a car travels due east with a speed of 30.0 km/h. Rain drops are falling at a constant speed celocity with respect to the earth. The traces of the rain on the side windows of the car make an angle of 42.0 degrees with the vertical. Find the velocity of the rain with respect to the car and the earth. (Enter the magnitude of the velocity.) (a) the car in km/h (B) the Earth in km/h

To find the velocity of the rain with respect to the car, we can use trigonometry. Let's assume the velocity of the rain with respect to the car is v_{\text{rain-car}} and the velocity of the car is v_{\text{car}}.

Using the angle of 42.0 degrees, we can write the following equation:

\tan(42.0^\circ) = \frac{{v_{\text{rain-car}}}}{{v_{\text{car}}}}

Now, let's solve for v_{\text{rain-car}}:

v_{\text{rain-car}} = v_{\text{car}} \cdot \tan(42.0^\circ)

Substituting the given value v_{\text{car}} = 30.0 \, \text{km/h} into the equation gives:

v_{\text{rain-car}} = 30.0 \, \text{km/h} \cdot \tan(42.0^\circ)

Now, we can use a calculator to evaluate this expression:

v_{\text{rain-car}} \approx 30.0 \, \text{km/h} \cdot 0.900404044 \approx 27.012 \, \text{km/h}

Therefore, the velocity of the rain with respect to the car is approximately 27.012 km/h.

To find the velocity of the rain with respect to the Earth, we can use the concept of relative velocity. Since the car is traveling due east, the velocity of the car with respect to the Earth is v_{\text{car}} = 30.0 \, \text{km/h} towards the east.

The velocity of the rain with respect to the Earth can be found by adding the velocity of the rain with respect to the car and the velocity of the car with respect to the Earth:

v_{\text{rain-earth}} = v_{\text{rain-car}} + v_{\text{car}}

Substituting the values gives:

v_{\text{rain-earth}} = 27.012 \, \text{km/h} + 30.0 \, \text{km/h}

Simplifying this equation gives:

v_{\text{rain-earth}} = 57.012 \, \text{km/h}

Therefore, the velocity of the rain with respect to the Earth is 57.012 km/h.

Answer:
(a) The velocity of the rain with respect to the car is approximately 27.012 km/h.
(b) The velocity of the rain with respect to the Earth is 57.012 km/h.