- #1

Mr Davis 97

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## 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?