Problem 75 - Fundamentals of Physics

Question

A train travels due south at \(30 \mathrm{~m} / \mathrm{s}\) (relative to the ground) in a rain that is blown toward the south by the wind. The path of each raindrop makes an angle of \(70^{\circ}\) with the vertical, as measured by an observer stationary on the ground. An observer on the train, however, sees the drops fall perfectly vertically. Determine the speed of the raindrops relative to the ground.

Step-by-Step Solution

Answer

The speed of the raindrops relative to the ground is approximately 31.93 m/s.

Step 1: Understand the scenario

We need to determine the speed of raindrops relative to the ground when a train is moving southward. The raindrops are influenced by both gravity and wind, causing them to fall at an angle of \(70^\circ\) to the vertical from the perspective of someone on the ground, while appearing vertical to someone on the train.

Step 2: Set up the velocity components

Let the speed of the raindrops relative to the ground be \(v_r\), and the horizontal component of this speed (caused by the wind) as \(v_{r,h}\), and the vertical component (due to gravity) as \(v_{r,v}\). Since the observer on the train sees the raindrops falling vertically, the horizontal wind speed equals the train's speed, \(v_{r,h} = 30\, \mathrm{m/s}\).

Step 3: Use trigonometry to find vertical component

The path of the raindrop makes a \(70^\circ\) angle with the vertical, meaning:\[v_{r,h} = v_r \sin(70^\circ) \]\[v_{r,v} = v_r \cos(70^\circ) \]Using \(v_{r,h} = 30 \, \mathrm{m/s}\), we solve for \(v_r\).

Step 4: Solve for raindrop speed using known component

Since \(v_{r,h} = v_r \sin(70^\circ)\) and \(v_{r,h} = 30 \, \mathrm{m/s}\), substitute to find \(v_r\):\[ v_r \sin(70^\circ) = 30 \] \[ v_r = \frac{30}{\sin(70^\circ)} \] Calculate \(v_r\).

Step 5: Calculate raindrop speed

Calculate the numerical value using the sine function:\[ \sin(70^\circ) \approx 0.9397 \]\[ v_r = \frac{30}{0.9397} \approx 31.93 \, \mathrm{m/s} \]Thus, the speed of the raindrops relative to the ground is approximately \(31.93\, \mathrm{m/s}\).

Key Concepts

Velocity Components

In physics, any velocity can be split into components. It helps to visualize motion from different perspectives and understand the influence of multiple forces. For example, a raindrop falling through the air is subjected to both gravity and wind.

Let's break it down using the problem of a train moving south at 30 m/s through rain. When viewed by an observer on the ground, the rain appears to be falling at an angle. This is due to the wind adding a horizontal velocity component to the raindrop's overall motion.

When resolving velocity into components, two main directions are considered: vertical and horizontal.

• The **vertical component** (\(v_{r,v}\)) results from gravity pulling it down.
• The **horizontal component** (\(v_{r,h}\)) is impacted by the wind blowing the raindrops sideways.

In this example, the horizontal component equals the velocity of the train, aligning with the observer's perspective onboard who sees the raindrops fall straight down.

Trigonometry in Physics

Trigonometry is a powerful tool in physics to resolve vectors like those of velocity into their components.

In our raining train scenario, the path of the raindrops makes an angle of \(70^\circ\) with the vertical. Understanding this angle is crucial for deploying trigonometric functions, particularly sine and cosine, to find velocity components.

Here's how it works:

• The **sine function** relates the angle to the horizontal component: \(v_{r,h} = v_r \sin(70^\circ)\).
• The **cosine function** relates the angle to the vertical component: \(v_{r,v} = v_r \cos(70^\circ)\).

By organizing these trigonometric relationships, one can utilize known values, like \(v_{r,h} = 30\, \mathrm{m/s}\), to find unknowns such as the total speed of the raindrops relative to the ground, including their direction and magnitude.

Angle of Motion

The angle of motion is essential in determining how an object moves relative to different frames of reference. It reveals how multiple forces interact to determine an object's path.

In this example, the raindrops fall at an angle of \(70^\circ\) from the vertical, meaning this is the angle at which the wind forces them to deviate from a straight fall. For an observer on the ground, the mix of wind and gravity gives the appearance of a diagonal path.

This angle is crucial because:

• It allows observers to use trigonometry to determine exact component speeds.
• It describes how environmental forces like wind modify object motion.

Understanding an angle of motion like this helps clarify how forces influence movement, offering insights into both velocity direction and magnitude, enabling broader realizations about motion across varying frames of reference.