A car travels due east with a speed of 40.0 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 54.0° with the vertical. Find the velocity of the rain with respect to the car and the earth. (Enter the magnitude of the velocity.) the earth

This problem involves two frames of reference: the car's perspective and the Earth's perspective. We are tasked with finding the velocity of the rain relative to both the car and the Earth.

### Given:
- Speed of the car, \( v_{\text{car}} = 40.0 \, \text{km/h} \) (due east).
- The angle of the rain traces on the side windows of the car, \( \theta = 54.0^\circ \).
- The rain falls vertically with respect to the Earth, meaning its velocity is downward (no horizontal component in Earth's reference frame).

### Step-by-step solution:

1. **Analyze from the car's reference frame**:
   - From the car's perspective, the rain appears to move at an angle of \( 54.0^\circ \) with the vertical. This is due to the relative motion between the car and the rain.
   - The horizontal component of the rain's velocity in the car's reference frame, \( v_{\text{horizontal}}^{\text{car}} \), is due to the car's motion at 40.0 km/h.

   Therefore, using trigonometry, we can relate the components of the rain's velocity:
   \[
   \tan(54.0^\circ) = \frac{v_{\text{horizontal}}^{\text{car}}}{v_{\text{vertical}}^{\text{car}}}
   \]
   where \( v_{\text{horizontal}}^{\text{car}} = v_{\text{car}} = 40.0 \, \text{km/h} \).

2. **Calculate the vertical component of the rain's velocity in the car's reference frame**:
   Using the above equation:
   \[
   v_{\text{vertical}}^{\text{car}} = \frac{v_{\text{horizontal}}^{\text{car}}}{\tan(54.0^\circ)}
   \]
   Substituting the known values:
   \[
   v_{\text{vertical}}^{\text{car}} = \frac{40.0}{\tan(54.0^\circ)}
   \]

   Let's calculate this:

\[
v_{\text{vertical}}^{\text{car}} = \frac{40.0}{\tan(54.0^\circ)} = \frac{40.0}{1.376} \approx 29.07 \, \text{km/h}
\]

3. **Find the velocity of the rain with respect to the Earth**:
   In the Earth's reference frame, the rain has no horizontal component; it falls vertically. So the vertical velocity we calculated is actually the rain's speed with respect to the Earth:
   \[
   v_{\text{rain}}^{\text{earth}} = v_{\text{vertical}}^{\text{car}} = 29.07 \, \text{km/h}
   \]

Thus, the magnitude of the rain's velocity with respect to the Earth is approximately **29.1 km/h** (downward).

Let me know if you would like a breakdown of the rain's velocity with respect to the car, or if you'd like further details! Here are five related questions for practice:

1. If the car were moving westward at the same speed, how would the rain's traces on the window appear?
2. How would the rain’s velocity change if the car sped up to 60.0 km/h?
3. What if the rain was falling at an angle instead of vertically—how would you solve the problem?
4. Can you find the rain’s velocity relative to the car if the angle was 30° instead of 54°?
5. How would the rain's velocity be affected if the car was stationary?

**Tip:** Always break down vectors into horizontal and vertical components for clarity when solving motion problems in two dimensions!

A car travels due east with a speed of 40.0 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 54.0° with the vertical. Find the velocity of the rain with respect to the car and the Earth. (Enter the magnitude of the velocity.)

Find the velocity of rain with respect to the car and the Earth as a car travels east at 40.0 km/h and raindrops form a 54.0° angle with the vertical. Learn how to apply kinematic concepts and trigonometric formulas for relative motion in this step-by-step solution. Suitable for students in Grades 10-12.

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