Question

What would be the internal energy, kinetic energy, and total energy of a 1 kg block moving at 0.8c?

Short answer

The internal energy, kinetic energy and total energy of block are $9\times {10}^{16}\text{J}$, $6\times {10}^{16}\text{J}$ and $15\times {10}^{16}\text{J}$.

Step-by-step solution

Identification of given data

The mass of block is $m=1\text{kg}$

The speed of block is $v=0.8c$

Momentum, Energy and Kinetic energy

The total energy of a particle moving at high speed consists of kinetic energy and rest energy. The rest energy is the energy same as produced due to complete conversion of mass of particle into energy. The momentum of moving particle is relativistic momentum.

Determination of internal energy of block

The internal energy of block is given as:

$E_i=m{c}^2$

Here, c is the speed of light and its value is$3\times {10}^8\text{m}/\text{s}$ .

Substitute all the values in the equation.

$$\begin{gathered} E_i=\left(1\text{kg}\right){\left(3\times {10}^8\text{m}/\text{s}\right)}^2 \\ {E}_i=9\times {10}^{16}\text{J} \end{gathered}$$

Determination of total energy of block

The total energy of block is given as:

$E=\frac{m{c}^2}{\sqrt{1-{\left(\frac{v}{c}\right)}^2}}$

Substitute all the values in the equation.

$$\begin{gathered} E=\frac{\left(1\text{kg}\right){\left(3\times {10}^8\text{m}/\text{s}\right)}^2}{\sqrt{1-{\left(\frac{0.8c}{c}\right)}^2}} \\ E=15\times {10}^{16}\text{J} \end{gathered}$$

Determination of kinetic energy of block

The energy of block is given as:

$K=E-E_i$

Substitute all the values in the equation.

$$\begin{gathered} K=15\times {10}^{16}\text{J}-9\times {10}^{16}\text{J} \\ K=6\times {10}^{16}\text{J} \end{gathered}$$

Therefore, the internal energy, kinetic energy and total energy of block are $9\times {10}^{16}\text{J}$, $6\times {10}^{16}\text{J}$ and $15\times {10}^{16}\text{J}$.