Question

The kinetic energy of a body with mass \(m\) and velocity \(v\) is \(K = \frac{1}{2}m{v^2}\). Such that \(\frac{{\partial K}}{{\partial m}}\frac{{{\partial ^2}K}}{{\partial {V^2}}} = K\).

Short answer

A partial differential equation (PDE) is a mathematical equation that establishes relationships between the partial derivatives of a multivariable function.

Step-by-step solution

Differentiate with respect to m.

Given: \(K = \frac{1}{2}m{V^2}\)

\(\frac{{\partial K}}{{\partial m}} = \frac{1}{2}{V^2}\)

Differentiate with respect to v.

\(\frac{{{\partial ^2}K}}{{\partial {V^2}}} = m\)

Consider \(\frac{{\partial K}}{{\partial m}}\frac{{{\partial ^2}K}}{{\partial {V^2}}} = \left( {\frac{1}{2}{V^2}} \right)\left( m \right)\)

\( = \frac{1}{2}m{V^2}\)

\( = K\)

Hence, \(\frac{{\partial K}}{{\partial m}}\frac{{{\partial ^2}K}}{{\partial {V^2}}} = K\).