# Simple Vector Boat Problem, Conceptual Misunderstanding

• WRS
In summary: So you can use ##||\mathbf{v}||## to get the time from your equation:##t = \frac{\Delta y}{||\mathbf{v}||}##
WRS
Homework Statement
Yes
Relevant Equations
Pythagoras, Trigonometry
Hi there,

I have attached the problem I'm working with.

I believe I must have the wrong idea of how to approach this question.

My issue is with the stated width and calculating how long the boat will take to cross the river.

It's using width; 110m and the boats velocity to determine how long the boat will take to cross the river. Later in the section it states that the boat will have drifted 71m downstream.

My assumption is that if the river was 110m (shore to shore) and the boat drifted 71m down stream the actual distance traveled by the boat would be found using c^2 = a^2 + b^2

Distance traveled = 110^2 + 71^2
Distance traveled = 130.9m (1.dp)

And therefore the time taken to travel would be founding using width 130.9m, rather than 110m?

Thank you very much

#### Attachments

• physicsboat.PNG
110.2 KB · Views: 153
But the speed of the boat along the diagonal path is increased by the current. That increase is exactly enough to not affect the crossing time. If you try to fight the current and cross directly by angling upstream, your crossing time will increase. This of course can all be shown most easily using vectors.

etotheipi
Like @hutchphd said, working it through with vectors might help to better your understanding. All we are really saying is that, in a frame fixed to the shore,

##\Delta \mathbf{r} = \mathbf{v} \Delta t##

Where ##\mathbf{v} = \mathbf{v}_{BS}##. From this you can get two other component equations,

##\Delta y = v_y \Delta t##
##\Delta x = v_x \Delta t##

Luckily in this case it's easy to see from a diagram that ##v_y## is just that of the boat with respect to the water, and ##v_x## is the velocity of the river (make sure to insert appropriate negative signs depending on how you setup your coordinate axes!).

So the easiest way to get the time is just to use ##\Delta t = \frac{\Delta y}{v_y}##, as your textbook did. But you can do your method also, just so long as you use the right speed! For instance, take the magnitude of both sides of the first equation I wrote:

##||\Delta \mathbf{r}|| = ||\mathbf{v}|| \Delta t##

So now ##||\Delta \mathbf{r}||## is the full (##a^2 + b^2##) distance, but ##||\mathbf{v}||## is also the magnitude of the total velocity with respect to the shore.

## 1. What is a simple vector boat problem?

A simple vector boat problem is a physics problem that involves calculating the resultant velocity of a boat moving in a body of water, taking into account both the boat's own velocity and the velocity of the water it is moving through. This type of problem often involves using vector addition and trigonometry to find the boat's overall velocity and direction.

## 2. What are some common misconceptions about simple vector boat problems?

One common misconception is that the boat's velocity and the water's velocity are added together to find the resultant velocity. In reality, the boat's velocity is added to the negative of the water's velocity, since the water's velocity is acting in the opposite direction of the boat's motion.

## 3. How can I solve a simple vector boat problem?

To solve a simple vector boat problem, you will need to break down the boat's velocity and the water's velocity into their x and y components. Then, use vector addition to find the overall x and y components of the boat's resultant velocity. Finally, use trigonometry to calculate the magnitude and direction of the resultant velocity.

## 4. What are some real-world applications of simple vector boat problems?

Simple vector boat problems have practical applications in navigation and marine transportation. For example, a boat captain may need to calculate the resultant velocity of their boat in order to accurately plan their route and arrival time.

## 5. What are some tips for avoiding conceptual misunderstandings when solving simple vector boat problems?

One tip is to carefully label the directions and velocities in the problem, and to consistently use positive and negative signs to indicate direction. It is also helpful to draw a diagram to visualize the problem and the vectors involved. Additionally, double-check your calculations and make sure to use the correct formulas for vector addition and trigonometry.

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