Question

Calculate the total kinetic energy, in Btu, of an object with a mass of \(10 \mathrm{lbm}\) when its velocity is \(50 \mathrm{ft} / \mathrm{s}\).

Short answer

Answer: The total kinetic energy of the object is approximately 249.04 Btu.

Step-by-step solution

Convert mass to slugs

To convert the mass from pounds-mass to slugs, use the conversion factor: \(1\,\text{slug} = 32.174\,\text{lbm}\) Find the mass in slugs by dividing the given mass by the conversion factor. \(10\,\text{lbm} \times \frac{1\,\text{slug}}{32.174\,\text{lbm}} \approx 0.3107\,\text{slugs}\)

Calculate kinetic energy

Now use the kinetic energy formula: \(KE = \frac{1}{2}mv^2\) Plug in the mass (in slugs) and the given velocity (ft/s): \(KE = \frac{1}{2} \times 0.3107\,\text{slugs} \times (50\,\text{ft/s})^2\) Calculate the kinetic energy in ft-lbf: \(KE \approx 193925\,\text{ft-lbf}\)

Convert ft-lbf to Btu

Convert kinetic energy from ft-lbf to Btu using the conversion factor: \(1\,\text{Btu} = 778.169\,\text{ft-lbf}\) Find the kinetic energy in Btu: \(193925\,\text{ft-lbf} \times \frac{1\,\text{Btu}}{778.169\,\text{ft-lbf}} \approx 249.04\,\text{Btu}\) After calculating and converting, the total kinetic energy of the object with a mass of \(10\,\text{lbm}\) and a velocity of \(50\,\text{ft/s}\,\) is approximately \(249.04\,\text{Btu}\).

Key concepts

Energy Conversion Units

When studying different forms of energy, such as kinetic energy, it's crucial to be fluent in converting units to accurately measure and compare energy values. For instance, in thermodynamics, energy is often measured in British Thermal Units (Btu), which is a traditional unit of heat. It is defined as the amount of heat required to raise the temperature of one pound of water by one degree Fahrenheit.

It's common for various exercises and problems to require the conversion from one unit of energy to another, such as from foot-pounds (ft-lbf) to Btu, to interpret the results correctly. The step-by-step solution above converts the calculated kinetic energy from ft-lbf to Btu using a conversion factor of 778.169 ft-lbf per Btu. Being confident in using these conversion factors is essential for anyone studying physics or engineering.

Slug Mass Unit

Understanding mass units is key to solving many problems in physics, particularly in the context of kinetic energy. The slug is a unit of mass used primarily in the British Engineering system. It is based on accelerating a mass by one foot per second squared with a force of one pound-force (lbf).

To obtain a slug's mass from a mass given in pounds-mass (lbm), one must divide the mass in lbm by the acceleration due to gravity in ft/s², which is approximately 32.174 ft/s². This conversion is necessary because the formula for kinetic energy requires mass to be in consistent units with the force due to gravity. By providing a clear explanation of the conversion process, as seen in the exercise solution, we ensure students understand both the concept of the slug and the method to convert to it.

Velocity in Thermodynamics

Velocity plays an important role in many areas of physics, including thermodynamics, where it's a measure of the speed and direction of an object. In the context of kinetic energy, velocity is squared in the formula, indicating that an increase in speed has a significant impact on the kinetic energy of an object.

It is essential for students to grasp that even though thermodynamics is generally concerned with heat and temperature, concepts like velocity are interrelated when analyzing the movement and energy of particles in a system. The precise measurement of velocity, which in this exercise is given in feet per second (ft/s), directly affects the calculation of kinetic energy, emphasizing the importance of accurate measurements and understanding their implications on energy values.

Kinetic Energy Formula

The kinetic energy formula is fundamental in physics and engineering disciplines. It is calculated as half the product of an object's mass and the square of its velocity, expressed as \(KE = \frac{1}{2}mv^2\). This formula reveals that kinetic energy is directly proportional to the mass and, even more so, to the velocity of an object, as the velocity is squared.

Understanding this relationship helps explain scenarios in our physical world. For example, why heavier or faster vehicles have greater energy impacts during collisions. In the solution provided, we see this formula applied to an object's mass in slugs and its velocity in feet per second, yielding the kinetic energy in foot-pounds before converting to Btu. This clear step-by-step application aids students in understanding and applying the kinetic energy formula accurately in various problems.