A car travels due east with a speed of 36.0 km/h. Raindrops are falling at a constant speed vertically with respect to the Earth. The traces of the rain on the side windows of the car make an angle of 76.0° with the vertical. Find the speed of the rain (in km/h) with respect to the car and the Earth.

We are given the following information:

- The car is traveling due east with a speed of \( v_{\text{car}} = 36.0 \, \text{km/h} \).
- The raindrops are falling vertically with respect to the Earth.
- The angle made by the traces of the rain on the car's side window with the vertical is \( \theta = 76.0^\circ \).

### Step 1: Understand the relationship between the speeds
The rain appears to be coming at an angle with respect to the car because the car is moving east while the rain is falling vertically. This creates a relative velocity between the car and the rain.

Let:
- \( v_{\text{rain, car}} \) be the speed of the rain with respect to the car.
- \( v_{\text{rain, earth}} \) be the speed of the rain with respect to the Earth.

We can break down the relative velocity of the rain with respect to the car into two components:
1. A vertical component (due to the rain falling straight down), which is equal to \( v_{\text{rain, earth}} \).
2. A horizontal component (due to the car moving east), which is equal to \( v_{\text{car}} \).

Using trigonometry, we can set up the following relationship:
\[
\tan(\theta) = \frac{v_{\text{car}}}{v_{\text{rain, earth}}}
\]
Substituting \( \theta = 76^\circ \) and \( v_{\text{car}} = 36.0 \, \text{km/h} \):
\[
\tan(76^\circ) = \frac{36.0}{v_{\text{rain, earth}}}
\]

### Step 2: Solve for \( v_{\text{rain, earth}} \)
Now, solving for \( v_{\text{rain, earth}} \):
\[
v_{\text{rain, earth}} = \frac{36.0}{\tan(76^\circ)}
\]

### Step 3: Calculate the value
Let me compute this.The speed of the rain with respect to the Earth is approximately \( 8.98 \, \text{km/h} \).

### Step 4: Find the speed of the rain with respect to the car
Now, using the Pythagorean theorem, the speed of the rain with respect to the car is given by:
\[
v_{\text{rain, car}} = \sqrt{v_{\text{car}}^2 + v_{\text{rain, earth}}^2}
\]

Let me compute that.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to guide you through solving this calculation manually or help with something else?

Here are some related questions you might find interesting:
1. How would the angle change if the car's speed increased?
2. What happens if the rain has a horizontal wind component as well?
3. How do we calculate the rain's velocity if the car is moving in a different direction?
4. Can the speed of the rain be calculated if the angle is closer to horizontal?
5. How would these calculations change if we considered air resistance?

**Tip:** Always break down relative velocity problems into their horizontal and vertical components to solve efficiently.

Solve a relative velocity problem involving a car traveling east at 36.0 km/h and raindrops falling vertically. Learn how to calculate the speed of the rain with respect to the car and the Earth using trigonometric ratios and vector components. Suitable for high school students in Grades 10-12.

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