Question

(a) Calculate the kinetic energy, in joules, of a \(1200-\mathrm{kg}\) automobile moving at \(18 \mathrm{~m} / \mathrm{s}\). (b) Convert this energy to calories. (c) What happens to this energy when the automobile brakes to a stop?

Short answer

The automobile's kinetic energy is \(194400\,\mathrm{J}\) or approximately \(46470.97\,\mathrm{cal}\). When it brakes to a stop, this energy is mostly converted into heat due to friction and dissipates into the environment. Some energy may also be converted into sound, and in electric or hybrid vehicles, some may be recovered and stored through regenerative braking systems.

Step-by-step solution

Calculate the Kinetic Energy in Joules

To calculate the kinetic energy, we will use the following formula: \[KE = \frac{1}{2}mv^2\] where \(KE\) represents the kinetic energy, \(m\) is the mass of the object, and \(v\) is its velocity. In our case, the mass of the automobile is \(1200\,\mathrm{kg}\), and its speed is \(18\,\mathrm{m/s}\). Plugging these values into the equation, we get: \[KE = \frac{1}{2}(1200\,\text{kg})(18\,\text{m/s})^2\]

Solve for the kinetic energy

Now, simply solve the equation: \[KE = \frac{1}{2}(1200\,\mathrm{kg})(324\,\mathrm{m^2/s^2})\] \[KE = 600\,\mathrm{kg}\cdot(324\,\mathrm{m^2/s^2})\] \[KE = 194400\,\mathrm{J}\] The automobile's kinetic energy is \(194400\,\mathrm{J}\).

Convert the energy to calories

To convert the energy from Joules to calories, you can use the following conversion factor: \[1\,\mathrm{cal} = 4.184\,\mathrm{J}\] Divide the given energy value by the conversion factor to obtain the energy in calories: \[\text{Energy in calories} = \frac{194400\,\mathrm{J}}{4.184\,\mathrm{J/cal}}\]

Calculate the energy in calories

Solve the equation: \[\text{Energy in calories} = 46470.97\,\mathrm{cal}\] The automobile's kinetic energy is approximately \(46470.97\,\mathrm{cal}\).

Discuss what happens to the kinetic energy when the car comes to a stop

When the automobile brakes to a stop, the kinetic energy is transformed into other forms of energy. A significant portion of this energy is converted into heat due to the friction between the braking system and the wheels. This heat dissipates into the surrounding environment. Some energy may also be converted into sound, and depending on the automobile's technology, some energy may be recovered and stored to be used later (as in regenerative braking systems in electric or hybrid vehicles).

Key concepts

Kinetic Energy Formula

Kinetic energy can be thought of as the energy of motion. Every moving object has kinetic energy, and it is proportional to both its mass and the square of its velocity. Using the kinetic energy formula, \( KE = \frac{1}{2}mv^2 \), where \( KE \) represents kinetic energy, \( m \) is the mass, and \( v \) is the velocity, we can determine how much energy is stored in a moving object. For example, to calculate the kinetic energy of a \( 1200-\text{kg} \) automobile traveling at \( 18 \text{m/s} \), one would plug the values into the equation and solve. This calculation is essential for understanding the dynamics of motion in physics and engineering applications, informing safety protocols in vehicle designs, and leveraging this energy in various technologies.

By educating about this process and ensuring students understand how to manipulate the formula, they become equipped to solve a wide array of real-world problems where the motion of objects plays a crucial role.

Energy Conversion

Energy conversion is the process of changing one form of energy into another. This concept is critical because it underpins many natural phenomena and technological processes. In our textbook exercise, when an automobile brakes to stop from a speed of \( 18 \text{m/s} \) for example, its kinetic energy is not lost but transformed mainly into heat due to friction. This process is part of the basis of energy conservation, which has practical implications in technologies like regenerative braking systems in electric cars, which recapture some of this energy to recharge the battery.

An easy analogy for students to understand energy conversion is to think of it as currency exchange—just as dollars can be converted into euros, kinetic energy can be converted into heat or other forms of energy. Emphasizing the concept of energy conversion helps students grasp that while the form of energy may change, the total amount remains the same, leading to deeper comprehension of how energy powers everything from cellular processes to massive mechanical systems.

Units of Energy

Energy can be quantified in various units, with the joule (J) being the standard unit of energy in the International System of Units (SI). However, other units like calories (cal), kilowatt-hours (kWh), and British thermal units (Btu) are also commonly used in different contexts. For instance, converting the kinetic energy of the automobile, which was initially calculated in joules, into calories, provides a sense of the energy in terms more familiar in contexts like nutrition and chemistry.

Understanding different units of energy helps students perform essential conversions, enabling them to compare energy quantities across various scientific and real-world scenarios. For example, calories are often used to express the energy content in foods, whereas joules may be more relevant in physics or mechanical engineering. Teaching the ability to convert between these units, as in \( 1 \text{cal} = 4.184 \text{J} \), is analogous to learning multiple languages, opening up broader communication and comprehension across disciplines.

Law of Conservation of Energy

The law of conservation of energy is a foundational principle in physics stating that energy cannot be created or destroyed, only transformed from one form to another. This bedrock concept is evident in our exercise example when the car's kinetic energy does not simply disappear but converts into heat and other forms of energy upon braking. It provides the basis for understanding all energy conversions and transfers, including those that occur within closed systems.

As educators, emphasizing the importance of the conservation of energy can help students recognize the continuity and immutable quantity of energy, even amid change. This principle is crucial for their comprehension of not only physical phenomena but also the constraints and potentials in engineering, environmental science, and even economics. Teaching students about this fundamental law of physics encourages them to think critically about the perpetual flow and transformation of energy in everything from simple machines to complex ecosystems.