Question

How fast must an object be moving for its kinetic energy to equal its internal energy?

Short answer

The object must be moving at a speed of$0.866c$

Step-by-step solution

Define Internal energy

The total energy of an object which is stationary in an isolated system is,

${\mathit{E}}_{\mathbf{i}\mathbf{n}\mathbf{t}}\mathbf{=}\mathit{m}{\mathit{c}}^{\mathbf{2}}$

Here, m is the mass and c is the speed of light.

Define Kinetic energy equation in relativistic terms

Kinetic energy is associated with the motion of the object It could be determined by substracting object’s energy (Internal) at rest from energy of moving object.

Therefore,

$\begin{array}{c}KE=energyofmotion-energyatrest\\ =\gamma m{c}^2-m{c}^2\\ =(\gamma -1)m{c}^2\end{array}$

$Here,\gamma istherelativisticfactor.$

Equating both of the above questions and solving,

$$\begin{gathered} KE=E_{int} \\ (\gamma -1)m{c}^2=m{c}^2 \\ \gamma m{c}^2=2m{c}^2 \\ \gamma =2 \end{gathered}$$

Write the equation for relativistic factor as given below.

$$\begin{gathered} \gamma =\frac{1}{\sqrt{1-{\displaystyle \raisebox{1ex}{$v^2$}\!\left/ \!\raisebox{-1ex}{$c^2$}\right.}}} \\ 2=\frac{1}{\sqrt{1-{\displaystyle \raisebox{1ex}{$v^2$}\!\left/ \!\raisebox{-1ex}{$c^2$}\right.}}} \\ 1-\raisebox{1ex}{$v^2$}\!\left/ \!\raisebox{-1ex}{$c^2$}\right.=\frac{1}{4} \\ \raisebox{1ex}{$v^2$}\!\left/ \!\raisebox{-1ex}{$c^2$}\right.=1-0.25 \end{gathered}$$

$$\begin{gathered} v^2=0.75c^2 \\ v=0.866c \end{gathered}$$

Hence, for an object’s kinetic energy to be equal to its internal energy, it must travel at a speed of $0.866$times the speed of light.