# Minimum time for rowing across a river and running

• Mr Davis 97
In summary: I have ##t_1 = \frac{d}{vcos\theta}##But, that's with ##\theta...=114.9## degrees which is not the correct answer.
Mr Davis 97

## Homework Statement

In hot pursuit of a suspect, Agent Logan of the FBI must get directly across
a 1200m wide river in minimum time. The river's current is 0.80m/s, she can row
a boat at 1.60m/s and she can run 3.00m/s. Describe the path she should take
(rowing + running along the shore) for the crossing to take the minimum amount
of time and determine the minimum time

## Homework Equations

Relative velocity and SUVAT equations

## The Attempt at a Solution

Let's make it so that the river goes horizontally from right to left, and the angle that the boat takes with the positive x-axis going across is ##\theta##.

Then it's position vector across the river is ##\vec{r} = (v_{row} \cos \theta - v_{riv})t \hat{i} + (v_{row} \sin \theta)t \hat{j}##.

Let ##d = 1200 ~m##.
Then ##d = v_{row} \sin \theta t_1##
and
##\displaystyle t_1 = \frac{d}{v_{row} \sin \theta}##

Also, the change in the x-direction in the water is ##\delta x = (v_{row} \cos \theta - v_{riv})t_1##
and the change in the x-direction on land is ##\delta x = v_{run} t_2##

Next, I let ##\displaystyle T = t_1 + t_2 = \frac{d}{v_{row} \sin \theta} + \frac{\delta x}{v_{run}}##

then ##\displaystyle T = \frac{d}{v_{row} \sin \theta} + \frac{(v_{row} \cos \theta - v_{riv})d}{v_{row} v_{run} \sin \theta}##

Finally, I take ##\displaystyle \frac{dT}{d \theta} = 0##, to get the minimum ##T##, and find that it is at a minimum when ##\displaystyle \cos \theta = \frac{v_{row}}{v_{run} - v_{riv}}##.

When I plug in the velocities, I find that ##\theta = 114.9## degrees. This is not the right answer, so what am I doing wrong?

Mr Davis 97 said:
When I plug in the velocities, I find that ##\theta = 114.9## degrees. This is not the right answer, so what am I doing wrong?

And what precisely is wrong with 114.9 degrees?

PeroK said:
And what precisely is wrong with 114.9 degrees?
I look at the book, and it says that the correct answer is 24.9 degrees...

Mr Davis 97 said:
I look at the book, and it says that the correct answer is 24.9 degrees...

Hmm 114.9 degrees or 24.9 degrees? I wonder ...

billy_joule
PeroK said:
Hmm 114.9 degrees or 24.9 degrees? I wonder ...
But what would justify me subtracting 90 degrees? Also, even if I plug 24.9 back into my equations, I am not getting the correct numbers for the other parts of the problem, which is that the minimum time is 862 seconds, and that the person runs along the bank for 104 meters...

PeroK said:
Hmm 114.9 degrees or 24.9 degrees? I wonder ...
Yeah, I do, too. Could there be any relationship between those two numbers?

Mr Davis 97 said:
But what would justify me subtracting 90 degrees? Also, even if I plug 24.9 back into my equations, I am not getting the correct numbers for the other parts of the problem, which is that the minimum time is 862 seconds, the the person runs along the bank for 104 meters...
I haven't studied the detail of your solution, but 114.9 (measured from parallel with the river bank) is the same angle as 24.9 (measured from a line at right angles across the river). I think you've measured the angle from parallel to the riverbank; or, at least, that's the angle your calculations assume.

PeroK said:
I haven't studied the detail of your solution, but 114.9 (measured from parallel with the river bank) is the same angle as 24.9 (measured from a line at right angles across the river). I think you've measured the angle from parallel to the riverbank; or, at least, that's the angle your calculations assume.
Well then I guess that's fine, I'll take it that somehow I am measuring parallel from the bank, and that the solution entails that I measure from the right angle to the bank. However, this still doesn't explain why inserting either 24.9 or 114.9 into the original equations does not result in me computing the minimum time, 862 seconds. For example, ##t_1 + t_2 = 635.621##, which is too small.

Mr Davis 97 said:
Well then I guess that's fine, I'll take it that somehow I am measuring parallel from the bank, and that the solution entails that I measure from the right angle to the bank. However, this still doesn't explain why inserting either 24.9 or 114.9 into the original equations does not result in me computing the minimum time, 862 seconds. For example, ##t_1 + t_2 = 635.621##, which is too small.

I get ##t_1 = 827s##, ##x =104m##, ##t_2 = 34.8s##

I suspect you just made a mistake in the calculations.

PeroK said:
I get ##t_1 = 827s##, ##x =104m##, ##t_2 = 34.8s##

I suspect you just made a mistake in the calculations.
Could you please show me the expression you use to get ##t_1##? For the life of me I keep getting the wrong answer even though I am using the right angle...

Mr Davis 97 said:
Could you please show me the expression you use to get ##t_1##? For the life of me I keep getting the wrong answer even though I am using the right angle...

I have ##t_1 = \frac{d}{vcos\theta}##

But, that's with ##\theta = 24.9##, upstream, as measured from the line across the river.

My ##cos\theta## should be the same as your ##sin\theta##

PeroK said:
I have ##t_1 = \frac{d}{vcos\theta}##

But, that's with ##\theta = 24.9##, upstream, as measured from the line across the river.

My ##cos\theta## should be the same as your ##sin\theta##
Finally, it all checks out now. Thanks for the help!

## 1. What is the minimum time required for rowing across a river and running?

The minimum time required for rowing across a river and running depends on various factors such as the distance of the river, the speed of the current, and the fitness level of the person rowing and running. Generally, it can take anywhere from 20 minutes to an hour to complete this task.

## 2. How can one minimize the time for rowing across a river and running?

To minimize the time for rowing across a river and running, one can focus on improving their rowing and running techniques, as well as their overall fitness level. Additionally, choosing a shorter distance to row and run can also help reduce the time.

## 3. What is the impact of the current on the time for rowing across a river and running?

The current of a river can significantly impact the time required for rowing across and running. A strong current can make rowing more difficult and slow down the overall pace, while a weaker current can make it easier to row and run faster.

## 4. Is it better to row across the river first or run first?

The order in which one rows across the river and runs can vary depending on personal preference and the specific conditions of the river. However, it is generally recommended to row first, as it requires more energy and it is easier to run with tired arms than to row with tired legs.

## 5. What are some tips for completing this task in the minimum time?

Some tips for completing this task in the minimum time include practicing regularly to improve rowing and running techniques, maintaining a good level of fitness, choosing a shorter distance to row and run, and paying attention to the current of the river. It is also important to stay hydrated and well-nourished before and during the task to maintain energy levels.

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