3. You drive a car 1500 ft to the east, then 2500 ft to the north. If the trip takes 3.0 minutes, what is the direction and magnitude of your average velocity?

To find the magnitude and direction of your average velocity, we'll first calculate the displacement vector and then use it to determine the velocity.

### 1. **Calculate the Displacement Vector**
- You drive 1500 ft to the east and 2500 ft to the north.
- These can be represented as components of the displacement vector:
  - \( \text{Eastward displacement} = 1500 \text{ ft} \)
  - \( \text{Northward displacement} = 2500 \text{ ft} \)

The displacement vector \( \vec{d} \) can be found using the Pythagorean theorem:

\[
|\vec{d}| = \sqrt{(1500)^2 + (2500)^2} \text{ ft}
\]

### 2. **Calculate the Magnitude of the Displacement**
\[
|\vec{d}| = \sqrt{2250000 + 6250000} \text{ ft} = \sqrt{8500000} \text{ ft} = 2915.48 \text{ ft}
\]

### 3. **Calculate the Direction of the Displacement**
The direction \( \theta \) (measured from the east direction towards north) can be found using the arctangent function:

\[
\theta = \tan^{-1}\left(\frac{2500 \text{ ft}}{1500 \text{ ft}}\right)
\]

\[
\theta = \tan^{-1}\left(\frac{5}{3}\right) \approx 59.04^\circ
\]

### 4. **Calculate the Magnitude of the Average Velocity**
The average velocity \( \vec{v}_{avg} \) is the displacement divided by the time taken:

\[
|\vec{v}_{avg}| = \frac{|\vec{d}|}{t} = \frac{2915.48 \text{ ft}}{3.0 \text{ min}}
\]

Convert minutes to seconds to get the velocity in feet per second:

\[
t = 3.0 \text{ min} \times 60 \text{ s/min} = 180 \text{ s}
\]

\[
|\vec{v}_{avg}| = \frac{2915.48 \text{ ft}}{180 \text{ s}} \approx 16.20 \text{ ft/s}
\]

### 5. **Final Answer**
- **Magnitude of the average velocity:** \( 16.20 \text{ ft/s} \)
- **Direction of the average velocity:** \( 59.04^\circ \) north of east

Would you like further details or have any questions?

Here are some related questions you might find interesting:
1. How would the average velocity change if you traveled back to your starting point?
2. How would the time taken affect the magnitude of the average velocity?
3. What would the result be if you drove west and south instead?
4. How do you calculate the resultant displacement if you drive in more than two directions?
5. What if the units of distance were in meters instead of feet?

**Tip:** Always remember to convert all time units to seconds when calculating velocity in feet per second.

Learn how to calculate average velocity when driving 1500 ft east and 2500 ft north in 3 minutes. Understand the application of vectors, displacement, and velocity concepts. Detailed step-by-step solution provided with calculations using the Pythagorean theorem and arctangent function.

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