A person looking out the window of a stationary train notices that raindrops are falling vertically down at a speed of 6.17 m/s relative to the ground. When the train moves at a constant velocity, the raindrops make an angle of 25° when they move past the window, as the drawing shows. How fast is the train moving?

Question

A person looking out the window of a stationary train notices that raindrops are falling vertically down at a speed of 6.17 m/s relative to the ground. When the train moves at a constant velocity, the raindrops make an angle of 25° when they move past the window, as the drawing shows. How fast is the train moving?

Answer 1

To determine the speed of the train, we can use trigonometry and consider the motion of the raindrops relative to the train. Let's break down the problem step by step:

1. Draw a diagram: Sketch a diagram representing the situation described in the question. Label the vertical direction (downwards) as "y" and the horizontal direction (parallel to the ground) as "x." Draw the raindrops falling at an angle of 25° to the vertical, and the train moving horizontally.

2. Identify the given information: The question provides the vertical speed of the raindrops relative to the ground, which is 6.17 m/s. We also know the angle at which the raindrops move past the window, which is 25°.

3. Analyze the motion: Since the raindrops are falling vertically relative to the ground, the vertical component of their velocity is 6.17 m/s. We can find the horizontal component of their velocity by using trigonometry. The horizontal velocity component can be calculated as follows:
Horizontal velocity = Vertical velocity * tan(angle)

4. Calculate the horizontal velocity: Substitute the known values into the formula:
Horizontal velocity = 6.17 m/s * tan(25°)

Using a scientific calculator, find the tangent of 25° and multiply it by the vertical velocity to obtain the horizontal velocity.

5. Determine the train's velocity: Once we have the horizontal velocity component of the raindrops, we can conclude that it is equal to the velocity of the train. Therefore, the speed of the train is the same as the calculated horizontal velocity.

6. Calculate the speed of the train: Multiply the horizontal velocity of the raindrops by the tangent of 25°.

By following these steps and performing the necessary calculations, you can determine the speed at which the train is moving.