Question

A train travels due south at \(28 \mathrm{~m} / \mathrm{s}\) (relative to the ground) in rain that is blown to the south by the wind. The path of each raindrop makes an angle of \(64^{\circ}\) with the vertical, as measured by an observer stationary on the Earth. An observer on the train, however, sees perfectly vertical tracks of rain on the windowpane. Determine the speed of the drops relative to the Earth.

Short answer

To find the speed of the raindrops relative to the Earth, substitute the value as calculated in the final step. Calculate \( v_{re} = 28 \mathrm{~m/s} / \cos(64^{\circ}) \) to get the speed of the raindrops relative to the Earth.

Step-by-step solution

Decompose the velocity of the rain

The velocity of the rain can be broken down into two components: vertical and horizontal. The velocity of the rain with respect to Earth, let's call this \( v_{re} \), is the vector sum of these two components. Given that the rain falls at an angle of \( 64^{\circ} \), the horizontal component \( v_{x} \) is \( v_{re}\cos(64^{\circ}) \) and the vertical component \( v_{y} \) is \( v_{re}\sin(64^{\circ}) \).

Determine the horizontal velocity

According to the observer on the train, the rain appears to fall vertically, meaning the horizontal component of the rain's velocity is equal to the train's velocity. So, we have \( v_{x} = v_{train} = 28 \mathrm{~m/s} \). Therefore, the rain's speed with respect to Earth is \( v_{re} = v_{x}/\cos(64^{\circ}) \).

Calculate the speed of the raindrops

Now substitute the known values into the equation from step 2 to find \( v_{re} \). This will give us the speed of the raindrops relative to Earth.

Key concepts

Vector Components

Vectors are essential in understanding motion, especially when dealing with multiple directions. For any vector, including velocity vectors, we can break it down into its horizontal and vertical components. This helps to simplify the problem by treating each direction separately.
Consider the raindrop in our exercise. It moves in a direction that forms a 64° angle with the vertical. This means we can use trigonometry to find separate components of the raindrop's velocity:

• The **horizontal component** is given by the formula \( v_{x} = v_{re} \times \cos(64^{\circ}) \).
• The **vertical component** is given by the formula \( v_{y} = v_{re} \times \sin(64^{\circ}) \).

These components help us understand how fast the raindrop moves in each direction. By understanding vector components, we gain insight into how motion is experienced in different frames of reference.

Relative Velocity

Relative velocity comes into play when observing an object from different frames of reference. It helps us understand how an object's velocity appears to change from different viewpoints. In the case of the train and the raindrops, the relative velocity is key.
The observer on the ground sees the raindrop moving at an angle, while the observer on the train sees it falling vertically. This difference is because the train itself is moving south. To the train observer, the horizontal component of the raindrop's velocity matches the train's velocity, which is 28 m/s.
Because of this, the horizontal component \( v_{x} \) is equal to the train’s velocity. We can then use this information to determine the speed of the raindrops relative to Earth:

• The horizontal velocity of the raindrops is equal to the train's speed, \( v_{x} = 28 \) m/s.
• We use the equation \( v_{re} = \frac{v_{x}}{\cos(64^{\circ})} \) to find the raindrop's speed relative to Earth.

Relative velocity makes sense of how motion varies depending on the observer's own movement.

Trigonometry in Physics

Trigonometry is a powerful tool in physics, especially when dealing with angles and vectors. It helps us determine the various components of a motion when direction is involved. In this rain velocity problem, trigonometry is used to break down the raindrop's velocity into its components.
When you have an angle like the 64° between the raindrop’s path and the vertical direction, trigonometric functions such as sine and cosine are vital. Here’s how they work:

• The **cosine function** relates the horizontal component of the rain's velocity to the velocity of the raindrop and the angle: \( v_{x} = v_{re} \times \cos(64^{\circ}) \).
• The **sine function** relates the vertical component of the rain's velocity in a similar way: \( v_{y} = v_{re} \times \sin(64^{\circ}) \).

Using these trigonometric functions, you can precisely calculate each component of a vector, helping to solve physics problems effectively. Trigonometry simplifies complex motions and enables a better understanding of how angles affect movement.