Tru Physics

Kinetic Energy

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Tru Physics
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Kinetic energy is a type of energy that an object possesses due to its motion. When an object is in motion, it has the potential to do work, and kinetic energy is the energy associated with this potential.

Wind turbines, like these in Italy, operate on the fundamental principles of kinetic energy. As wind turns the turbines, a small percentage of the rotational energy can be harnessed as usable energy.
Wind turbines, like these in Italy, operate on the fundamental principles of kinetic energy. As wind turns the turbines, a small percentage of the rotational energy can be harnessed as usable energy.

Formulas for Kinetic Energy

The formula for translational kinetic energy is:

KE_t = \dfrac{1}{2}mv^2

where KE is the translational kinetic energy, m is the mass of the object, and v is the velocity of the object. This formula states that the kinetic energy of an object is directly proportional to its mass and the square of its velocity. This means that an object with a larger mass or higher velocity will have a greater amount of kinetic energy than an object with a smaller mass or lower velocity.

It is also common in higher level physics classes to write kinetic energy in terms of momentum. Noting that p=mv, we can write kinetic energy as:

KE = \dfrac{p^2}{2m}

We also note that the rotational kinetic energy is expressed quite similarly but in terms of rotational quantities such that:

KE_r = \dfrac{1}{2}I \omega^2

where I is the object’s moment of inertia about the rotational axis and \omega is the object’s angular velocity.

We also briefly mention the formula for relativistic kinetic energy which, following a Taylor Series expansion, is approximated as:

E_k \approx \dfrac{p^2}{2m} - \dfrac{p^4}{8m^3c^2}

where the first term is clearly the general Newtonian expression of kinetic energy which indicates that the second expression is the relativistic correction term. Here, c is the speed of light.

And, finally, there is the quantum mechanics formulation of momentum which is given as:

\hat{T} = \dfrac{\hat{p}^2}{2m}

which can be further simplified in the Schrodinger approach to quantum mechanics by replacing \hat{p} with -i \hbar \nabla such that:

\hat{T} = -\dfrac{\hbar^2}{2m} \nabla^2

Units of Energy

The units of kinetic energy are Joules (J), which are the same units used for work and energy. One Joule is equal to the amount of energy required to move an object with a force of one Newton over a distance of one meter.

Note: In the English System, the unit of energy is the foot-pound.

Bullets like this one, though small, can carry high amounts of kinetic energy due to their speed. One estimate for a 9-mm bullet (14 grams with a muzzle velocity of 1200 ft/s) is a kinetic energy of nearly 2000 Joules. That's enough energy to lift a small African elephant 6 inches off the ground, more or less.
Bullets like this one, though small, can carry high amounts of kinetic energy due to their speed. One estimate for a 9-mm bullet (14 grams with a muzzle velocity of 1200 ft/s) is a kinetic energy of nearly 2000 Joules. That’s enough energy to lift a small African elephant 6 inches off the ground, more or less.

Applications of Kinetic Energy

The concept of kinetic energy is applied in many areas of science and engineering, including mechanics, thermodynamics, and electromagnetism. For example, it is used in the design of cars, airplanes, and spacecraft, as well as in the study of collisions between objects. Kinetic energy is also used in the field of renewable energy, such as wind and hydroelectric power, where the motion of wind or water is converted into electrical energy.

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