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 following reference frames. (Enter the magnitude of the velocity.)
To find the velocity of the rain with respect to different reference frames, we need to consider the relative velocities and apply vector addition.
First, let's define the reference frames:
1. Earth Frame (EF): This is the frame in which the raindrops are falling vertically.
2. Car Frame (CF): This is the frame in which the car is stationary.
3. Road Frame (RF): This is the frame in which the car travels due east.
Now, let's find the velocity of the rain in each reference frame:
1. Velocity of the Rain with respect to Earth Frame (VR/EF):
Since the raindrops are falling vertically with respect to the Earth, their velocity with respect to the Earth Frame is simply the vertical velocity component:
VR/EF = 0 km/h (Raindrops fall vertically on the Earth)
2. Velocity of the Rain with respect to Car Frame (VR/CF):
In the Car Frame, the car is stationary, so the only velocity of the raindrops that contributes is the vertical component. However, this component is now inclined due to the angle between the traces of raindrops on the side windows and the vertical direction. This angle is 54.0°.
Therefore, VR/CF = (Vertical Component)/(Cosine of Angle)
= 0 km/h / cos(54.0°)
≈ 1.18 km/h
3. Velocity of the Rain with respect to Road Frame (VR/RF):
In the Road Frame, the car is moving due east at a speed of 40.0 km/h.
Therefore, VR/RF = VR/EF + VR/CF (Applying vector addition)
= 0 km/h + 1.18 km/h
≈ 1.18 km/h
So, the velocity of the rain with respect to the Road Frame is approximately 1.18 km/h.