Calculate the ratio of (a) energy to momentum for a photon, (b) kinetic energy to momentum for a relativistic massive object of speed u, and (e) total energy to momentum for a relativistic massive object. (d) There is a qualitative difference between the ratio in part (a) and the other two. What is it? (e) What are the ratios of kinetic and total energy to momentum for an extremelyrelativistic massive object, for which$\text{u}\cong c\text{?}\underset{\delta x\to 0}{lim}$ What about the qualitative difference now?

We have E = pc for massless particles, which means that the ratio of energy to momentum will be equal to c. However, if you start with the energy relation and combine it with the de-Broglie relation, you'll get the same result.

$\text{E}=\text{hf}=\frac{\text{hc}}{\text{λ}}=\text{pc}$……………..(1)

Therefore the ratio of the energy and momentum from equation (1) is

$\frac{\text{E}}{\text{p}}=\text{c}$

Now, apply the relativistic kinetic energy and momentum formulas.

$\begin{array}{c}\frac{\text{KE}}{\text{p}}=\frac{(\gamma -\text{1}){\text{mc}}^{\text{2}}}{\gamma \text{mu}}\\ =(\text{1}-\frac{\text{1}}{\gamma })\frac{{\text{c}}^{\text{2}}}{\text{u}}\end{array}$

Here, $\gamma =\frac{1}{\sqrt{1-u^2/c^2}}istheLorentzfactor.$

Instead of kinetic energy as in section (b), the total energy is now used.

$\begin{array}{c}\frac{\text{E}}{\text{p}}=\frac{(\gamma -\text{1}){\text{mc}}^{\text{2}}+{\text{mc}}^{\text{2}}}{\gamma \text{mu}}\\ =\frac{{\text{mc}}^{\text{2}}}{\gamma \text{mu}}\\ =\frac{{\text{c}}^{\text{2}}}{\text{u}}\end{array}$

The ratios in parts (b) and (c) are affected by the particle's velocity u, whereas the ratio in part (a) for massless particles is constant.

Because the speed of light is constant for massless particles in part (c).

For an extremely relativistic object, we may take the limit of the expressions in parts(b) and (c) as $u\to c$, and therefore, $\gamma \to \infty$.

$\begin{array}{l}\frac{\text{KE}}{\text{p}}=\text{c}\\ \frac{\text{E}}{\text{p}}=\text{c}\end{array}$

As we approach c, the internal energy of the relativistic decreases, thus the two ratios are equal. Where all of the energy is in the form of kinetic energy, and large particles act more like massless particles, with about the same c/c ratio.

Result.

(a)

de-Broglie relation result c

(b)

kinetic energy and momentum formulas $(1-\frac{1}{\gamma })\frac{{\mathbf{c}}^2}{u}$.

(c)

Instead of kinetic energy as in section b is$\frac{{\text{c}}^{\text{2}}}{\text{u}}$

(d)

Unlike massless particles, the ratios in parts b) and c) are affected by the velocity of the particle u.

(e)

Massive particles act similarly to massless particles, resulting in the same c ratio