Asked by `
On a trip to the store around the block a bike rider goes 75 m toward the West and 75 m toward the
North at a constant speed of 1.5 m/s. What is the magnitude of the average velocity?
1.5 m/s
0.75 m/s
1.6 m/s
1.1 m/s
North at a constant speed of 1.5 m/s. What is the magnitude of the average velocity?
1.5 m/s
0.75 m/s
1.6 m/s
1.1 m/s
Answers
Answered by
GPT-4o mini
To find the magnitude of the average velocity, we need to calculate the total displacement and total time taken.
1. **Total Displacement**:
The bike rider travels 75 m West and then 75 m North. We can find the resultant displacement using the Pythagorean theorem (\(d = \sqrt{x^2 + y^2}\)) where:
- \(x = 75 \, \text{m}\) (West)
- \(y = 75 \, \text{m}\) (North)
\[
d = \sqrt{(75 \, \text{m})^2 + (75 \, \text{m})^2} = \sqrt{5625 + 5625} = \sqrt{11250} \approx 106.07 \, \text{m}
\]
2. **Total Time**:
The total distance traveled is the sum of the two legs of the trip:
\[
\text{Total distance} = 75 \, \text{m} + 75 \, \text{m} = 150 \, \text{m}
\]
The speed of the bike rider is 1.5 m/s. To find the time, we use the formula:
\[
\text{Time} = \frac{\text{Total distance}}{\text{Speed}} = \frac{150 \, \text{m}}{1.5 \, \text{m/s}} = 100 \, \text{s}
\]
3. **Average Velocity**:
The average velocity is defined as the total displacement divided by the total time.
\[
\text{Average velocity} = \frac{\text{Total displacement}}{\text{Total time}} = \frac{106.07 \, \text{m}}{100 \, \text{s}} = 1.0607 \, \text{m/s}
\]
Rounded to two decimal places, the magnitude of the average velocity is approximately **1.06 m/s**.
None of the provided options exactly match this result, but it closely approximates **1.1 m/s**. Therefore, the best choice among the given options is:
**1.1 m/s**.
1. **Total Displacement**:
The bike rider travels 75 m West and then 75 m North. We can find the resultant displacement using the Pythagorean theorem (\(d = \sqrt{x^2 + y^2}\)) where:
- \(x = 75 \, \text{m}\) (West)
- \(y = 75 \, \text{m}\) (North)
\[
d = \sqrt{(75 \, \text{m})^2 + (75 \, \text{m})^2} = \sqrt{5625 + 5625} = \sqrt{11250} \approx 106.07 \, \text{m}
\]
2. **Total Time**:
The total distance traveled is the sum of the two legs of the trip:
\[
\text{Total distance} = 75 \, \text{m} + 75 \, \text{m} = 150 \, \text{m}
\]
The speed of the bike rider is 1.5 m/s. To find the time, we use the formula:
\[
\text{Time} = \frac{\text{Total distance}}{\text{Speed}} = \frac{150 \, \text{m}}{1.5 \, \text{m/s}} = 100 \, \text{s}
\]
3. **Average Velocity**:
The average velocity is defined as the total displacement divided by the total time.
\[
\text{Average velocity} = \frac{\text{Total displacement}}{\text{Total time}} = \frac{106.07 \, \text{m}}{100 \, \text{s}} = 1.0607 \, \text{m/s}
\]
Rounded to two decimal places, the magnitude of the average velocity is approximately **1.06 m/s**.
None of the provided options exactly match this result, but it closely approximates **1.1 m/s**. Therefore, the best choice among the given options is:
**1.1 m/s**.