To solve this problem, we need to find two things:

1. The speed of the rain with respect to the car.
2. The speed of the rain with respect to the Earth.

Let's denote the following:

• $v_c = 38.0$ km/h: the speed of the car moving due east.
• $v_r$: the speed of the rain with respect to the Earth (vertically downward).
• $v_{rc}$: the speed of the rain with respect to the car.
• The angle of the rain traces on the car's windows with the vertical is $\theta = 64.0^\circ$.

Step 1: Understanding the Relationship Between the Speeds

When the car is moving due east and the rain is falling vertically with respect to the Earth, the raindrops appear to follow a slanted path relative to the car. This is due to the relative motion of the car and the rain. The rain appears to the car to fall at an angle because of the car's horizontal velocity.

Given the angle $\theta$ between the rain traces and the vertical, we can relate the speeds using trigonometry:

$\tan(\theta) = \frac{\text{horizontal component of rain's velocity (with respect to the car)}}{\text{vertical component of rain's velocity (with respect to the car)}}$

Since the rain falls vertically with respect to the Earth, the horizontal component of the rain's velocity with respect to the car is equal to the speed of the car:

$\tan(64.0^\circ) = \frac{v_c}{v_r}$

Solving for $v_r$:

$v_r = \frac{v_c}{\tan(64.0^\circ)}$

Now, let's calculate $v_r$.The speed of the rain with respect to the Earth, $v_r$, is approximately 18.53 km/h.

Step 2: Finding the Speed of the Rain with Respect to the Car

To find the speed of the rain with respect to the car, $v_{rc}$, we use the Pythagorean theorem. The rain's speed with respect to the car is the vector sum of the rain's vertical speed (with respect to the Earth) and the car's horizontal speed:

$v_{rc} = \sqrt{v_c^2 + v_r^2}$

Now, let's compute $v_{rc}$.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can guide you through the remaining calculation!

Step 2: Calculating the Speed of the Rain with Respect to the Car

To find the speed of the rain with respect to the car ($v_{rc}$), use the Pythagorean theorem:

$v_{rc} = \sqrt{v_c^2 + v_r^2}$

Substitute the known values:

$v_{rc} = \sqrt{(38.0)^2 + (18.53)^2}$

Compute the expression to find the speed of the rain with respect to the car.

Regarding the "Speed of the Earth"

The "speed of the Earth" isn't directly relevant in this context. The problem focuses on the speed of the rain relative to both the car and the Earth.

Would you like help with any other aspect, or do you have any questions about this solution?

Related Questions to Explore:

1. How does relative velocity work in different frames of reference?
2. How would the rain's angle change if the car's speed increased?
3. What would happen if the rain had a horizontal component of velocity?
4. How would you calculate the time taken for the rain to reach the car if it started from a specific height?
5. Can this concept be applied to other real-life scenarios, like walking in the rain with an umbrella?

Tip: Understanding relative velocity can help in solving problems involving moving objects, such as planes encountering wind or boats in current.