Answer the following questions about the factor$\mathit{\gamma }$ (gamma) in the full relativistic equation for momentum:

(a) Is$\mathit{\gamma }$ a scalar or a vector quantity?

(b) What is the minimum possible value of $\mathit{\gamma }$?

(c) Does$\mathit{\gamma }$ reach its minimum value when an object’s speed is high or low?

(d) Is there a maximum possible value for $\mathit{\gamma }$?

(e) Does$\mathit{\gamma }$ become large when an object’s speed is high or low?

(f) Does the approximation$\mathit{\gamma }\mathbf{\approx }\mathbf{1}$ apply when an object’s speed is low or when it is high?

For a moving particle the momentum can be defined as the product of mass of the object, velocity of the moving object and the proportional factor gamma.

$\overrightarrow{p}=\gamma m\overrightarrow{v}$

The proportionality factor gamma can be defined as the,

$\gamma =\frac{1}{\sqrt{1-{\left({\displaystyle \frac{\left|\overrightarrow{v}\right|}{c}}\right)}^2}}$

In the above equation the

$$\begin{gathered} \overrightarrow{p}=Momentum \\ \gamma =Gamma\left(propotionalfactor\right) \\ m=Massoftheobject \\ \overrightarrow{v}=Velocity \\ c=Velocityoflight \end{gathered}$$

The above definition is the Relativistic equation for momentum.

The$\gamma =Gamma\left(proportionalfactor\right)$ is a scaler quantity as it is just a coefficient. It is showing the change in the value or magnitude change of the relativistic velocities and it does not show directional change.

The minimum possible value of gamma $\gamma =Gamma\left(proportionalfactor\right)$will be when v= 0

$$\begin{gathered} \gamma =\frac{1}{\sqrt{1-{\left({\displaystyle \frac{\left|\overrightarrow{v}\right|}{c}}\right)}^2}} \\ \gamma =\frac{1}{\sqrt{1-{\left({\displaystyle \frac{\left|\overrightarrow{0}\right|}{c}}\right)}^2}} \\ \gamma =1 \end{gathered}$$

So the gamma $\gamma =1$.

The gamma will be minimum when the value of the object’s speed will be minimum as the object’s speed is the separating factor for the denominator part of the gamma equation.

Highest the object’s velocity will create lowest the denominator value and the value of gamma will be highest and lowest the object’s speed will create highest the denominator value and the value of gamma will be lowest.

The maximum possible value of gamma $\gamma =Gamma\left(proportionalfactor\right)$will be when v = c

$$\begin{gathered} \gamma =\frac{1}{\sqrt{1-{\left({\displaystyle \frac{\left|\overrightarrow{v}\right|}{c}}\right)}^2}} \\ \gamma =\frac{1}{\sqrt{1-{\left({\displaystyle \frac{\left|\overrightarrow{c}\right|}{c}}\right)}^2}} \\ \gamma =\infty \end{gathered}$$

So maximum possible value of the gamma $\gamma =\infty$.

The gamma will be maximum when the value of the object’s speed will be maximum as the object’s speed is the separating factor for the denominator part of the gamma equation.

Highest the object’s velocity will create lowest the denominator value and the value of gamma will be highest and lowest the object’s speed will create highest the denominator value and the value of gamma will be lowest.

The minimum possible value of gamma $\gamma =Gamma\left(proportionalfactor\right)$will be when object velocity v= 0 or minimum

$$\begin{gathered} \gamma =\frac{1}{\sqrt{1-{\left({\displaystyle \frac{\left|\overrightarrow{v}\right|}{c}}\right)}^2}} \\ \gamma =\frac{1}{\sqrt{1-{\left({\displaystyle \frac{\left|\overrightarrow{0}\right|}{c}}\right)}^2}} \\ \gamma =1 \end{gathered}$$

So, the approximation$\gamma \approx 1$ apply when an object’s speed is low or zero.