In reaching her destination, a backpacker walks with an average velocity of 1.20 m/s, due west. This average velocity results, because she hikes for 5.94 km with an average velocity of 3.28 m/s due west, turns around, and hikes with an average velocity of 0.778 m/s due east. How far east did she walk (in kilometers)?

To find the distance the backpacker walked east, we need to find the displacement caused by the eastward hike.

First, let's find the displacement caused by the westward hike. The equation for displacement is:

displacement = velocity × time

For the westward hike:
velocity = 3.28 m/s
time = 5.94 km / (3.28 m/s) = 1807.32 s

Therefore, the displacement caused by the westward hike is:
displacement_west = 3.28 m/s × 1807.32 s = 5932.7696 m west

Next, let's find the displacement caused by the eastward hike. Similarly:

velocity = 0.778 m/s
time = t (unknown)

To find the displacement, we need to know the time for the eastward hike. However, we can use the fact that average velocity is defined as the total displacement divided by the total time.

The total time is the sum of the times for both hikes:
total time = 1807.32 s + t

The total displacement is the sum of displacements caused by each hike:
total displacement = displacement_west + displacement_east

Given that the total displacement is zero (since she ended up back at her starting point), we can write the equation:

0 = 5932.7696 m west + displacement_east

And plug in the known values:
0 = 5932.7696 m + (0.778 m/s) * t

Simplifying the equation, we get:
-5932.7696 m = 0.778 m/s * t

Now we can solve for t:
t = -5932.7696 m / 0.778 m/s ≈ -7631.56 s

Since time cannot be negative in this context, it means that the person did not have enough time to fully hike east. Therefore, the displacement caused by the eastward hike is zero, and the answer to the question is that she did not walk any distance east (in kilometers).