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Translational kinetic energy
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The kinetic energy of translation is a function of the velocity achieved through the application of a force over a given time while traveling a given path.
Thus, the kinetic energy of translation is proportional to the mass of the object and the square of the velocity.
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Equations
$ K_t =\displaystyle\frac{1}{2} m \vec{v} \cdot \vec{v} $
K_t = m * &v . &v / 2
$ K_t =\displaystyle\frac{1}{2} m_i v ^2$
K_t = m_i * v ^2/2
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Translational Kinetic Energy
Equation
In the case of studying translational motion, the definition of energy
| $ dW = \vec{F} \cdot d\vec{s} $ |
is applied to Newton's second law
| $ F = m_i a $ |
resulting in the expression
| $ K_t =\displaystyle\frac{1}{2} m_i v ^2$ |
The energy required for an object to transition from velocity $v_1$ to velocity $v_2$ can be calculated using the definition with
| $ dW = \vec{F} \cdot d\vec{s} $ |
Using the second law of Newton, this expression can be rewritten as
$\Delta W = m a \Delta s = m\displaystyle\frac{\Delta v}{\Delta t}\Delta s$
Employing the definition of velocity with
| $ \bar{v} \equiv\displaystyle\frac{ \Delta s }{ \Delta t }$ |
we obtain
$\Delta W = m\displaystyle\frac{\Delta v}{\Delta t}\Delta s = m v \Delta v$
where the difference in velocities is
$\Delta v = v_2 - v_1$
Furthermore, the velocity itself can be approximated by the average velocity
$v = \displaystyle\frac{v_1 + v_2}{2}$
Using both expressions, we arrive at
$\Delta W = m v \Delta v = m(v_2 - v_1)\displaystyle\frac{(v_1 + v_2)}{2} = \displaystyle\frac{m}{2}(v_2^2 - v_1^2)$
Thus, the change in energy is given by
$\Delta W = \displaystyle\frac{m}{2}v_2^2 - \displaystyle\frac{m}{2}v_1^2$
In this way, we can define kinetic energy
| $ K_t =\displaystyle\frac{1}{2} m_i v ^2$ |
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Translational kinetic energy (vector)
Equation
The kinetic energy of one-dimensional translation is equal to
| $ K_t =\displaystyle\frac{1}{2} m_i v ^2$ |
so in the multidimensional case,
| $ K_t =\displaystyle\frac{1}{2} m \vec{v} \cdot \vec{v} $ |
The kinetic energy of one-dimensional translation
| $ K_t =\displaystyle\frac{1}{2} m_i v ^2$ |
can be generalized in vector form by replacing the square with a dot product
$\vec{v}^2=\vec{v}\cdot\vec{v}$
resulting in
| $ K_t =\displaystyle\frac{1}{2} m \vec{v} \cdot \vec{v} $ |
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