A person looking out the window of a stationary train notices that raindrops are falling vertically down at a speed of 6.6 m/s relative to the ground. When the train moves at a constant velocity, the raindrops make an angle of θ = 36° when they move past the window as the drawing shows. How fast is the train moving?
To find the speed of the train, we can use trigonometry and the concept of relative velocity.
Let's consider the scenario when the train is moving. The raindrops, when observed from the train, appear to be falling diagonally instead of vertically as seen from the stationary train.
We have the following information:
- Speed of raindrops relative to the ground: 6.6 m/s
- Angle between the direction of raindrops (relative to the ground) and the direction of raindrops observed from the train (θ): 36°
Now, we need to analyze the velocity components of the raindrops as observed from the train. We can break this velocity into two components:
1. Vertical component: The vertical component of the train's velocity is the same as the vertical speed of the raindrops as observed from the train.
2. Horizontal component: The horizontal component of the train's velocity is what causes the raindrops to appear to move diagonally from the perspective of the train.
Using trigonometry, we can relate the components of the velocity:
Vertical component = Speed of raindrops relative to the ground = 6.6 m/s
Horizontal component = Vertical component / tan(θ)
Now, we can calculate the horizontal component and find the speed of the train:
Horizontal component = 6.6 m/s / tan(36°)
Speed of the train = √(Vertical component^2 + Horizontal component^2)
Substituting the values and calculating, we can determine the speed of the train.