Problem 86 - Physics

Question

Consider the freight train in Figure 16 -6. Suppose 15 boxcars pass by in a time of \(12.0 \mathrm{~s}\) and each has a length of \(14.0 \mathrm{~m}\). (a) What is the frequency at which each boxcar passes? (b) What is the speed of the train?

Step-by-Step Solution

Answer

(a) Frequency: 1.25 Hz, (b) Speed: 17.5 m/s.

Step 1: Understanding Frequency

Frequency is defined as the number of occurrences of a repeated event per unit of time. Here, it refers to how often the boxcars pass a fixed point. To find the frequency of the boxcars passing by, we can use the formula: \( f = \frac{N}{T} \), where \( N \) is the number of boxcars, and \( T \) is the time period. In this case, \( N = 15 \) boxcars and \( T = 12.0 \mathrm{~s} \).

Step 2: Calculating Frequency

Using the formula from Step 1, substitute the given values: \( f = \frac{15}{12.0} \mathrm{~Hz} \). Calculating this gives \( f \approx 1.25 \mathrm{~Hz} \). This means that approximately 1.25 boxcars pass by per second.

Step 3: Determining the Train's Speed

Speed is defined as the distance traveled per unit of time. Since each boxcar has a length of \( 14.0 \mathrm{~m} \), the total length of the train passing a point per second can be used to find the speed. The speed formula is \( v = f \times L \), where \( L \) is the length of each boxcar.

Step 4: Calculating Train's Speed

Substitute the values into the speed formula: \( v = 1.25 \mathrm{~Hz} \times 14.0 \mathrm{~m} \), giving \( v = 17.5 \mathrm{~m/s} \). Thus, the speed of the train is \( 17.5 \mathrm{~m/s} \).

Key Concepts

Frequency Calculation

When solving physics problems, understanding frequency can be pivotal. Frequency, in simple terms, is how often something happens over a period of time. In the case of the boxcars, frequency refers to the number of boxcars that pass a point per second. The formula for frequency is given as:

• Frequency (\( f \)) = Number of Events (\( N \)) / Time (\( T \)).

For our exercise, 15 boxcars pass by in 12 seconds. By substituting these values, we calculate the frequency:

• \( f = \frac{15\, \, \text{boxcars}}{12\,\text{s}} = 1.25\,\text{Hz} \).

This means approximately 1.25 boxcars pass the observer each second. Understanding frequency helps in situations involving repeated events over time.

Speed of Train

Calculating the speed of the train involves determining how fast it moves relative to a fixed point. Speed, in physics, is the rate of distance covered over time. Here, we use the relationship between frequency and the length of an individual boxcar. The speed of the train is thus the frequency of the boxcars passing multiplied by the length of each boxcar:

• Speed (\( v \)) = Frequency (\( f \)) \( \times \) Boxcar Length (\( L \)).

In the exercise, each boxcar is 14 meters long and the frequency is 1.25 Hz:

• \( v = 1.25\,\text{Hz} \times 14\,\text{m} = 17.5\,\text{m/s} \).

This solution tells us the train travels at a speed of 17.5 meters per second. Understanding such calculations is valuable for resolving kinematics-related tasks.

Kinematics

Kinematics is a branch of physics that describes motion without considering its causes. It focuses on quantities such as displacement, velocity, acceleration, and time. In this exercise:

• Displacement refers to the length of the boxcars passing a point.
• Velocity is illustrated by calculating the train's speed.
• Time factor is evidenced in the 12 seconds duration for the boxcars to pass.

The fundamentals of kinematics help us relate these variables through formulas and reasoning. The speed of 17.5 m/s, for example, is calculated by understanding both the kinematic equation for velocity and how continuous events contribute to this rate. Real-world applications of kinematics can be seen in mechanics, robotics, and transportation analysis.
By grasping these principles, students can tackle a range of physics problems involving various types of motion.