A car is moving east at 20 km/h. The wind is also moving in the same direction at 4 km/h. At this time rain starts falling steeply downhill at 6 km/h. At what angle will the rain hit the car with the vertical?

To solve this problem, we need to determine the angle at which the rain appears to hit the car. This requires calculating the relative velocity of the rain with respect to the car, and then finding the angle between this relative velocity and the vertical.

### Step 1: Define the velocities
- **Car's velocity**: The car is moving east at 20 km/h.
- **Wind's velocity**: The wind is also moving east at 4 km/h.
- **Rain's velocity**: The rain is falling vertically downward at 6 km/h.

### Step 2: Find the relative velocity of the rain with respect to the car

The relative velocity of the rain with respect to the car takes into account both the horizontal and vertical components:

- **Vertical component**: Since the rain is falling vertically downward, its vertical component is simply \( V_{rv} = 6 \, \text{km/h} \).

- **Horizontal component**: The car’s velocity relative to the rain’s velocity in the east direction is the sum of the car’s velocity and the wind’s velocity. Therefore, the total horizontal velocity is:

\[
V_{rh} = 20 \, \text{km/h} + 4 \, \text{km/h} = 24 \, \text{km/h}
\]

### Step 3: Find the angle of impact with the vertical

The angle \( \theta \) at which the rain hits the car with respect to the vertical can be found using trigonometry. Specifically, we can use the tangent of the angle \( \theta \), which is the ratio of the horizontal to the vertical component of the relative velocity:

\[
\tan(\theta) = \frac{V_{rh}}{V_{rv}} = \frac{24}{6} = 4
\]

Now, calculate the angle \( \theta \):

\[
\theta = \tan^{-1}(4)
\]

Using a calculator:

\[
\theta \approx 75.96^\circ
\]

### Final Answer:
The rain will hit the car at an angle of approximately **76°** with the vertical.

Would you like more details on this, or do you have any questions?

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Here are five follow-up questions related to this concept:

1. How would the angle change if the car increased its speed?
2. What if the rain had a horizontal velocity component as well? How would that affect the angle?
3. How can we calculate the angle if the car was moving in a different direction (not east)?
4. What happens to the angle if the wind's velocity changes direction and opposes the car's movement?
5. How would the angle of the rain's impact differ if the car was stationary?

**Tip**: Always consider the relative velocities of objects when calculating the angle at which one moving object appears to another. This is key in problems involving rain, wind, and moving vehicles!

Learn how to calculate the angle at which rain hits a car moving east at 20 km/h while the wind blows at 4 km/h, and rain falls vertically at 6 km/h. This problem covers relative velocity, trigonometry, and angle calculation using tan(θ). Suitable for Grades 10-12.

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