A car travels due east with a speed of 50.0 km/h. Raindrops are falling at a constant speed vertically with respect to the Earth.The traces of the rain on the windows of the car make an angle of 60 degrees with the vertical. Find the velocity of the rain with respect to (a) the car and (b) the Earth

(a) Well, if the raindrops are falling vertically with respect to the Earth and the car is traveling east, then the raindrops would look like they are coming at an angle to a person in the car. It's like Mother Nature is playing a trick on you, making it seem like the rain is purposely trying to hit your windshield. So, in this case, the velocity of the rain with respect to the car would be equal to the velocity of the raindrops falling vertically minus the velocity of the car. But since the raindrops are falling vertically, and the car is moving horizontally, the angle between them will change as the car moves. So, if the angle between the raindrops and the car is 60 degrees, you might want to consider taking an umbrella for some extra protection!

(b) Now, let's consider the velocity of the rain with respect to the Earth. Since the car is also moving on the Earth, we need to take that into account. You can think of it like this: the rain is falling vertically with respect to the Earth, and the car is driving horizontally. So, to find the velocity of the rain with respect to the Earth, we add the velocity of the car to the velocity of the rain with respect to the car. In this case, my friend, you have a little bit of a rain dance going on. The velocity of the rain with respect to the Earth is like a symphony of motion – the vertical raindrop dance plus the horizontal car tango!

And that's the funny thing about rain, it can make your car seem like a target and turn a simple drive into a wet adventure!

To find the velocity of the rain with respect to the car and the Earth, we can break down the given information and apply vector addition.

Let's consider the following velocities:

1. Velocity of the car: The car is traveling due east with a speed of 50.0 km/h. Since it's traveling only in the horizontal direction, its velocity can be represented as (50.0 km/h, 0).

2. Velocity of the raindrops with respect to the car: The traces of the raindrops on the windows of the car make an angle of 60 degrees with the vertical. This means the raindrops are falling at an angle of 60 degrees with respect to the car. Since the car is moving horizontally, the vertical component of the raindrops' velocity will only affect their direction without changing the magnitude. The horizontal component can be considered as the velocity of the raindrops with respect to the car. Thus, we need to find this horizontal component.

To find the horizontal component of the raindrops' velocity, we can use trigonometry. The angle between the vertical direction and the raindrops' velocity is 60 degrees, and the speed (magnitude) of the raindrops' velocity is unknown (let's call it v).

Using trigonometry, we can find the horizontal component of the raindrops' velocity:

Horizontal component = v * cos(60 degrees) = v * 0.5

Therefore, the velocity of the raindrops with respect to the car is (0.5v, 0).

3. Velocity of the raindrops with respect to the Earth: To find this velocity, we need to consider the velocity of the car and the velocity of the raindrops with respect to the car. The velocity of the raindrops with respect to the Earth can be obtained by adding the velocity of the car and the velocity of the raindrops with respect to the car.

Velocity of the raindrops with respect to the Earth = Velocity of the car + Velocity of the raindrops with respect to the car

= (50.0 km/h, 0) + (0.5v, 0)

= (50.0 km/h + 0.5v, 0)

Therefore, the velocity of the raindrops with respect to the Earth is (50.0 km/h + 0.5v, 0).

To summarize:

(a) The velocity of the rain with respect to the car is (0.5v, 0), where v is the magnitude of the raindrops' speed.

(b) The velocity of the rain with respect to the Earth is (50.0 km/h + 0.5v, 0), where v is the magnitude of the raindrops' speed.

To find the velocity of the rain with respect to the car and the Earth, we'll need to break down the rain's velocity into its horizontal and vertical components.

Let's denote the velocity of the car as Vcar = 50.0 km/h, and the angle made by the raindrops with the vertical as θ = 60 degrees.

(a) Velocity of the rain with respect to the car:
To find the velocity of the rain with respect to the car (Vrain_car), we only need to consider the horizontal component since raindrops fall vertically.

Vrain_car = Vrain * cosθ

Given that Vrain = Vrain_car = 50 * cos(60°) = 25 km/h

Therefore, the velocity of the rain with respect to the car is 25 km/h in the horizontal direction (eastward).

(b) Velocity of the rain with respect to Earth:
To find the velocity of the rain with respect to the Earth (Vrain_earth), we need to consider both the horizontal and vertical components.

Vrain_earth = √(Vrain_horizontal^2 + Vrain_vertical^2)

The raindrops fall vertically, so the vertical component of their velocity is:

Vrain_vertical = Vrain * sinθ

Vrain_vertical = 50 * sin(60°) = 43.3 km/h

The raindrops are falling vertically, which means that the horizontal component of their velocity is the same as the velocity of the car:

Vrain_horizontal = Vcar = 50 km/h

Now, we can calculate the velocity of the rain with respect to the Earth:

Vrain_earth = √(50^2 + 43.3^2) = 64.6 km/h

Therefore, the velocity of the rain with respect to the Earth is 64.6 km/h, making an angle of 60 degrees with the vertical.