While driving in the rain at 30 m/s you notice that the raindrops appear to be going straight down. After dropping your little sister off at daycare you turn around and drive the opposite direction at the same speed. However now you notice that the rain is making an angle of 35o relative to the ground. What is the angle and the speed of the rain relative to the ground?
To find the angle and speed of the rain relative to the ground, we can use the concept of vector addition.
Let's break down the situation into components:
1. First, let's consider the raindrop's vertical component. When driving in the rain at 30 m/s, the raindrops appear to be going straight down. So, the vertical velocity of the raindrop relative to the car is 0 m/s.
2. Now, when driving in the opposite direction at the same speed, we observe the raindrops making an angle of 35 degrees relative to the ground. The raindrop's vertical velocity relative to the ground can then be calculated using trigonometry. Since the vertical component (0 m/s) remains the same, we can use the following equation:
Vertical velocity relative to the ground = Vertical velocity relative to the car * sin(angle)
Vertical velocity relative to the ground = 0 m/s * sin(35 degrees) = 0 m/s
From this, we can deduce that the raindrop's vertical velocity remains unchanged after turning around.
3. Finally, let's consider the raindrop's horizontal component. Regardless of the direction of the car, raindrops appear to be falling straight down. Therefore, the horizontal velocity of the raindrop relative to the ground remains the same as the car's speed, which is 30 m/s.
Now, to find the angle and speed of the rain relative to the ground, we can use the following equation:
tan(angle) = Vertical velocity relative to the ground / Horizontal velocity relative to the ground
Since the vertical velocity relative to the ground is 0 m/s and the horizontal velocity relative to the ground is 30 m/s, we have:
tan(angle) = 0 m/s / 30 m/s = 0
Taking the inverse tangent (arctan) of both sides, we find:
angle = arctan(0) = 0 degrees
From this, we determine that the angle of the rain relative to the ground is 0 degrees, indicating that the raindrops appear to be falling vertically.
The speed of the rain relative to the ground remains the same as the car's speed, which is 30 m/s.