never reaches point Q. Was not it constant speed with respect to the river?

You are right, treat the position and velocity of the boat as vectors. The velocity of the boat with respect to the river points towards Q. What is the velocity of the boat with respect to the shore?

ehild

## 1. What factors affect the trajectory of a boat in a river?

The trajectory of a boat in a river is affected by several factors, including the speed and direction of the current, the shape and size of the boat, and the force and direction of the wind.

## 2. How do you calculate the trajectory of a boat in a river?

To calculate the trajectory of a boat in a river, you need to consider the boat's velocity, the velocity of the current, and the angle at which the boat is moving relative to the current. This can be determined using mathematical equations and vector analysis.

## 3. Can the trajectory of a boat in a river be predicted accurately?

While it is possible to make predictions about the trajectory of a boat in a river, it is not always possible to do so with complete accuracy. This is because there are many variables that can affect the trajectory, such as changing wind patterns or unexpected obstacles in the river.

## 4. How can the trajectory of a boat in a river be adjusted or controlled?

The trajectory of a boat in a river can be adjusted or controlled through various means, such as using a rudder or steering mechanism, adjusting the sail or motor, or using oars to manually change the direction of the boat. The exact method will depend on the type of boat and the conditions of the river.

## 5. What are some common challenges when determining the trajectory of a boat in a river?

Some common challenges when determining the trajectory of a boat in a river include unpredictable changes in current or wind patterns, varying depths and obstacles in the river, and the skill and experience of the boat's operator. It is important to regularly monitor and adjust the boat's trajectory to navigate safely and effectively.

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