Question

If the average speed of a car is 45 mph to the west and 31 mph to the north, what is the average velocity
of the car?
answers:
60.8 mph to the northwest

18.1 mph to the southwest

54.6 mph to the northwest

27.7 mph to the southwest

Answer 1

To find the average velocity of the car given its average speeds in the west and north directions, we can treat the speeds as components of a vector.

1. **Determine the velocity components**:
- Westward velocity component (V_west) = -45 mph (negative because it's to the west)
- Northward velocity component (V_north) = 31 mph

2. **Calculate the resultant velocity**:
The average velocity \( V \) can be calculated using the Pythagorean theorem:

\[
V = \sqrt{V_{west}^2 + V_{north}^2}
\]

Substituting in the values:

\[
V = \sqrt{(-45)^2 + (31)^2}
\]

\[
V = \sqrt{2025 + 961} = \sqrt{2986} \approx 54.6 \text{ mph}
\]

3. **Determine the direction**:
The direction (θ) can be calculated using the tangent function:

\[
\tan(\theta) = \frac{V_{north}}{|V_{west}|}
\]

\[
\tan(\theta) = \frac{31}{45}
\]

Now, calculate θ:

\[
\theta = \arctan\left(\frac{31}{45}\right) \approx 36.87^\circ
\]

This angle is measured from the westward direction towards the north, meaning that the overall direction is northwest.

So, the average velocity of the car is approximately **54.6 mph to the northwest**.

The correct answer is:
**54.6 mph to the northwest**.