Questions: A car travels due east with a speed of 44.0 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 62.0° with the vertical. Find the speed of the rain (in km/h) with respect to the car and the Earth. (a) the car km/h (b) the Earth km/h

Solution Steps

We need to find the speed of the rain with respect to the car and the Earth. The car is moving east at \(44.0 \, \text{km/h}\), and the rain traces make an angle of \(62.0^\circ\) with the vertical on the car's windows.

The angle of \(62.0^\circ\) with the vertical indicates that the horizontal component of the rain's velocity relative to the car is due to the car's motion. Let \(v_r\) be the speed of the rain with respect to the Earth, and \(v_{rc}\) be the speed of the rain with respect to the car.

The horizontal component of the rain's velocity relative to the car is: \[ v_{rc} \sin(62.0^\circ) = 44.0 \, \text{km/h} \]

Using the equation from Step 2, solve for \(v_{rc}\): \[ v_{rc} = \frac{44.0}{\sin(62.0^\circ)} \]

Calculate \(v_{rc}\): \[ v_{rc} = \frac{44.0}{0.8829} \approx 49.85 \, \text{km/h} \]

The vertical component of the rain's velocity relative to the car is: \[ v_{rc} \cos(62.0^\circ) \]

Since the rain falls vertically with respect to the Earth, this vertical component is the actual speed of the rain with respect to the Earth: \[ v_r = v_{rc} \cos(62.0^\circ) \]

Calculate \(v_r\): \[ v_r = 49.85 \times 0.4695 \approx 23.41 \, \text{km/h} \]

Final Answer