Problem 10 - Applied Physics

Question

A piece of broken glass with mass \(15.0 \mathrm{~kg}\) falls from the side of a building \(8.00 \mathrm{~m}\) above the street. (a) What is the kinetic energy of the glass as it hits the street? (b) What is the speed of the glass as it hits the street?

Step-by-Step Solution

Answer

(a) KE = 1177.2 J; (b) Speed = 12.53 m/s.

Step 1: Identify the known variables

First, we need to identify the known variables from the problem. The mass of the glass is \( m = 15.0 \text{ kg} \), and it falls from a height of \( h = 8.00 \text{ m} \). The acceleration due to gravity is \( g = 9.81 \text{ m/s}^2 \).

Step 2: Calculate the potential energy at the initial height

The potential energy (PE) at the height from which the glass falls is given by the formula \( PE = mgh \). Here,\[ PE = 15.0 \times 9.81 \times 8.00 \]Compute to find:\[ PE = 1177.2 \text{ J} \].

Step 3: Understand energy conservation principle

As the glass falls, its potential energy is converted into kinetic energy. When it hits the ground, all the potential energy is converted into kinetic energy. Hence, the kinetic energy (KE) just before impact is the same as the initial potential energy: \( KE = 1177.2 \text{ J} \).

Step 4: Use kinetic energy to find speed

The kinetic energy formula is \( KE = \frac{1}{2}mv^2 \). Set \( KE = 1177.2 \) and solve for \( v \):\[ 1177.2 = \frac{1}{2} \times 15.0 \times v^2 \]Solving for \( v^2 \):\[ v^2 = \frac{1177.2 \times 2}{15.0} = 156.96 \].

Step 5: Calculate the speed

Take the square root of both sides to solve for \( v \):\[ v = \sqrt{156.96} \approx 12.53 \text{ m/s} \]. This is the speed of the glass as it hits the street.

Key Concepts

Potential Energy

Potential energy is the energy stored in an object due to its position relative to Earth. For objects near the Earth's surface, this is primarily gravitational potential energy. The formula to calculate gravitational potential energy is \( PE = mgh \), where:

• \( m \) is the mass of the object in kilograms.
• \( g \) is the acceleration due to gravity, approximately \( 9.81 \text{ m/s}^2 \) on Earth's surface.
• \( h \) is the height above the ground in meters.

In our exercise, a piece of glass has a mass of 15.0 kg, and it is at a height of 8.00 m. By substituting these values into the formula, \( PE = 15.0 \times 9.81 \times 8.00 \), we calculate the potential energy to be 1177.2 joules (J). This stored energy is crucial as it determines the kinetic energy of the glass just before it hits the ground. Understanding potential energy helps explain how energy is conserved in physics problems that involve falling objects.

Energy Conservation

The principle of energy conservation is a fundamental concept in physics that states that energy cannot be created or destroyed, only transformed from one form to another. In our problem, as the piece of glass falls from the building, the potential energy it initially possesses converts into kinetic energy as it descends.
When the glass reaches the street, all of its initial potential energy has been transformed into kinetic energy - meaning its kinetic energy at that point is equal to the potential energy it had at the start. Therefore, the kinetic energy (KE) of the glass just before it impacts the street is 1177.2 J, the same as the initial potential energy (1177.2 J).

• This transformation demonstrates how energy conservation works in practical scenarios.
• The exact interchange from potential to kinetic energy underlies many kinematics problems.

Recognizing how energy conservation operates empowers you to solve numerous physics problems.

Physics Problems

Physics problems such as the falling glass scenario are core examples of how the principles of physics apply in real-life situations. They require identifying known values, applying relevant physics formulas, and understanding concepts like potential and kinetic energy, and energy conservation.
Here’s how to approach such problems:

• Identify all given variables and quantities.
• Choose the correct physics formula to apply.
• Compute each step carefully, observing the units.
• Ensure comprehension of the underlying concepts such as energy conversions.

Aside from practicing calculations, delving into the theory enriches your understanding. For example, knowing why potential energy becomes kinetic energy enhances problem-solving skills and promotes a deeper grasp of physics concepts. By tackling physics problems diligently, you not only find solutions but also refine your analytical and critical thinking abilities.

Kinematics

Kinematics involves the study of motion, specifically how objects move and travel through space and time without considering the forces that cause this motion. When calculating the speed of the glass as it hits the ground, we use kinematics principles combined with energy concepts.
From our problem, once potential energy is converted into kinetic energy, we need to find the velocity or speed of the glass. Using the kinetic energy formula \( KE = \frac{1}{2}mv^2 \), where:

• \( KE \) is kinetic energy (1177.2 J).
• \( m \) is the mass (15.0 kg).
• \( v \) is the velocity we need to calculate.

Spacing of numerical substitution simplifies into solving \( v \): \[ v^2 = \frac{2 \times 1177.2}{15.0} \rightarrow v = \sqrt{156.96} \approx 12.53 \text{ m/s}. \]Kinematics provides the techniques to determine not only speed and distance but also acceleration and time in myriad physics problems, forming a critical base for understanding motion.