Question

What are the momentum, energy, and kinetic energy of a proton moving at 0.8c?

Short answer

The momentum, energy and kinetic energy of proton are $6.68\times {10}^{-19}\text{kg}·\text{m}/\text{s}$, $2.505\times {10}^{-10}\text{J}$ and $1.002\times {10}^{-10}\text{J}$.

Step-by-step solution

Identification of given data

The speed of proton 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 momentum of proton

The momentum of proton is given as:

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

Here is the mass of proton and its value is $1.67\times {10}^{-27}\text{kg}$, 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.

$$\begin{gathered} P=\frac{\left(1.67\times {10}^{-27}\text{kg}\right)\left(0.8c\right)\left(\frac{3\times {10}^8\text{m}/\text{s}}{c}\right)}{\sqrt{1-{\left(\frac{0.8c}{c}\right)}^2}} \\ P=6.68\times {10}^{-19}\text{kg}·\text{m}/\text{s} \end{gathered}$$

Determination of energy of proton

The energy of proton is given as:

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

Substitute all the values in the equation.

$$\begin{gathered} E=\frac{\left(1.67\times {10}^{-27}\text{kg}\right){\left(3\times {10}^8\text{m}/\text{s}\right)}^2}{\sqrt{1-{\left(\frac{0.8c}{c}\right)}^2}} \\ E=2.505\times {10}^{-10}\text{J} \end{gathered}$$

Determination of kinetic energy of proton

The energy of proton is given as:

$K=E-m{c}^2$

Substitute all the values in the equation.

$$\begin{gathered} K=2.505\times {10}^{-10}\text{J}-\left(1.67\times {10}^{-27}\text{kg}\right){\left(3\times {10}^8\text{m}/\text{s}\right)}^2 \\ K=2.505\times {10}^{-10}\text{J}-1.503\times {10}^{-10}\text{J} \\ K=1.002\times {10}^{-10}\text{J} \end{gathered}$$

Therefore, the momentum, energy and kinetic energy of proton are $6.68\times {10}^{-19}\text{kg}·\text{m}/\text{s}$, $2.505\times {10}^{-10}\text{J}$ and $1.002\times {10}^{-10}\text{J}$.