QuestionA car travels due east with a speed of $42.0 \mathrm{~km} / \mathrm{h}$ . 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 $74.0^{\circ}$ with the vertical. Find the velocity of the rain with respect to the following reference frames. (a) the car $\square$ $\mathrm{km} / \mathrm{h}$ downward and $\square$ 0 west of vertical (b) the Earth $\qquad$ $\mathrm{km} / \mathrm{h}$ vertically downward.

Studdy Solution

STEP 1

1. The car is traveling due east with a constant speed of $42.0 \ \mathrm{km/h}$ .
2. The rain is falling vertically downward with a constant speed relative to the Earth.
3. The traces of the rain on the car's side windows form an angle of $74.0^\circ$ with the vertical.
4. We need to find the velocity of the rain with respect to the car and the Earth.

STEP 2

1. Analyze the relative motion of the rain with respect to the car to find the rain's velocity components.
2. Use trigonometric relationships to determine the rain's vertical and horizontal components.
3. Calculate the velocity of the rain with respect to the Earth.

STEP 3

Let's denote the velocity of the rain with respect to the Earth as $v_r$ (downward) and the velocity of the car as $v_c = 42.0 \ \mathrm{km/h}$ (eastward).
The traces of the rain on the car's side windows form an angle of $74.0^\circ$ with the vertical. This indicates that the rain has both a vertical and a horizontal component with respect to the car.
Given that the angle is with the vertical, the horizontal component of the rain's velocity with respect to the car is $v_c$ (eastward), and the vertical component is $v_r$ (downward).

STEP 4

Using trigonometric relationships, we can find the rain's vertical and horizontal components with respect to the car.
The tangent of the angle $74.0^\circ$ gives the ratio of the horizontal component (which is the car's speed) to the vertical component (rain's speed downward):
$\tan(74.0^\circ) = \frac{v_c}{v_r}$

STEP 5

Calculate the rain's vertical speed $v_r$ using the given car speed $v_c = 42.0 \ \mathrm{km/h}$ :
$v_r = \frac{v_c}{\tan(74.0^\circ)}$
Using the value $\tan(74.0^\circ) \approx 3.487$ :
$v_r = \frac{42.0 \ \mathrm{km/h}}{3.487} \approx 12.05 \ \mathrm{km/h}$

STEP 6

To find the horizontal component of the rain's velocity with respect to the car $v_rh$ , we note that it must be equal to the car's speed $v_c$ since the rain is falling vertically with respect to the Earth:
$v_rh = v_c = 42.0 \ \mathrm{km/h}$

STEP 7

To find the velocity of the rain with respect to the Earth, we combine the vertical speed of the rain $v_r$ and the horizontal speed of the car $v_c$ .
Since the rain is falling vertically with respect to the Earth, its horizontal component with respect to the Earth is zero, and its vertical component is equal to what we calculated:
$\text{Velocity of rain with respect to the Earth} = 12.05 \ \mathrm{km/h} \ \text{downward}$
Solution: (a) The velocity of the rain with respect to the car is $42.0 \ \mathrm{km/h}$ downward and $0 \ \mathrm{km/h}$ west of vertical.
(b) The velocity of the rain with respect to the Earth is $12.05 \ \mathrm{km/h}$ vertically downward.

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