Problem 77 - Fundamentals of Physics

Question

Snow is falling vertically at a constant speed of \(8.0 \mathrm{~m} / \mathrm{s}\). At what angle from the vertical do the snowflakes appear to be falling as viewed by the driver of a car traveling on a straight, level road with a speed of \(50 \mathrm{~km} / \mathrm{h} ?\)

Step-by-Step Solution

Answer

Approximately \(60.9^\circ\) from the vertical.

Step 1: Understanding the problem

We need to find the angle at which the snowflakes appear to be falling as viewed by a driver traveling with a certain speed. This involves visualizing the relative motion of the snowflakes with respect to the driver.

Step 2: Convert car speed to consistent units

First, let's convert the speed of the car from kilometers per hour to meters per second to match the units of the snow's falling speed. 1 km/h = 0.27778 m/s, thus:\[ 50 \text{ km/h} = 50 \times 0.27778 \text{ m/s} = 13.89 \text{ m/s} \]

Step 3: Determine relative motion components

In the driver's frame of reference, the snow is falling vertically at 8 m/s and has a horizontal component due to the driver's own motion at 13.89 m/s. We need to think of this as a vector with a vertical (let's call it \(V_s\)) and horizontal (let's call it \(V_c\)) component.\(V_s = 8 \text{ m/s (vertical)}\)\(V_c = 13.89 \text{ m/s (horizontal)}\)

Step 4: Calculate the angle of apparent motion

The angle \(\theta\) from the vertical can be found using the tangent function, which relates the opposite (horizontal speed) and adjacent (vertical speed) sides of a right triangle:\[ \tan(\theta) = \frac{V_c}{V_s} = \frac{13.89}{8} \]Solving for \(\theta\):\[ \theta = \tan^{-1}\left(\frac{13.89}{8}\right) \approx \tan^{-1}(1.736) \approx 60.9^\circ \]

Step 5: Conclusion

The snowflakes appear to be falling at an angle of approximately \(60.9^\circ\) from the vertical when viewed by the driver.

Key Concepts

Understanding Vector Components

In the context of relative motion, vector components play a crucial role in analyzing and visualizing how objects move. When you're in a moving car, the movement of other objects, like falling snowflakes, appears altered. This is due to the combined motion of both the car and the snowflakes.

In our example, the snowflakes are falling vertically with a speed of 8 m/s. However, to a driver traveling horizontally at 13.89 m/s (after converting the car's speed from km/h to m/s), these snowflakes appear to travel in a different direction. To fully understand this relative motion, we split the snowflakes' movement into two vector components:

• Vertical component (V_s): This represents the true motion of the snow, 8 m/s downwards.
• Horizontal component (V_c): This is due to the car's speed, 13.89 m/s, which affects how the snow appears to move from the driver's perspective.

By thinking of these motions as vectors, we can more easily calculate their effects on the perceived movement of the snowflakes.

Performing Angle Calculation

Now that we have our vector components, the next step is to determine the direction or angle at which the snowflakes seem to fall to the driver. This requires some knowledge of trigonometry, specifically the tangent function.

Imagine a right triangle where the horizontal component of motion (due to the car's speed) forms the opposite side, and the vertical speed of the snowflakes makes up the adjacent side. The angle from the vertical can be determined using the tangent function, given by:

\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{V_c}{V_s} = \frac{13.89}{8} \]

The angle \(\theta\) can then be found by taking the inverse tangent (also known as arctangent) of the ratio:

\[ \theta = \tan^{-1}\left(\frac{13.89}{8}\right) \approx \tan^{-1}(1.736) \approx 60.9^\circ \]

The driver would see the snowflakes falling at an angle of approximately 60.9 degrees from the vertical.

Converting Speed Units

In any physics problem, consistency in units is crucial. This is especially true when dealing with speed, a key part of understanding relative motion. our example from above highlights this with two different speed units: the snow falls at 8 m/s, and the car initially travels at 50 km/h.

To ensure clear, accurate calculations, we convert the car's speed from kilometers per hour to meters per second. The conversion factor is that 1 km/h equals roughly 0.27778 m/s. Using this, we find:

• Car's speed: 50 km/h \(= 50 \times 0.27778 \text{ m/s} \approx 13.89 \text{ m/s}\)

This simple conversion allows you to compare the speeds directly and use them together in calculations. By maintaining dimensional consistency, you can avoid errors and better understand the described motion.