Question

Derive a relationship between kinetic energy and linear momentum.

Answer 1

Step 1: Expression of kinetic energy and momentum

Kinetic energy (K.E) = $\frac{1}{2}m{v}^2$

Momentum (p) = $mv$

where $m$ = mass and $v$ = velocity

Step 2: Relation between momentum and kinetic energy

Considering the expression of kinetic energy,

$K.E=\frac{1}{2}m{v}^2$

Multiplying mass $\left(m\right)$ on numerator and denominator,

$$\begin{gathered} K.E=\frac{1}{2}m{v}^2\left(\frac{m}{m}\right) \\ K.E=\frac{m^2{v}^2}{2m} \\ K.E=\frac{{\left(mv\right)}^2}{2m} \end{gathered}$$

Substituting $mv=p$,

$K.E=\frac{p^2}{2m}$

Hence, the relationship between kinetic energy and linear momentum is given by $Kineticenergy=\frac{{(linearmomentum)}^2}{\left(2\times mass\right)}$