A person looking out the window of a stationary train notices that raindrops are falling vertically down at a speed of 8.1 m/s relative to the ground. When the train moves at a constant velocity, the raindrops make an angle of θ = 25° when they move past the window as the drawing shows. How fast is the train moving?

To calculate the speed of the train, we need to consider the motion of the raindrops relative to the train. Let's break down the problem step by step:

Step 1: Understand the given information.
- The raindrops fall vertically at a speed of 8.1 m/s relative to the ground.
- When the train moves at a constant velocity, the raindrops make an angle of θ = 25° as they move past the window.

Step 2: Analyze the motion of the raindrops relative to the train.
The raindrops appear to move horizontally from the perspective of a person in the train. This horizontal motion results from the combination of their vertical motion relative to the ground and the horizontal motion of the train.

Step 3: Break down the raindrop's motion.
Let's consider the horizontal and vertical components of the raindrop's motion.

- Vertical motion: The raindrops fall vertically at 8.1 m/s relative to the ground. This velocity remains constant.
- Horizontal motion: The raindrops move horizontally due to the motion of the train, resulting in an angle of 25°.

Step 4: Determine the horizontal and vertical components of the raindrops' motion.
To calculate the horizontal and vertical components, we can use trigonometry. The vertical component is the same as the raindrop's speed relative to the ground, which is 8.1 m/s. The horizontal component can be calculated as follows:

Horizontal component = Raindrop speed * cos(θ)
Horizontal component = 8.1 m/s * cos(25°)

Step 5: Determine the train's speed.
The horizontal component of the raindrop's motion is equal to the train's velocity because the raindrops appear to move horizontally from the perspective of a person in the train.

Thus, the speed of the train is:

Train speed = Horizontal component
Train speed = 8.1 m/s * cos(25°)

Now, you can calculate the train speed using a calculator or software:

Train speed ≈ 7.311 m/s

Therefore, the train is moving at a speed of approximately 7.311 m/s.