A car travels due east with a speed of 35.0 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 40.0° with the vertical. Find the velocity of the rain with respect to the following reference frames.
a) car
b) earth
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car: 35km/h divided by tan (40) = 41.7 sqrt of 35^2+41.7^2=54.4
earth: use Pythagorean theorem sqrt 54.4^2-35^2=41.6
use the same 40 degrees in the answer
Unsure if earth is right, but I had 38.0km/h at 62 degrees for my problem and used those steps to get 43.0 km/h for the car and 20.1km/h for Earth. They were correct so here is my input
a) In the reference frame of the car, we need to find the velocity of the raindrops. Since the car is moving due east, we need to consider the horizontal component of the velocity of the raindrops.
Using trigonometry, we can find the horizontal component of the rain velocity. The angle between the raindrops' velocity and the vertical is given as 40.0°. Since the vertical component of the rain's velocity is directly downwards, we can use the cosine function to find the horizontal component:
horizontal component = rain velocity * cosine(angle)
The cosine of 40.0° is approximately 0.766, so we can plug in the values:
horizontal component = rain velocity * 0.766
Since the raindrops are falling vertically with respect to the Earth, their velocity in the car's reference frame is equal to the horizontal component:
velocity of rain with respect to the car = horizontal component = rain velocity * 0.766
b) In the reference frame of the Earth, we need to find the velocity of the raindrops relative to the Earth's surface. The vertical component of the rain's velocity will contribute to the total velocity of the raindrops with respect to the Earth.
Using trigonometry again, we can find the vertical component of the rain velocity. The angle between the raindrops' velocity and the vertical is 40.0°, and the vertical component is directly downwards, so we can use the sine function:
vertical component = rain velocity * sine(angle)
The sine of 40.0° is approximately 0.643, so we can plug in the values:
vertical component = rain velocity * 0.643
The total velocity of the raindrops with respect to the Earth is the vector sum of the horizontal and vertical components. Since we already found the horizontal component to be rain velocity * 0.766, we can add:
velocity of rain with respect to the Earth = horizontal component + vertical component
= rain velocity * 0.766 + rain velocity * 0.643
= rain velocity * (0.766 + 0.643)
So in the reference frame of the Earth, the velocity of the rain with respect to the Earth is rain velocity times the sum of 0.766 and 0.643.
To find the velocity of the rain with respect to different reference frames, we need to consider the relative motion between the car and the Earth.
a) Velocity of the rain with respect to the car:
Since we want to find the velocity of the raindrops relative to the car, we can use vector addition. The velocity of the raindrops relative to the car will be the vector sum of the raindrops' velocity relative to the Earth and the car's velocity relative to the Earth.
Let's denote the velocity of the rain with respect to Earth as v₁ and the velocity of the car with respect to Earth as v₂. We want to find the velocity of the rain with respect to the car, which we'll call v₃.
First, we need to find the horizontal component of the rain's velocity relative to Earth. Since the raindrops are falling vertically, the horizontal component is simply 0 km/h.
Next, we'll find the vertical component of the rain's velocity relative to Earth:
The angle between the raindrops' path and the vertical is given as 40.0°. The vertical component of the rain's velocity can be expressed as v₁ * sin(40.0°).
Now, we have:
Horizontal component of v₃ = Horizontal component of v₁ + Horizontal component of v₂ = 0 km/h + 35.0 km/h = 35.0 km/h
Vertical component of v₃ = Vertical component of v₁ + Vertical component of v₂ = v₁ * sin(40.0°) + 0 km/h
Therefore, the velocity of the rain with respect to the car is a vector with a horizontal component of 35.0 km/h and a vertical component of v₁ * sin(40.0°).
b) Velocity of the rain with respect to the Earth:
In this case, we want to find the velocity of the raindrops relative to the Earth. Since we already know the velocity of the raindrops with respect to the car (v₃) and the velocity of the car with respect to Earth (v₂), we can use vector subtraction to find the velocity of the rain with respect to the Earth.
Horizontal component of v₃ = 35.0 km/h
Vertical component of v₃ = v₁ * sin(40.0°)
Horizontal component of the rain's velocity with respect to Earth = Horizontal component of v₃ - Horizontal component of v₂ = 35.0 km/h - 35.0 km/h = 0 km/h
Vertical component of the rain's velocity with respect to Earth = Vertical component of v₃
Therefore, the velocity of the rain with respect to the Earth is a vector with a horizontal component of 0 km/h and a vertical component of v₁ * sin(40.0°).
a) They have told you that the rain is falling vertically. Call its vertical velocity Vy. In the reference frame of the car, the rain also has a horizontal component of Vx = 35 km/h and a vertical component Vy. Because of the 40 degree tracks of the drops,
Vx/Vy = tan 40 = 0.839
Vy = 1.19*Vx = 41.7 km/h
Magnitude of velocity = sqrt (Vx^2 + Vy^2) = 54.5 km/h
b) In the earth references frame, the only velocity componenet is the vertical one, Vx, which is the same as in part (a)