In reaching her destination, a backpacker walks with an average velocity of 1.22 m/s, due west. This average velocity results, because she hikes for 5.61 km with an average velocity of 2.69 m/s due west, turns around, and hikes with an average velocity of 0.367 m/s due east. How far east did she walk (in kilometers)?
To find out how far east the backpacker walked, we need to determine the distance she hiked due east and subtract it from the total distance she hiked due west.
First, let's convert the distance she hiked from kilometers to meters.
Distance due west = 5.61 km = 5.61 × 1000 = 5610 m
Next, we use the formula:
Average velocity = total distance / total time
Let's calculate the total time she took during the entire hike:
Total time = time due west + time due east
For the time due west:
Velocity due west = 2.69 m/s
Distance due west = 5610 m
Time due west = Distance due west / Velocity due west
Time due west = 5610 / 2.69
Time due west ≈ 2085.87 seconds
For the time due east:
Velocity due east = 0.367 m/s
Distance due east = unknown (let's call it x)
Time due east = Distance due east / Velocity due east
Time due east = x / 0.367
Since the total time is the sum of the times due east and due west, we have:
Total time = 2085.87 seconds + x/0.367
Now, let's rearrange the average velocity formula to solve for total distance:
Total distance = average velocity × total time
Total distance = 1.22 m/s × Total time
Substituting in the expression for the total time:
Total distance = 1.22 m/s × (2085.87 seconds + x/0.367)
Since the total distance is the sum of the distances due east and due west, we have:
Total distance = Distance due west - Distance due east
Total distance = 5610 m - x
Now, we can set up an equation:
5610 - x = 1.22 × (2085.87 + x/0.367)
Simplifying the equation:
5610 - x = 1.22 × 2085.87 + 1.22 × x/0.367
5610 - x = 2549.14 + 3.33x
Grouping the x terms:
x - 3.33x = 2549.14 - 5610
-2.33x = -3060.86
Dividing both sides by -2.33:
x = -3060.86 / -2.33
x ≈ 1313.76
So, the distance due east is approximately 1313.76 meters.
Converting this to kilometers:
Distance due east = 1313.76 m = 1313.76 ÷ 1000 = 1.314 km
Therefore, the backpacker walked approximately 1.314 kilometers due east.
To find out how far east the backpacker walked, we can break down the journey into two parts: the distance traveled west and the distance traveled east.
First, let's find the distance traveled west. We are given the average velocity for this part of the journey, which is 2.69 m/s. We also know the total distance traveled during this part of the journey, which is 5.61 km. To find the time taken for this part of the journey, we can use the formula:
Time = Distance / Velocity
Substituting the given values, we get:
Time = 5.61 km / 2.69 m/s
To simplify the calculation, we need to convert the distance from kilometers to meters and the velocity from meters per second to kilometers per second:
1 km = 1000 m
1 m/s = 0.001 km/s
So, we get:
Time = 5610 m / (2.69 m/s * 1000)
Time = 2.085 seconds
Now that we know the time taken for the westward journey, we can find the distance traveled east. We are given the average velocity for this part of the journey, which is 0.367 m/s. Using the formula again:
Distance = Velocity * Time
Substituting the known values:
Distance = 0.367 m/s * 2.085 s
Calculating this, we get:
Distance = 0.766 km
Therefore, the backpacker walked approximately 0.766 kilometers east.