Problem 94 - Physics for Scientists and Engineers with Modern Physics

Question

You are driving south on a highway at \(25 \mathrm{~m} / \mathrm{s}\) (approximately \(55 \mathrm{mi} / \mathrm{h}\) ) in a snowstorm. When you last stopped, you noticed that the snow was coming down vertically, but it is passing the windows of the moving car at an angle of \(37^{\circ}\) to the horizontal. Estimate the speed of the snowflakes relative to the car and relative to the ground.

Step-by-Step Solution

Answer

The snow's speed relative to the car is approximately 31.3 m/s, and relative to the ground is 18.8 m/s.

Step 1: Visualize and Set-Up the Problem

Imagine you are driving south at a constant speed, and the snow is falling straight down relative to the ground. But because you are moving, the snow appears to come at an angle relative to you. We need to find the snow's actual speed (relative to the ground) and apparent speed (relative to the car).

Step 2: Draw a Free Body Diagram

Draw a right triangle. The horizontal component is the speed of your vehicle, which is 25 m/s. The angle between the horizontal (highway) and the direction of snowflakes as they appear to you is 37°. The vertical side of this triangle represents the vertical speed of snowflakes.

Step 3: Calculate Snowflakes' Apparent Speed Relative to the Car

Use trigonometry to calculate the apparent speed. In the right triangle, the adjacent side is the car's speed, and the hypotenuse is the apparent speed of snowflakes relative to the car.\[\text{Let } v_c = \text{apparent speed}. \cos(37^{\circ}) = \frac{25}{v_c} \Rightarrow v_c = \frac{25}{\cos(37^{\circ})}\]

Step 4: Solve for Apparent Speed

Calculate the apparent speed:\[v_c = \frac{25}{0.7986} \approx 31.3 \, \text{m/s}\]

Step 5: Calculate the Vertical Component of the Snow's Velocity

Use the sine function to determine the vertical component of the snow's velocity (\(v_s\)) relative to the car:\[\sin(37^{\circ}) = \frac{v_s}{31.3} \Rightarrow v_s = 31.3 \times \sin(37^{\circ})\]

Step 6: Solve for Vertical Speed

Calculate the vertical speed of snowflakes:\[v_s = 31.3 \times 0.6018 \approx 18.8 \, \text{m/s}\]

Step 7: Find the Snow's Speed Relative to the Ground

Since the snow falls vertically relative to the ground, its speed relative to the ground is the vertical component calculated. Thus, the speed of the snow relative to the ground is 18.8 m/s.

Key Concepts

Understanding Relative Velocity

Relative velocity is the speed of an object as observed from a particular frame of reference. In our example, you're driving through the snow, making two frames of reference: one is from your car, and the other from the ground.
When you're inside the car, the snow seems to move at an angle and seems faster because you are moving too. To calculate the snow's apparent speed relative to the car, we used the relationship between the actual speed and the angle at which it appears.

• The apparent speed, also known as relative speed, depends on both the velocity of the car and the snow.
• By using trigonometric functions, like cosine, we determined that the apparent speed of snowflakes relative to the car is about 31.3 m/s.
• To an observer standing on the ground, however, the snow falls straight down, and its speed relative to the ground is just the vertical component, which is 18.8 m/s.

By considering these two perspectives, you understand how motion affects perceived speed.

Trigonometry's Role in Physics Problems

Trigonometry helps solve physics problems by dealing with angles and their relationships to side lengths of triangles. In the snow exercise, it turns a description of motion into a mathematical relationship.
Here, we use a right triangle to represent the motion of the car and the snow. The car's velocity is a horizontal line, and the hypotenuse is the apparent velocity of snow.

• The angle given, 37º, helps us calculate unknown speeds with trigonometric functions.
• The cosine function relates the car's speed and apparent speed: \[\cos(37^{\circ}) = \frac{25}{v_c}\]
• Sine helps find the vertical velocity of snow: \[\sin(37^{\circ}) = \frac{v_s}{31.3}\]

With these equations, we solve the triangle to find various components of the motion, clarifying how snow and car speeds relate based on direction and angle.

Exploring Rectilinear Motion

Rectilinear motion refers to movement along a straight line. In this physics problem, both the car and snowflakes move in straight lines.
As the car moves south along a highway, it maintains a constant velocity of 25 m/s, a classic example of rectilinear motion.
On the other hand, the snow is falling perpendicularly to the ground, making its movement also rectilinear, but vertically.

• Understanding this helps us set up the problem correctly by assuming linear paths.
• Rectilinear motion simplifies calculations as it involves consistent speeds and directions.
• In our problem, the apparent change in snow's path from the car's perspective connects the idea of relative velocity with rectilinear motion.

By analyzing each object's rectilinear motion, we get foundational insights needed to understand more complex motion scenarios.