
A car travels due east with a speed of 46.0 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 74.0° with the vertical. Find the velocity of the rain with respect to the following reference frames. (a) the car (b) the earth

We can solve this problem by analyzing the velocities using vector components. Here's how:

### Given:
- Speed of the car (\(v_{\text{car}}\)) = 46.0 km/h
- Angle of the rain traces on the window (\(\theta\)) = 74.0° with the vertical
- Rain is falling vertically with respect to the Earth.

We'll break the problem into two parts.

---

### Part (a): **Velocity of the rain with respect to the car**

The angle \( \theta = 74.0^\circ \) gives us a clue about the relationship between the horizontal and vertical components of the rain's velocity with respect to the car. The rain appears to make an angle with the vertical, so from the car's perspective, the rain has both horizontal and vertical components of velocity.

Let \( v_{\text{rain, car}} \) be the rain’s velocity with respect to the car. We can resolve this into:
- \( v_{\text{vertical}} \) (vertical component of the rain's velocity relative to Earth).
- \( v_{\text{horizontal, car}} \) (horizontal component due to the car's motion).

From the geometry of the problem, using the trigonometric relation:

\[
\tan(\theta) = \frac{v_{\text{horizontal, car}}}{v_{\text{vertical}}}
\]

Where:
- \( v_{\text{horizontal, car}} = v_{\text{car}} = 46.0 \, \text{km/h} \).

Thus:
\[
\tan(74.0^\circ) = \frac{46.0 \, \text{km/h}}{v_{\text{vertical}}}
\]

Solving for \(v_{\text{vertical}}\):

\[
v_{\text{vertical}} = \frac{46.0 \, \text{km/h}}{\tan(74.0^\circ)}
\]

We’ll calculate this in the next step.

---

### Part (b): **Velocity of the rain with respect to the Earth**

Since the rain is falling vertically with respect to the Earth, the horizontal component of the rain’s velocity with respect to the Earth is zero. So, the velocity of the rain with respect to the Earth is just the vertical component \( v_{\text{vertical}} \), which we have calculated from part (a).

Let’s now compute these values.The vertical component of the rain's velocity with respect to the Earth is approximately \( 13.19 \, \text{km/h} \).

Now we can determine the velocity of the rain in both frames of reference:

### (a) Velocity of the rain with respect to the car:
The horizontal component is the speed of the car, \( 46.0 \, \text{km/h} \), and the vertical component is \( 13.19 \, \text{km/h} \). To find the magnitude of the rain's velocity relative to the car, we use the Pythagorean theorem:

\[
v_{\text{rain, car}} = \sqrt{v_{\text{horizontal}}^2 + v_{\text{vertical}}^2}
\]

Let’s compute this.

### (b) Velocity of the rain with respect to the Earth:
This is simply the vertical component, which is \( 13.19 \, \text{km/h} \), as the rain is falling vertically with respect to the Earth.

Now, let's calculate the rain's velocity with respect to the car.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can calculate the velocity of the rain with respect to the car using the following formula:

\[
v_{\text{rain, car}} = \sqrt{v_{\text{horizontal}}^2 + v_{\text{vertical}}^2}
\]

Where:
- \(v_{\text{horizontal}} = 46.0 \, \text{km/h}\)
- \(v_{\text{vertical}} = 13.19 \, \text{km/h}\)

Simply plug these values into the formula to find the result.

### Relative questions:
1. How do velocity components relate to the overall velocity?
2. What is the significance of resolving velocities into horizontal and vertical components?
3. Can this method be applied to objects in free fall?
4. What does the angle of rain traces tell us about relative motion?
5. How would the problem change if the car were traveling at a different speed?

**Tip:** When analyzing motion, always consider both magnitude and direction for a complete understanding of velocity.

A car travels due east with a speed of 46.0 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 74.0° with the vertical. Find the velocity of the rain with respect to the following reference frames: (a) the car, (b) the earth.

Solve a physics problem involving a car traveling east at 46.0 km/h and raindrops falling vertically with respect to the Earth. Learn how to calculate the velocity of rain with respect to both the car and the Earth using trigonometry and vector components. Suitable for Grades 10-12.

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