91Ó°ÊÓ

Kinetic energy is the difference between total energy and rest energy.

The total energyE of a particle is

$$\begin{gathered} E=K+m{c}^2 \\ =\frac{m{c}^2}{\sqrt{1-v^2/c^2}} \\ =\mathrm{γ}m{c}^2 \end{gathered}$$

Where, K is the relativistic kinetic energy. The expression of K is:

$K=\frac{m{c}^2}{\sqrt{1-v^2/c^2}}-m{c}^2$

$\mathrm{γ}$is the Lorentz factor. The expression of$\mathrm{γ}$ is:

$\mathrm{γ}=\frac{1}{\sqrt{1-v^2/c^2}}$

The time interval$\left(∆t\right)$ in the frame is

$∆t=\frac{∆t_0}{\sqrt{1-u^2/c^2}}$

The energy$\mathit{m}{\mathit{c}}^{\mathbf{2}}$ associated with rest mass m rather than motion is known as rest energy of the particle.

For$\mathrm{Δ}{t}_0=1.80s$

$$\begin{gathered} ∆t=\frac{∆t_0}{\sqrt{1-u^2/c^2}} \\ =\frac{1.80}{\sqrt{1-{\left(3/4\right)}^2}} \\ =2.72s \end{gathered}$$

For$\mathrm{Δ}t=1.80s$

$$\begin{gathered} ∆t=\frac{∆t_0}{\sqrt{1-u^2/c^2}} \\ 1.80=\frac{∆t_0}{\sqrt{1-{\left(3/4\right)}^2}} \\ ∆t_0=1.80\sqrt{1-{\left(3/4\right)}^2} \\ ∆t_0=1.19s \end{gathered}$$

Hence, if swing takes 1.80s as measured by an astronaut performing an experiment inside the spaceship, 2.72s will the swing take as measured by a person at mission control who is watching the experiment and if the swing takes 1.80s as measured by a person at mission control, 1.19s will take as measured by the astronaut in the spaceship.