Problem 72 - University Physics with Modern Physics

Question

When a train's velocity is 12.0 m/s eastward, raindrops that are falling vertically with respect to the earth make traces that are inclined 30.0\(^\circ\) to the vertical on the windows of the train. (a) What is the horizontal component of a drop's velocity with respect to the earth? With respect to the train? (b) What is the magnitude of the velocity of the raindrop with respect to the earth? With respect to the train?

Step-by-Step Solution

Answer

The horizontal velocity with respect to the earth is 18.0 m/s; the total velocity (earth) is 20.8 m/s. With respect to the train, horizontal velocity is 6.0 m/s; total velocity (train) is 12.0 m/s.

Step 1: Understanding Velocity Components

The raindrop is falling with a velocity that is inclined at 30.0\(^\circ\) to the vertical. This inclination implies that the raindrop has both vertical and horizontal velocity components. The task requires calculating these components relative to both the earth and the train.

Step 2: Finding the Horizontal Component with respect to the Train

We first determine the horizontal component of the velocity of the raindrop with respect to the train. Given the angle, we use the component formula: \( v_{horizontal} = v_{total} \times \sin(\theta) \), where \( \theta = 30.0\, ^\circ \) is the angle with respect to the vertical.

Step 3: Calculating Horizontal Component with respect to the Earth

The horizontal component of the raindrop's velocity with respect to the earth is equal to the horizontal component relative to the train plus the velocity of the train (12.0 m/s). Thus, \( v_{horizontal,\,earth} = v_{horizontal,\,train} + 12.0\, \text{m/s} \).

Step 4: Finding the Total Velocity of the Raindrop with respect to the Train

The total velocity of the raindrop with respect to the train can be calculated using trigonometric relationships. Since the vertical component is along the lines of the window (perpendicular to the horizontal ground), it doesn't change with respect to the earth. Use \( v_{total} = \frac{v_{horizontal,\,train}}{\sin(30.0\, ^\circ)} \).

Step 5: Calculating Total Velocity of the Raindrop with respect to the Earth

The total velocity of the raindrop with respect to the earth is: \( v_{total,\,earth} = \sqrt{v_{horizontal,\,earth}^2 + v_{vertical}^2} \), where \( v_{horizontal,\,earth} = v_{horizontal,\,train} + 12.0\, \text{m/s} \) and \( v_{vertical} = v_{total} \cos(30.0\, ^\circ) \).

Step 6: Calculating the Magnitude of the Velocity with respect to the Train

With respect to the train, the horizontal velocity component is aligned with the raindrop traces. Hence, use \( v_{total,\,train} = \frac{2 \times v_{horizontal,\,train}}{\sqrt{3}} \) from previously deduced relationships.

Step 7: Final Calculation

Using the above steps, determine the exact numerical values of the horizontal and total velocities with respect to both the earth and the train.

Key Concepts

Projectile Motion

Projectile motion involves analyzing objects propelled into the air and influenced only by gravity. Raindrops, when viewed from a moving train, exhibit projectile motion. This is due to their seemingly angular paths relative to the train’s windows. This motion can be dissected into two components: horizontal and vertical. Both components work together to give raindrops their full trajectory.

When you observe raindrops through the train's window, they seem to move in a slant rather than the vertical path they take in reality. The train's velocity moving eastward combines with the drop’s initial vertical descent to create this perceived angled motion. This blended movement is a fundamental aspect of projectile motion—where multiple velocity components define an object's complete movement despite gravity being the only acting force.

Trigonometry in Physics

Trigonometry plays a vital role in physics, particularly when breaking down forces and velocities into usable components. The angle of 30° given in this problem guides us to use trigonometric functions to find individual velocity components of the raindrops.

For instance, the horizontal component can be found with the sine function:

• Calculate using the formula: \( v_{horizontal} = v_{total} \times \sin(\theta) \)

Here, the angle \( \theta \) is 30°. This calculation is crucial as it helps determine how the velocity is distributed in a horizontal plane from the perspective of the train.

Similarly, the vertical component aligns with the use of the cosine function since it remains aligned with the gravitational pull. Understanding how to apply trigonometry allows one to dissect complex movements like raindrops slanting due to external motion, making otherwise difficult calculations straightforward and precise.

Velocity Components

Understanding velocity components is key to solving problems involving relative motion, like those seen in raindrop analysis from a moving train. These components help explain how different velocities interact from various frames of reference.

In this scenario, a raindrop has both horizontal and vertical velocity components:

• **Horizontal component**: Represented as part of the raindrop's velocity affected by the train's eastward movement. This requires adding the train’s velocity to the calculated horizontal component of the raindrop relative to the train.
• **Vertical component**: This component stays consistent, as it is independent of horizontal motion from the train. Calculated using: \( v_{vertical} = v_{total} \times \cos(\theta) \).

These components combined give the total velocity of the raindrop in two reference frames: relative to the earth, and relative to the moving train. Such analyses are crucial in physics to account for all forces and directions impacting an object's path.