This problem involves relative velocity, where we need to find the speed of the rain both relative to the car and the Earth. We'll break it down step by step:

Step 1: Analyzing the Velocity Components

Let:

• $v_c = 60 \, \text{km/h}$ be the speed of the car due east (horizontal direction).
• $v_r$ be the speed of the rain with respect to the Earth, which is falling vertically (in the vertical direction).
• $v_{r/c}$ be the velocity of the rain relative to the car.

Given:

• The rain traces make an angle $\theta = 64.0^\circ$ with the vertical on the car window.

Since the rain is falling vertically with respect to the Earth and the car is moving horizontally, the angle formed by the rain on the car window is due to the combination of these two motions (the horizontal motion of the car and the vertical motion of the rain).

We can express the relationship between the car's velocity, the rain's velocity relative to the car, and the angle using trigonometry:

$\tan(\theta) = \frac{v_c}{v_r}$

Step 2: Solving for the Rain's Speed with Respect to the Earth

Rearranging the equation to solve for $v_r$:

$v_r = \frac{v_c}{\tan(\theta)}$

Substituting the known values:

$v_r = \frac{60.0 \, \text{km/h}}{\tan(64.0^\circ)}$

We will now compute this value.The speed of the rain with respect to the Earth is approximately $29.26 \, \text{km/h}$.

Step 3: Finding the Speed of the Rain with Respect to the Car

Now, to find the rain's speed with respect to the car $v_{r/c}$, we use the Pythagorean theorem since the velocities form a right triangle:

$v_{r/c} = \sqrt{v_r^2 + v_c^2}$

We will compute this next.The speed of the rain with respect to the car is approximately $66.76 \, \text{km/h}$.

Summary:

• The speed of the rain with respect to the Earth: $29.26 \, \text{km/h}$.
• The speed of the rain with respect to the car: $66.76 \, \text{km/h}$.

Would you like a detailed explanation of any part of the process? Here are some related questions to explore further:

1. How is relative velocity used in real-world applications?
2. What would happen if the car's speed increased?
3. How can we apply vector addition in this scenario?
4. Why is the angle important in determining the rain’s velocity?
5. What is the significance of reference frames in physics?

Tip: When solving problems involving angles and velocity, always break the velocity into its components to make the calculations simpler.